Research ArticleOscillation for a Class of Right Fractional DifferentialEquations on the Right Half Line with Damping

Hui Liu and Run Xu

School of Mathematical Sciences Qufu Normal University Qufu 273165 Shandong China

Correspondence should be addressed to Run Xu xurun 2005163com

Received 24 January 2019 Revised 15 March 2019 Accepted 17 March 2019 Published 1 April 2019

Academic Editor Chris Goodrich

Copyright copy 2019 Hui Liu and Run XuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we discuss a class of fractional differential equations of the form 119863120572+1minus 119910(119905) sdot 119863120572minus119910(119905) minus 119901(119905)119891(119863120572minus119910(119905)) +119902(119905)ℎ(intinfin119905

(119904 minus 119905)minus120572119910(119904)119889119904) = 0119863120572minus119910(119905) is the Liouville right-sided fractional derivative of order120572 isin (0 1)We obtain some oscillationcriteria for the equation by employing a generalized Riccati transformation technique Some examples are given to illustrate thesignificance of our results

1 Introduction

The theory of fractional derivatives was originated fromGW Leibnizrsquos conjecture To this day the theory aboutfractional calculus and fractional differential equation havebeen well developed see [1ndash7] In the beginning thetheory of fractional derivatives developed mainly as a puretheoretical filed of mathematics which can be used only formathematicians However in the past few decades fractionaldifferential equations were widely used in many fields suchas fluid flow rheology electrical networks and many otherbranches of science Great attention was paid to study theproperties of solutions of fractional differential equations

Because only few differential equations can be solvedmany researches focus on the analysis of qualitative theoryfor fractional differential equations such as the existenceuniqueness of solutions numerical solutions stability andoscillation of solutions see [8ndash34] and the references thereinAmong them there have beenmany results for the oscillationof solutions for fractional differential equations

In 2013 Chen [16] studied oscillatory behavior of thefractional differential equation in the form of

119863120572+1minus 119910 (119905) minus 119901 (119905)119863120572minus119910 (119905)+ 119902 (119905) 119891(intinfin

119905(V minus 119905)minus120572 119910 (V) 119889V) = 0 (1)

for 119905 gt 0 where 119863120572minus119910 is the Liouville right-sided fractionalderivative of order 120572 isin (0 1)

In 2013 Han [17] brought up the oscillation of fractionaldifferential equations

[119903 (119905) 119892 ((119863120572minus119910) (119905))]1015840 minus 119901 (119905) 119891(intinfin119905

(119904 minus 119905)minus120572 119910 (119904))= 0 (2)

for 119905 gt 0 where 0 lt 120572 lt 1 is a real number and 119863120572minus119910 is theLiouville right-sided fractional derivative of 119910

In 2013 Xu [18] studied the oscillation of nonlinearfractional differential equations of the form

119886 (119905) [(119903 (119905) 119863120572minus119909 (119905))1015840]1205781015840minus 119865(119905 intinfin

119905(V minus 119905)minus120572 119909 (V) 119889V) = 0 119905 ge 1199050 gt 0 (3)

where 120572 isin (0 1) is a constant and 120578 is a ratio of two oddpositive integers

In 2013 based on themodified Riemann-Liouville deriva-tive Qin and Zheng [19] discussed the oscillation of a class

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 4902718 9 pageshttpsdoiorg10115520194902718

2 Discrete Dynamics in Nature and Society

of fractional differential equations with damping term asfollows

119863120572119905 [119886 (119905) 119863120572119905 (119903 (119905) 119863120572119905 119909 (119905))] + 119901 (119905) 119863120572119905 (119903 (119905) 119863120572119905 119909 (119905))+ 119902 (119905) 119909 (119905) = 0 (4)

for 119905 ge 1199050 gt 0 0 lt 120572 lt 1 where 119863120572119905 (sdot) denotes the modifiedRiemann-Liouville derivative regarding the variable 119905 thefunction 119886 isin 119862120572([1199050infin) 119877+) 119903 isin 1198622120572([1199050infin) 119877+) 119901 119902 isin119862120572([1199050infin) 119877+) and 119862120572 denotes continuous derivative oforder

In 2014 Jehad Alzabut andThabet Abdeljawad [20] stud-ied the oscillatory theory of fractional difference equations inthe form

nabla119902119886(119902)minus1

119909 (119905) + 1198911 (119905 119909 (119905) = 119903 (119905) + 1198912 (119905 119909 (119905)) 119905 isin N119886(119902)

nablaminus(1minus119902)119886(119902)minus1

119909 (119905)100381610038161003816100381610038161003816119905=119886(119902) = 119909 (119886 (119902)) = 119888 119888 isin R(5)

where119898 minus 1 lt 119902 lt 119898119898 isin N nabla119902119886(119902)

is the Riemann-Liouvilledifference operator of order 119902 and nablaminus119902

119886(119902)is the Riemann-

Liouville sum operator where N119886(119902) = 119886(119902) + 1 119886(119902) +2 119886(119902) = 119886 + 119898 minus 1119898 = [119902] + 1 and [119902] is the greatestinteger less than or equal to 119902

In 2017 B Abdalla K Abodayeh T Abdeljawad JAlzabut [21] studied the oscillation of solutions of nonlinearforced fractional difference equations in the form

nabla119902119886(119902)minus1

119909 (119905) + 1198911 (119905 119909 (119905) = 119903 (119905) + 1198912 (119905 119909 (119905)) 119905 isin N119886(119902)

nablaminus(119898minus119902)119886(119902)minus1

119909 (119905)100381610038161003816100381610038161003816119905=119886(119902) = 119909 (119886 (119902)) = 119888 119888 isin R(6)

where 119902 gt 0119898 = [119902] + 1119898 isin N [119902] is the greatest integerless than or equal to 119902 N119886(119902) = 119886(119902) + 1 119886(119902) + 2 119886(119902) =119886 + 119898 minus 1 119891119894 N119886(119902) times R 997888rarr R(119894 = 1 2) and nablaminus119902

119886(119902)and nabla119902

119886(119902)

are the Riemann-Liouville sum and difference operatorsIn 2018 Bai and Xu [22] discussed the oscillation problem

of a class of nonlinear fractional difference equations with thedamping term in the from

Δ(119888 (119905) [Δ (119903 (119905) Δ120572119909 (119905))]120574) + 119901 (119905) [Δ (119903 (119905) Δ120572119909 (119905))]120574+ 119902 (119905) 119891(119905minus1+120572sum

119904=1199050

(119905 minus 119904 minus 1)(minus120572) 119909 (119904)) = 0119905 isin 1198731199050

(7)

where 120574 ge 1 is a quotient of two odd positive integers 0 lt 120572 le1 is a constant Δ120572 denotes the Riemann-Liouville fractionaldifference operator of order 120572 and1198731199050 = 1199050 1199050+1 1199050+2

In 2018 Bahaaeldin Abdalla andThabet Abdeljawad [23]studied the oscillation of Hadamard fractional differentialequation of the form

119863120572119886119909 (119905) + 1198911 (119905 119909) = 119903 (119905) + 1198912 (119905 119909) 119905 gt 119886lim119905997888rarr119886+

119863120572minus119895119886 119909 (119905) = 119887119895 (119895 = 1 2 119899) (8)

where 119899 = lceil120572rceil 119863120572119886 is the left-fractional Hadamard derivativeof order 120572 isin C Re(120572) ge 0 in the Riemann-Liouville setting

In 2018 J Alzabut T Abdeljawad H Alrabaiah [24]considered the following forced and damped nabla fractionaldifference equation

(1 minus 119901 (119899)) nablanabla1205720 119910 (119899) + 119901 (119899) nabla1205720 119910 (119899) + 119902 (119899) 119891 (119910 (119899))= 119892 (119899) 119899 isin N1

nablaminus(1minus120572)0 119910 (1) = 119910 (1) = 119888(9)

where nabla1205720 119910 and nablaminus1205720 119910 are the Riemann-Liouville fractionaldifference and sum operators of 119910 of order 120572 respectively120572 is a real number 119888 is constant N1 = 1 2 and 119901 119902 arereal sequences from N1 997888rarr R 119901(119899) lt 1 119902 is a positive realsequence from N1 997888rarr R+ and 119891 R 997888rarr R such that 119891(119904)119904 gt 0 for all 119904 = 0

In 2018 B Abdalla J Alzabut T Abdeljawad [25] inves-tigated the oscillation of solutions for fractional differenceequations with mixed nonlinearities in forms

nabla120572119886(120572)minus1119909 (119905) minus 119901 (119905) 119909 (119905) + 119899sum119894=1

119902119894 (119905) |119909 (119905)|120582119894minus1 119909 (119905)= V (119905) 119905 isin N119886(120572)+1

nablaminus(119898minus119886)119886(120572minus1) 119909 (119905)10038161003816100381610038161003816119905=119886(120572) = 119909 (119886 (120572)) = 119888 119888 isin R(10)

and

119888nabla120572119886(120572)119909 (119905) minus 119901 (119905) 119909 (119905) + 119899sum119894=1

119902119894 (119905) |119909 (119905)|120582119894minus1 119909 (119905)= V (119905) 119905 isin N119886(120572)

nabla119896119909 (119886 (120572)) = 119887119896 119896 isin R 119896 = 0 1 2 119898 minus 1(11)

where 119898 = [120572] + 1 120572 gt 0 119901(119905) V(119905) and 119902119894(119905)(1 le 119894 le 119899) arefunctions defined fromN119886(120572) to119877 and 120582119894(1 le 119894 le 119899) are ratiosof odd positive integers with 1205821 gt gt 120582119897 gt 1 gt 120582119897+1 gt gt120582119899

Inspired by the above results in this paper we discuss theoscillatory behavior of the fractional differential equationalwith damping

119863120572+1minus 119910 (119905) sdot 119863120572minus119910 (119905) minus 119901 (119905) 119891 (119863120572minus119910 (119905))+ 119902 (119905) ℎ (intinfin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904) = 0 119905 gt 0 (12)

where 0 lt 120572 lt 1 is a real number 119863120572minus119910 is the Liouville right-sided fractional derivative of 119910 We always assume that thefollowing conditions are valid

Discrete Dynamics in Nature and Society 3

(1198601) 119901(119905) ge 0 and 119902(119905) ge 0 are continuous functions on119905 isin [1199050infin) 1199050 gt 0(1198602) ℎ 119891 R 997888rarr R are continuous functions with119909ℎ(119909) gt 0 119909119891(119909) gt 0 for 119909 = 0 and there exist positive

constants 1198961 1198962 such that ℎ(119909)119909 ge 1198961 119909119891(119909) ge 1198962 for all119909 = 0(1198603) 1198911015840(119906) le 119906 119891minus1(119906) isin 119862(RR) are continuousfunctions with 119891minus1(119906) gt 0 for 119906 = 0 and there exists somepositive constant 1205721 such that 119891minus1(119906V) ge 1205721119891minus1(119906)119891minus1(V) for119906V = 02 Preliminaries

For convenience some background materials from fractionalcalculus are given

From [4] we can get the definition for Liouville right-side fractional integral and Liouville right-side fractionalderivative on the whole axis R of order 120573 for a function119892 R+ 997888rarr R as follows

(119868120573minus119892) (119905) fl 1Γ (120573) intinfin

119905(V minus 119905)120573minus1 119892 (V) 119889V 119905 gt 0 (13)

(119863120573minus119892) (119905) fl (minus1)lceil120573rceil 119889lceil120573rceil119889119905lceil120573rceil (119868lceil120573rceilminus120573minus 119892) (119905)= (minus1)lceil120573rceil 1Γ (lceil120573rceil minus 120573) int

infin

119905(V minus 119905)lceil120573rceilminus120573minus1 119892 (V) 119889V

119905 gt 0(14)

provided the right hand side is pointwise defined on R+where Γ(sdot) is the gamma function defined by Γ(119905) flintinfin119905

119904119905minus1119890minus119904119889119904 and fl min119911 isin Z 119911 ge 120573 is the ceilingfunction

If 120573 isin (0 1) we have119863120572minus119910 (119905) fl minus 1Γ (1 minus 120572) 119889119889119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 (15)

for 119905 isin R+ fl (0infin)The following relations also existed

(119863(1+120572)minus 119910) (119905) = minus (119863120572minus119910)1015840 (119905) 120572 isin (0 1) 119905 gt 0 (16)

Set

119866 (119905) fl intinfin119905

(V minus 119905)minus120572 119910 (V) 119889V 120572 isin (0 1) (17)

and then

1198661015840 (119905) = minusΓ (1 minus 120572) (119863120572minus119910) (119905) 120572 isin (0 1) (18)

3 Main Results

First we study the oscillation of (12) under the followingcondition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 = infin (19)

Theorem 1 Suppose that (1198601) minus (1198603) and (19) hold fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that

lim sup119905997888rarrinfin

int1199051199050

V (119904)sdot [[1198961119903 (119904) 119902 (119904) minus

(1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = infin(20)

where 1198961 1198962 are defined as in (1198602) andV (119904) fl exp(int119904

1199050

119901 (V) 119889V) 119904 ge 1199050 (21)

Then every solution of (12) is oscillatory

Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Then there exists 1199051 isin[1199050infin] such that 119910(119905) gt 0 and 119866(119905) gt 0 for 119905 isin [1199051infin] where119866 is defined in (16) From (1198603) (12) and (16) we have

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 = minus1198911015840 (119863120572minus119910 (119905)) V (119905) 119863120572+1minus 119910 (119905)+ 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)

= 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) V (119905)

ge 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 119863120572+1minus 119910 (119905) 119863120572minus119910 (119905) V (119905)

= 119902 (119905) ℎ (119866 (119905)) V (119905) gt 0119905 isin [1199050infin]

(22)

Thus 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199050infin] SinceV(119905) gt 0 for 119905 isin [1199050infin] and from (1198603) we see that 119863120572minus119910(119905) iseventually of one sign Now we can claim

119863120572minus119910 (119905) lt 0 119905 isin [1199051infin] (23)

If not then there exists 1199052 isin [1199051infin] such that 119863120572minus119910(1199052) gt0 Since 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199051infin] it isclear that 119891(119863120572minus119910(119905))V(119905) ge 119891(119863120572minus119910(1199052))V(1199052) fl 119888 gt 0 for119905 isin [1199052infin] Therefore from (18) we have

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1 ( 119888V (119905))

= 119891minus1 (119888 sdot exp(minusint1199051199050

119901 (V) 119889V))ge 1205721119891minus1 (119888) 119891minus1 (exp(minusint119905

1199050

119901 (V) 119889V)) 119905 isin [1199052infin]

(24)

4 Discrete Dynamics in Nature and Society

Then we get

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V)) le minus 1198661015840 (119905)1205721119891minus1 (119888) Γ (1 minus 120572) 119905 isin [1199052infin] (25)

Integrating the above inequality from 1199052 to 119905 we haveint1199051199052

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904le minusint1199051199052

1198661015840 (119904)1205721119891minus1 (119888) Γ (1 minus 120572)119889119904= minus 119866 (119905) minus 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572)

119905 isin [1199052infin]

(26)

Letting 119905 997888rarr infin we see

intinfin1199052

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904le 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt infin (27)

This is in contradiction with (19) Hence (23) holdsDefine the function 119908 as generalized Riccati substitution

119908 (119905) = 119903 (119905) minusV (119905) 119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (28)

Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (18) (22) (28)and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)

sdot minus [119891 (119863120572minus119910 (119905)) V (119905)]1015840 119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905)le 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot [minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905) ]= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) V (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198961119903 (119905) 119902 (119905) V (119905) + 1199082 (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))119903 (119905) 119891 (119863120572minus119910 (119905)) V (119905)le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905) V (119905) minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905)

(29)

That is

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905) (30)

Taking 119909 = 119908(119905) 119886 = 1198962Γ(1 minus 120572)119903(119905)V(119905) (119886 = 0) and 119887 =1199031015840(119905)119903(119905) from 1198861199092+119887119909 le minus11988724119886 and (30)we could concludethat

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + V (119905) (1199031015840 (119905))241198962Γ (1 minus 120572) 119903 (119905) (31)

Integrating both sides of inequality (31) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)

ge int1199051199050

[[minus1198961119903 (119904) 119902 (119904) V (119904) +

V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119905)]]119889119904 (32)

Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get

lim sup119905997888rarrinfin

int1199051199050

[[minus1198961119903 (119904) 119902 (119904) V (119904)

+ V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 lt 119908 (1199050) lt infin(33)

which is in contradiction with (20)If 119910(119905) is an eventually negative solution of (12) the proof

is similar hence we omit itThe proof is complete

Theorem 2 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that

lim sup119905997888rarrinfin

int1199051199050

[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= infin(34)

where V(119904) are defined inTheorem 1Then every solution of (12)is oscillatory

Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Proceeding the same asin the proof of Theorem 1 we get (23) Define the function119908(119905) as follows

119908(119905) = 119903 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (35)

Discrete Dynamics in Nature and Society 5

Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840

= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)

(36)

That is

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)

Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude

1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)

Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050

[[minus1198961119903 (119904) 119902 (119904) +

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)

Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get

lim sup119905997888rarrinfin

int1199051199050

[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

lt 119908 (1199050) lt infin(40)

which is in contradiction with (34)

If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it

The proof is complete

We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)

and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable

Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904

= infin(42)

Then every solution of (12) is oscillatory

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)

Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get

int1199051199051

1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051

119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051

119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]

(43)

Using the formula integration by parts we obtain

minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)

+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)

(44)

6 Discrete Dynamics in Nature and Society

Substituting (44) with (43) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)

minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)

Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget

int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904

le int1199051199051

(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904

le int1199051199051

[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905

1199051

[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904

le int1199051199051

[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905

1199051

[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904

(46)

Substituting (46) in (45) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905

1199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)

Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get

int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)

119905 isin [1199051infin) (48)

Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have

1119867 (119905 1199050) int119905

1199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051

1199050

119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)

Letting 119905 997888rarr infin we obtain

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)

which yields a contradiction to (42) The proof is complete

Discrete Dynamics in Nature and Society 7

Second we study the oscillation of (12) under the follow-ing condition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 lt infin (51)

Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050

intinfin119879

119891minus1 [ 1V (119905) int

119905

119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)

where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)

If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)

(53)

Integrating both sides of the above inequality from 1199052 to 119905 wehave

119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904gt 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904119905 isin [1199052infin)

(54)

Hence from (17) we get

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )

ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )

119905 isin [1199052infin)

(55)

Integrating both sides of the last inequality from 1199052 to 119905 weobtain

119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052

119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906

119905 isin [1199052infin) (56)

Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete

Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem

4 Examples

Example 1 Consider the fractional differential equation

(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)

In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since

intinfin1199050

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

(exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

exp(1119905 minus 11199050)119889119905 ge intinfin1199050

exp(minus 11199050)119889119905 = infin(58)

then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin

int1199051

[[1198961119903 (119904) 119902 (119904) minus

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= lim sup119905997888rarrinfin

int1199051

[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]

]119889119904= lim sup119905997888rarrinfin

int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin

(59)

which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory

8 Discrete Dynamics in Nature and Society

Example 2 Consider the fractional differential equation

(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905

(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)

In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that

(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050

Since

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

= lim sup119905997888rarrinfin

1(119905 minus 1199050)14 int119905

1199050

119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin

sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]

= lim sup119905997888rarrinfin

(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin

(61)

which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory

5 Conclusion

In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper

Authorsrsquo Contributions

Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript

Acknowledgments

This research is supported by National Science Foundation ofChina (11671227)

References

[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962

[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999

[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006

[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993

[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002

[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010

[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012

[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012

[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

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Discrete Dynamics in Nature and Society 3

(1198601) 119901(119905) ge 0 and 119902(119905) ge 0 are continuous functions on119905 isin [1199050infin) 1199050 gt 0(1198602) ℎ 119891 R 997888rarr R are continuous functions with119909ℎ(119909) gt 0 119909119891(119909) gt 0 for 119909 = 0 and there exist positive

constants 1198961 1198962 such that ℎ(119909)119909 ge 1198961 119909119891(119909) ge 1198962 for all119909 = 0(1198603) 1198911015840(119906) le 119906 119891minus1(119906) isin 119862(RR) are continuousfunctions with 119891minus1(119906) gt 0 for 119906 = 0 and there exists somepositive constant 1205721 such that 119891minus1(119906V) ge 1205721119891minus1(119906)119891minus1(V) for119906V = 02 Preliminaries

For convenience some background materials from fractionalcalculus are given

From [4] we can get the definition for Liouville right-side fractional integral and Liouville right-side fractionalderivative on the whole axis R of order 120573 for a function119892 R+ 997888rarr R as follows

(119868120573minus119892) (119905) fl 1Γ (120573) intinfin

119905(V minus 119905)120573minus1 119892 (V) 119889V 119905 gt 0 (13)

(119863120573minus119892) (119905) fl (minus1)lceil120573rceil 119889lceil120573rceil119889119905lceil120573rceil (119868lceil120573rceilminus120573minus 119892) (119905)= (minus1)lceil120573rceil 1Γ (lceil120573rceil minus 120573) int

infin

119905(V minus 119905)lceil120573rceilminus120573minus1 119892 (V) 119889V

119905 gt 0(14)

provided the right hand side is pointwise defined on R+where Γ(sdot) is the gamma function defined by Γ(119905) flintinfin119905

119904119905minus1119890minus119904119889119904 and fl min119911 isin Z 119911 ge 120573 is the ceilingfunction

If 120573 isin (0 1) we have119863120572minus119910 (119905) fl minus 1Γ (1 minus 120572) 119889119889119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 (15)

for 119905 isin R+ fl (0infin)The following relations also existed

(119863(1+120572)minus 119910) (119905) = minus (119863120572minus119910)1015840 (119905) 120572 isin (0 1) 119905 gt 0 (16)

Set

119866 (119905) fl intinfin119905

(V minus 119905)minus120572 119910 (V) 119889V 120572 isin (0 1) (17)

and then

1198661015840 (119905) = minusΓ (1 minus 120572) (119863120572minus119910) (119905) 120572 isin (0 1) (18)

3 Main Results

First we study the oscillation of (12) under the followingcondition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 = infin (19)

Theorem 1 Suppose that (1198601) minus (1198603) and (19) hold fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that

lim sup119905997888rarrinfin

int1199051199050

V (119904)sdot [[1198961119903 (119904) 119902 (119904) minus

(1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = infin(20)

where 1198961 1198962 are defined as in (1198602) andV (119904) fl exp(int119904

1199050

119901 (V) 119889V) 119904 ge 1199050 (21)

Then every solution of (12) is oscillatory

Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Then there exists 1199051 isin[1199050infin] such that 119910(119905) gt 0 and 119866(119905) gt 0 for 119905 isin [1199051infin] where119866 is defined in (16) From (1198603) (12) and (16) we have

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 = minus1198911015840 (119863120572minus119910 (119905)) V (119905) 119863120572+1minus 119910 (119905)+ 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)

= 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) V (119905)

ge 119891 (119863120572minus119910 (119905)) V (119905) 119901 (119905)minus 119863120572+1minus 119910 (119905) 119863120572minus119910 (119905) V (119905)

= 119902 (119905) ℎ (119866 (119905)) V (119905) gt 0119905 isin [1199050infin]

(22)

Thus 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199050infin] SinceV(119905) gt 0 for 119905 isin [1199050infin] and from (1198603) we see that 119863120572minus119910(119905) iseventually of one sign Now we can claim

119863120572minus119910 (119905) lt 0 119905 isin [1199051infin] (23)

If not then there exists 1199052 isin [1199051infin] such that 119863120572minus119910(1199052) gt0 Since 119891(119863120572minus119910(119905))V(119905) is strictly increasing on [1199051infin] it isclear that 119891(119863120572minus119910(119905))V(119905) ge 119891(119863120572minus119910(1199052))V(1199052) fl 119888 gt 0 for119905 isin [1199052infin] Therefore from (18) we have

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1 ( 119888V (119905))

= 119891minus1 (119888 sdot exp(minusint1199051199050

119901 (V) 119889V))ge 1205721119891minus1 (119888) 119891minus1 (exp(minusint119905

1199050

119901 (V) 119889V)) 119905 isin [1199052infin]

(24)

4 Discrete Dynamics in Nature and Society

Then we get

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V)) le minus 1198661015840 (119905)1205721119891minus1 (119888) Γ (1 minus 120572) 119905 isin [1199052infin] (25)

Integrating the above inequality from 1199052 to 119905 we haveint1199051199052

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904le minusint1199051199052

1198661015840 (119904)1205721119891minus1 (119888) Γ (1 minus 120572)119889119904= minus 119866 (119905) minus 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572)

119905 isin [1199052infin]

(26)

Letting 119905 997888rarr infin we see

intinfin1199052

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904le 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt infin (27)

This is in contradiction with (19) Hence (23) holdsDefine the function 119908 as generalized Riccati substitution

119908 (119905) = 119903 (119905) minusV (119905) 119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (28)

Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (18) (22) (28)and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)

sdot minus [119891 (119863120572minus119910 (119905)) V (119905)]1015840 119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905)le 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot [minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905) ]= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) V (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198961119903 (119905) 119902 (119905) V (119905) + 1199082 (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))119903 (119905) 119891 (119863120572minus119910 (119905)) V (119905)le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905) V (119905) minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905)

(29)

That is

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905) (30)

Taking 119909 = 119908(119905) 119886 = 1198962Γ(1 minus 120572)119903(119905)V(119905) (119886 = 0) and 119887 =1199031015840(119905)119903(119905) from 1198861199092+119887119909 le minus11988724119886 and (30)we could concludethat

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + V (119905) (1199031015840 (119905))241198962Γ (1 minus 120572) 119903 (119905) (31)

Integrating both sides of inequality (31) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)

ge int1199051199050

[[minus1198961119903 (119904) 119902 (119904) V (119904) +

V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119905)]]119889119904 (32)

Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get

lim sup119905997888rarrinfin

int1199051199050

[[minus1198961119903 (119904) 119902 (119904) V (119904)

+ V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 lt 119908 (1199050) lt infin(33)

which is in contradiction with (20)If 119910(119905) is an eventually negative solution of (12) the proof

is similar hence we omit itThe proof is complete

Theorem 2 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that

lim sup119905997888rarrinfin

int1199051199050

[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= infin(34)

where V(119904) are defined inTheorem 1Then every solution of (12)is oscillatory

Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Proceeding the same asin the proof of Theorem 1 we get (23) Define the function119908(119905) as follows

119908(119905) = 119903 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (35)

Discrete Dynamics in Nature and Society 5

Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840

= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)

(36)

That is

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)

Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude

1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)

Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050

[[minus1198961119903 (119904) 119902 (119904) +

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)

Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get

lim sup119905997888rarrinfin

int1199051199050

[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

lt 119908 (1199050) lt infin(40)

which is in contradiction with (34)

If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it

The proof is complete

We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)

and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable

Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904

= infin(42)

Then every solution of (12) is oscillatory

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)

Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get

int1199051199051

1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051

119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051

119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]

(43)

Using the formula integration by parts we obtain

minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)

+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)

(44)

6 Discrete Dynamics in Nature and Society

Substituting (44) with (43) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)

minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)

Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget

int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904

le int1199051199051

(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904

le int1199051199051

[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905

1199051

[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904

le int1199051199051

[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905

1199051

[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904

(46)

Substituting (46) in (45) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905

1199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)

Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get

int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)

119905 isin [1199051infin) (48)

Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have

1119867 (119905 1199050) int119905

1199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051

1199050

119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)

Letting 119905 997888rarr infin we obtain

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)

which yields a contradiction to (42) The proof is complete

Discrete Dynamics in Nature and Society 7

Second we study the oscillation of (12) under the follow-ing condition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 lt infin (51)

Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050

intinfin119879

119891minus1 [ 1V (119905) int

119905

119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)

where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)

If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)

(53)

Integrating both sides of the above inequality from 1199052 to 119905 wehave

119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904gt 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904119905 isin [1199052infin)

(54)

Hence from (17) we get

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )

ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )

119905 isin [1199052infin)

(55)

Integrating both sides of the last inequality from 1199052 to 119905 weobtain

119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052

119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906

119905 isin [1199052infin) (56)

Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete

Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem

4 Examples

Example 1 Consider the fractional differential equation

(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)

In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since

intinfin1199050

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

(exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

exp(1119905 minus 11199050)119889119905 ge intinfin1199050

exp(minus 11199050)119889119905 = infin(58)

then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin

int1199051

[[1198961119903 (119904) 119902 (119904) minus

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= lim sup119905997888rarrinfin

int1199051

[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]

]119889119904= lim sup119905997888rarrinfin

int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin

(59)

which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory

8 Discrete Dynamics in Nature and Society

Example 2 Consider the fractional differential equation

(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905

(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)

In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that

(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050

Since

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

= lim sup119905997888rarrinfin

1(119905 minus 1199050)14 int119905

1199050

119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin

sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]

= lim sup119905997888rarrinfin

(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin

(61)

which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory

5 Conclusion

In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper

Authorsrsquo Contributions

Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript

Acknowledgments

This research is supported by National Science Foundation ofChina (11671227)

References

[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962

[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999

[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006

[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993

[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002

[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010

[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012

[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012

[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

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4 Discrete Dynamics in Nature and Society

Then we get

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V)) le minus 1198661015840 (119905)1205721119891minus1 (119888) Γ (1 minus 120572) 119905 isin [1199052infin] (25)

Integrating the above inequality from 1199052 to 119905 we haveint1199051199052

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904le minusint1199051199052

1198661015840 (119904)1205721119891minus1 (119888) Γ (1 minus 120572)119889119904= minus 119866 (119905) minus 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572)

119905 isin [1199052infin]

(26)

Letting 119905 997888rarr infin we see

intinfin1199052

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904le 119866 (1199052)1205721119891minus1 (119888) Γ (1 minus 120572) lt infin (27)

This is in contradiction with (19) Hence (23) holdsDefine the function 119908 as generalized Riccati substitution

119908 (119905) = 119903 (119905) minusV (119905) 119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (28)

Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (18) (22) (28)and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)

sdot minus [119891 (119863120572minus119910 (119905)) V (119905)]1015840 119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905)le 1199031015840 (119905) minus119891 (119863120572minus119910 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot [minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119891 (119863120572minus119910 (119905)) V (119905) 1198661015840 (119905)1198662 (119905) ]= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) minus119902 (119905) ℎ (119866 (119905)) V (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) V (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198961119903 (119905) 119902 (119905) V (119905) + 1199082 (119905) (minusΓ (1 minus 120572)119863120572minus119910 (119905))119903 (119905) 119891 (119863120572minus119910 (119905)) V (119905)le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905) V (119905) minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905)

(29)

That is

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + 1199031015840 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) V (119905) 1199082 (119905) (30)

Taking 119909 = 119908(119905) 119886 = 1198962Γ(1 minus 120572)119903(119905)V(119905) (119886 = 0) and 119887 =1199031015840(119905)119903(119905) from 1198861199092+119887119909 le minus11988724119886 and (30)we could concludethat

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) V (119905) + V (119905) (1199031015840 (119905))241198962Γ (1 minus 120572) 119903 (119905) (31)

Integrating both sides of inequality (31) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)

ge int1199051199050

[[minus1198961119903 (119904) 119902 (119904) V (119904) +

V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119905)]]119889119904 (32)

Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get

lim sup119905997888rarrinfin

int1199051199050

[[minus1198961119903 (119904) 119902 (119904) V (119904)

+ V (119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 lt 119908 (1199050) lt infin(33)

which is in contradiction with (20)If 119910(119905) is an eventually negative solution of (12) the proof

is similar hence we omit itThe proof is complete

Theorem 2 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] such that

lim sup119905997888rarrinfin

int1199051199050

[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= infin(34)

where V(119904) are defined inTheorem 1Then every solution of (12)is oscillatory

Proof Suppose that 119910(119905) is a nonoscillation solution of (12)without loss of generality we may assume that 119910(119905) is aneventually positive solution of (12) Proceeding the same asin the proof of Theorem 1 we get (23) Define the function119908(119905) as follows

119908(119905) = 119903 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) 119905 isin [1199051infin] (35)

Discrete Dynamics in Nature and Society 5

Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840

= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)

(36)

That is

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)

Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude

1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)

Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050

[[minus1198961119903 (119904) 119902 (119904) +

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)

Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get

lim sup119905997888rarrinfin

int1199051199050

[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

lt 119908 (1199050) lt infin(40)

which is in contradiction with (34)

If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it

The proof is complete

We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)

and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable

Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904

= infin(42)

Then every solution of (12) is oscillatory

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)

Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get

int1199051199051

1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051

119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051

119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]

(43)

Using the formula integration by parts we obtain

minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)

+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)

(44)

6 Discrete Dynamics in Nature and Society

Substituting (44) with (43) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)

minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)

Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget

int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904

le int1199051199051

(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904

le int1199051199051

[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905

1199051

[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904

le int1199051199051

[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905

1199051

[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904

(46)

Substituting (46) in (45) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905

1199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)

Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get

int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)

119905 isin [1199051infin) (48)

Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have

1119867 (119905 1199050) int119905

1199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051

1199050

119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)

Letting 119905 997888rarr infin we obtain

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)

which yields a contradiction to (42) The proof is complete

Discrete Dynamics in Nature and Society 7

Second we study the oscillation of (12) under the follow-ing condition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 lt infin (51)

Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050

intinfin119879

119891minus1 [ 1V (119905) int

119905

119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)

where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)

If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)

(53)

Integrating both sides of the above inequality from 1199052 to 119905 wehave

119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904gt 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904119905 isin [1199052infin)

(54)

Hence from (17) we get

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )

ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )

119905 isin [1199052infin)

(55)

Integrating both sides of the last inequality from 1199052 to 119905 weobtain

119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052

119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906

119905 isin [1199052infin) (56)

Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete

Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem

4 Examples

Example 1 Consider the fractional differential equation

(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)

In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since

intinfin1199050

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

(exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

exp(1119905 minus 11199050)119889119905 ge intinfin1199050

exp(minus 11199050)119889119905 = infin(58)

then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin

int1199051

[[1198961119903 (119904) 119902 (119904) minus

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= lim sup119905997888rarrinfin

int1199051

[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]

]119889119904= lim sup119905997888rarrinfin

int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin

(59)

which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory

8 Discrete Dynamics in Nature and Society

Example 2 Consider the fractional differential equation

(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905

(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)

In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that

(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050

Since

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

= lim sup119905997888rarrinfin

1(119905 minus 1199050)14 int119905

1199050

119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin

sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]

= lim sup119905997888rarrinfin

(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin

(61)

which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory

5 Conclusion

In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper

Authorsrsquo Contributions

Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript

Acknowledgments

This research is supported by National Science Foundation ofChina (11671227)

References

[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962

[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999

[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006

[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993

[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002

[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010

[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012

[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012

[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

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Discrete Dynamics in Nature and Society 5

Then we have 119908(119905) gt 0 for 119905 isin [1199051infin] From (16) (18) (22)(35) and (1198601) minus (1198603) it follows that1199081015840 (119905) = 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905) [minus119891 (119863120572minus119910 (119905))119866 (119905) ]1015840

= 1199031015840 (119905) minus119891 (119863120572minus119910 (119905))119866 (119905) + 119903 (119905)sdot 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905) 119866 (119905) + 119891 (119863120572minus119910 (119905)) 1198661015840 (119905)1198662 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905) 1198911015840 (119863120572minus119910 (119905))119863120572+1minus 119910 (119905)119866 (119905) + 119903 (119905)sdot 119891 (119863120572minus119910 (119905)) (minusΓ (1 minus 120572)119863120572minus119910 (119905))1198662 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905)+ 119903 (119905) 119863120572minus119910 (119905) 119863120572+1minus 119910 (119905)119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)= 1199031015840 (119905)119903 (119905) 119908 (119905) + 119903 (119905)sdot 119901 (119905) 119891 (119863120572minus119910 (119905)) minus 119902 (119905) ℎ (119866 (119905))119866 (119905) minus 1198962Γ (1 minus 120572)119903 (119905)sdot 1199082 (119905) le 1199031015840 (119905)119903 (119905) 119908 (119905) minus 119901 (119905) 119908 (119905) minus 1198961119903 (119905) 119902 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905)

(36)

That is

1199081015840 (119905) le minus1198961119903 (119905) 119902 (119905) + 1199031015840 (119905) minus 119903 (119905) 119901 (119905)119903 (119905) 119908 (119905)minus 1198962Γ (1 minus 120572)119903 (119905) 1199082 (119905) (37)

Taking 119909 = 119908(119905) 119886 = 1198962Γ(1minus120572)119903(119905) (119886 = 0) and 119887 = (1199031015840(119905) minus119903(119905)119901(119905))119903(119905) from 1198861199092 + 119887119909 le minus11988724119886 and (37) we conclude

1199081015840 (119905) le minus1198961119903 (119904) 119902 (119904) + [1199031015840 (119905) minus 119903 (119905) 119901 (119905)]241198962Γ (1 minus 120572) 119903 (119905) (38)

Integrating both sides of inequality (38) from 1199050 to 119905 we obtaininfin gt 119908(1199050) gt 119908 (1199050) minus 119908 (119905)ge int1199051199050

[[minus1198961119903 (119904) 119902 (119904) +

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904 (39)

Taking the limit supremum of both sides of the aboveinequality as 119905 997888rarr infin we get

lim sup119905997888rarrinfin

int1199051199050

[[1198961119903 (119904) 119902 (119904) minus[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

lt 119908 (1199050) lt infin(40)

which is in contradiction with (34)

If 119910(119905) is an eventually negative solution of (12) the proofis similar here we omit it

The proof is complete

We define a function class119866 setD fl (119905 119904) fl 119905 ge 119904 ge 1199050D0 fl (119905 119904) fl 119905 gt 119904 ge 1199050We say119867(119905 119904) isin 119866 if119867(119905 119904) satisfy119867(119905 119905) = 0 119905 ge 1199050119867 (119905 119904) gt 0 (119905 119904) isin D0 (41)

and 119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904) fl 120597119867(119905 119904)120597119904 on D0 with respect to the secondvariable

Theorem 3 Suppose that (1198601) minus (1198603) and (19) hold Fur-thermore assume that there exists a positive function 119903(119905) isin1198621[1199050infin] and a function119867 isin 119866 satisfies

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]] 119889119904

= infin(42)

Then every solution of (12) is oscillatory

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 2 we get (37)

Multiplying (37) by 119867(119905 119904) and integrating from 1199051 to 119905we get

int1199051199051

1198961119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904+ minusint1199051199051

119867(119905 119904) 119908 (119904) 1199031015840 (119904)119903 (119904) 119889119904minus int1199051199051

119867(119905 119904) 1199082 (119904) 1198962Γ (1 minus 120572)119903 (119904) 119889119904119905 isin [1199051infin]

(43)

Using the formula integration by parts we obtain

minusint1199051199051

119867(119905 119904) 1199081015840 (119904) 119889119904 = minus [119867 (119905 119904) 119908 (119904)]119904=119905119904=1199051+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904= 119867 (119905 1199051)119908 (1199051)

+ int1199051199051

1198671015840119904 (119905 119904) 119908 (119904) 119889119904119905 isin [1199051infin)

(44)

6 Discrete Dynamics in Nature and Society

Substituting (44) with (43) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)

minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)

Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget

int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904

le int1199051199051

(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904

le int1199051199051

[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905

1199051

[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904

le int1199051199051

[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905

1199051

[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904

(46)

Substituting (46) in (45) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905

1199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)

Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get

int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)

119905 isin [1199051infin) (48)

Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have

1119867 (119905 1199050) int119905

1199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051

1199050

119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)

Letting 119905 997888rarr infin we obtain

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)

which yields a contradiction to (42) The proof is complete

Discrete Dynamics in Nature and Society 7

Second we study the oscillation of (12) under the follow-ing condition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 lt infin (51)

Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050

intinfin119879

119891minus1 [ 1V (119905) int

119905

119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)

where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)

If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)

(53)

Integrating both sides of the above inequality from 1199052 to 119905 wehave

119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904gt 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904119905 isin [1199052infin)

(54)

Hence from (17) we get

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )

ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )

119905 isin [1199052infin)

(55)

Integrating both sides of the last inequality from 1199052 to 119905 weobtain

119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052

119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906

119905 isin [1199052infin) (56)

Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete

Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem

4 Examples

Example 1 Consider the fractional differential equation

(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)

In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since

intinfin1199050

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

(exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

exp(1119905 minus 11199050)119889119905 ge intinfin1199050

exp(minus 11199050)119889119905 = infin(58)

then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin

int1199051

[[1198961119903 (119904) 119902 (119904) minus

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= lim sup119905997888rarrinfin

int1199051

[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]

]119889119904= lim sup119905997888rarrinfin

int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin

(59)

which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory

8 Discrete Dynamics in Nature and Society

Example 2 Consider the fractional differential equation

(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905

(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)

In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that

(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050

Since

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

= lim sup119905997888rarrinfin

1(119905 minus 1199050)14 int119905

1199050

119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin

sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]

= lim sup119905997888rarrinfin

(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin

(61)

which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory

5 Conclusion

In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper

Authorsrsquo Contributions

Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript

Acknowledgments

This research is supported by National Science Foundation ofChina (11671227)

References

[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962

[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999

[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006

[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993

[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002

[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010

[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012

[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012

[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Discrete Dynamics in Nature and Society

Substituting (44) with (43) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 le 119867 (119905 1199051)119908 (1199051)+ int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904)

minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904(45)

Taking 119909 = 119908(119905) 119886 = minus1198962Γ(1 minus 120572)(119867(119905 119904)119903(119904)) (119886 = 0) and119887 = 1198671015840119904(119905 119904) + 119867(119905 119904)(1199031015840(119904)119903(119904)) from 1198861199092 + 119887119909 le minus11988724119886 weget

int1199051199051

[1198671015840119904 (119905 119904) + 119867 (119905 119904) 1199031015840 (119904)119903 (119904) ]119908 (119904) minus 1198962Γ (1 minus 120572) 119867 (119905 119904)119903 (119904) 1199082 (119904) 119889119904

le int1199051199051

(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904) + 21198671015840119904 (119905 119904)119867 (119905 119904) (1199031015840 (119904) 119903 (119904))41198962Γ (1 minus 120572)119867 (119905 119904) 119903 (119904) 119889119904

le int1199051199051

[(1198671015840119904 (119905 119904))2 + 1198672 (119905 119904) (1199031015840 (119904))2 1199032 (119904)] 119903 (119904)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 = int119905

1199051

[1198671015840119904 (119905 119904) 119903 (119904)]2 + [119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904

le int1199051199051

[1198671015840119904 (119905 119904) 119903 (119904) + 119867 (119905 119904) 1199031015840 (119904)]2)41198962Γ (1 minus 120572)119867 (119905 119904) 119889119904 le int119905

1199051

[119867 (119905 119904) 1199031015840 (119904)]241198962Γ (1 minus 120572)119867 (119905 119904)119889119904 = int1199051199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904

(46)

Substituting (46) in (45) we have

1198961 int1199051199051

119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904le 119867 (119905 1199051) 119908 (1199051) + int119905

1199051

119867(119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)119889119904(47)

Since 1198671015840119904(119905 119904) le 0 for 119905 gt 119904 ge 1199050 we have 0 lt 119867(119905 1199051) le119867(119905 1199050) for 119905 gt 1199051 ge 1199050 Therefore from the previousinequality we get

int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904 minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904le 119896minus1119867(119905 1199051) 119908 (1199051) le 119896minus1119867(119905 1199050) 119908 (1199051)

119905 isin [1199051infin) (48)

Since 0 lt 119867(119905 1199051) le 119867(119905 1199050) for 119905 gt 119904 ge 1199050 we have 0 lt119867(119905 119904)119867(119905 1199050) le 1 for 119905 gt 119904 ge 1199050 Hence it follows from (48)that we have

1119867 (119905 1199050) int119905

1199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 = 1119867 (119905 1199051)sdot int11990511199050

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 + 1119867 (119905 1199050)sdot int1199051199051

[[119903 (119904) 119902 (119904)119867 (119905 119904) 119889119904

minus 119867 (119905 119904) (1199031015840 (119904))241198962Γ (1 minus 120572) 119903 (119904)]]119889119904 le 1119867 (119905 1199051) int1199051

1199050

119903 (119904)sdot 119902 (119904)119867 (119905 119904) 119889119904 + 1119867 (119905 1199051)119896minus1119867(119905 1199050)119908 (1199051)le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) 119905 isin [1199051infin) (49)

Letting 119905 997888rarr infin we obtain

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

le int11990511199050

119903 (119904) 119902 (119904) 119889119904 + 119896minus1119908 (1199051) lt infin(50)

which yields a contradiction to (42) The proof is complete

Discrete Dynamics in Nature and Society 7

Second we study the oscillation of (12) under the follow-ing condition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 lt infin (51)

Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050

intinfin119879

119891minus1 [ 1V (119905) int

119905

119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)

where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)

If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)

(53)

Integrating both sides of the above inequality from 1199052 to 119905 wehave

119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904gt 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904119905 isin [1199052infin)

(54)

Hence from (17) we get

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )

ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )

119905 isin [1199052infin)

(55)

Integrating both sides of the last inequality from 1199052 to 119905 weobtain

119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052

119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906

119905 isin [1199052infin) (56)

Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete

Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem

4 Examples

Example 1 Consider the fractional differential equation

(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)

In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since

intinfin1199050

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

(exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

exp(1119905 minus 11199050)119889119905 ge intinfin1199050

exp(minus 11199050)119889119905 = infin(58)

then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin

int1199051

[[1198961119903 (119904) 119902 (119904) minus

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= lim sup119905997888rarrinfin

int1199051

[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]

]119889119904= lim sup119905997888rarrinfin

int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin

(59)

which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory

8 Discrete Dynamics in Nature and Society

Example 2 Consider the fractional differential equation

(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905

(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)

In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that

(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050

Since

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

= lim sup119905997888rarrinfin

1(119905 minus 1199050)14 int119905

1199050

119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin

sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]

= lim sup119905997888rarrinfin

(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin

(61)

which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory

5 Conclusion

In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper

Authorsrsquo Contributions

Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript

Acknowledgments

This research is supported by National Science Foundation ofChina (11671227)

References

[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962

[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999

[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006

[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993

[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002

[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010

[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012

[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012

[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 7

Second we study the oscillation of (12) under the follow-ing condition

intinfin1199050

119891minus1 (exp(minusint1199041199050

119901 (V) 119889V))119889119904 lt infin (51)

Theorem4 Suppose that (1198601)minus(1198603) and (51) hold and thereexists a positive function 119903(119905) isin 1198621[1199050infin] such that (20) holdsFurthermore assume that for every constant 119879 ge 1199050

intinfin119879

119891minus1 [ 1V (119905) int

119905

119879119902 (119904) V (119904) 119889119904] 119889119905 = infin (52)

where V(119904) are defined as in Theorem 1 Then every solution of(12) is oscillatory or satisfies lim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

Proof Suppose that 119910 is a nonoscillation solution of (12)without loss of generality we may assume that 119910 is aneventually positive solution of (12) Proceeding as in the proofof Theorem 1 we know that 119863120572minus119910(119905) is eventually one signthen there are two cases for the sign of119863120572minus119910(119905)

If 119863120572minus119910(119905) is eventually negative similar to Theorem 1we have the oscillation of (12) Next if 119863120572minus119910(119905) is eventuallypositive then there exists 1199052 ge 1199051 such that 119863120572minus119910(119905) gt 0 for119905 ge 1199052 From (18) we get 1198661015840(119905) lt 0 for 119905 ge 1199052 Thus we getlim119905997888rarrinfin119866(119905) fl 119871 ge0 and 119866(119905) ge 119871 We now claim that 119871 = 0Assuming not that is 119871 gt 0 then from (23) and (1198603) we get

[119891 (119863120572minus119910 (119905)) V (119905)]1015840 ge 119902 (119905) ℎ (119866 (119905)) V (119905)ge 1198961119902 (119905) 119866 (119905) V (119905)ge 1198961119871119902 (119905) V (119905) 119905 isin [1199052infin)

(53)

Integrating both sides of the above inequality from 1199052 to 119905 wehave

119891 (119863120572minus119910 (119905)) V (119905) ge 119891 (119863120572minus119910 (1199052)) V (1199052)+ 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904gt 1198961119871int119905

1199052

119902 (119904) V (119904) 119889119904119905 isin [1199052infin)

(54)

Hence from (17) we get

minus 1198661015840 (119905)Γ (1 minus 120572) = 119863120572minus119910 (119905) ge 119891minus1(1198961119871 int1199051199052 119902 (119904) V (119904) 119889119904V (119905) )

ge 1205721119891minus1 (1198961119871) 119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119905) )

119905 isin [1199052infin)

(55)

Integrating both sides of the last inequality from 1199052 to 119905 weobtain

119866 (119905) le 119866 (1199052) minus 1205721Γ (1 minus 120572) 119891minus1 (1198961119871)sdot int1199051199052

119891minus1(int1199051199052119902 (119904) V (119904) 119889119904V (119906) )119889119906

119905 isin [1199052infin) (56)

Letting 119905 997888rarr infin from (52) we get lim119905997888rarrinfin119866(119905) = minusinfin thisis in contradiction with 119866(119905) gt 0 Therefore we have 119871 = 0that is lim119905997888rarrinfin119866(119905) = 0 The proof is complete

Theorem 5 Suppose that (1198601) minus (1198603) and (51) hold Let 119903(119905)and 119867(119905 119904) be defined as in Theorem 3 such that (42) holdsFurthermore assume that for every constant 119879 ge 1199050 (52)holds Then every solution of (12) is oscillatory or satisfieslim119905997888rarrinfin intinfin

1199050(119904 minus 119905)minus120572119910(119904)119889119904 = 0

From Theorem 3 proceeding as in the proof ofTheorem 4 we get the results of the theorem

4 Examples

Example 1 Consider the fractional differential equation

(11986332minus 119910) (119905) sdot (11986312minus 119910) (119905) minus 11199052 (11986312minus 119910) (119905)+ 1119905 int

infin

119905(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (57)

In (57) 120572 = 12 119901(119905) = 11199052 119902(119905) = 1119905 and 119891(119909) = ℎ(119909) = 119909Since

intinfin1199050

119891minus1 (exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

(exp(minusint1199051199050

119901 (V) 119889V))119889119905= intinfin1199050

exp(1119905 minus 11199050)119889119905 ge intinfin1199050

exp(minus 11199050)119889119905 = infin(58)

then (19) holdsTaking 1199050 = 1 1198961 = 1198962 = 1 It is clear that conditions(1198601) minus (1198603) hold Furthermore taking 119903(119905) = 1199052 we havelim sup119905997888rarrinfin

int1199051

[[1198961119903 (119904) 119902 (119904) minus

[1199031015840 (119904) minus 119903 (119904) 119901 (119904)]241198962Γ (1 minus 120572) 119903 (119904) ]]119889119904

= lim sup119905997888rarrinfin

int1199051

[[1119904 1199042 minus (2119904 minus 1199042 (11199042))24radic1205871199042 ]

]119889119904= lim sup119905997888rarrinfin

int1199051(119904 minus 14radic120587 (2119904 minus 1119904 )2)119889119904 = infin

(59)

which shows that (34) holds Therefore by Theorem 2 everysolution of (12) is oscillatory

8 Discrete Dynamics in Nature and Society

Example 2 Consider the fractional differential equation

(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905

(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)

In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that

(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050

Since

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

= lim sup119905997888rarrinfin

1(119905 minus 1199050)14 int119905

1199050

119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin

sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]

= lim sup119905997888rarrinfin

(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin

(61)

which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory

5 Conclusion

In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper

Authorsrsquo Contributions

Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript

Acknowledgments

This research is supported by National Science Foundation ofChina (11671227)

References

[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962

[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999

[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006

[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993

[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002

[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010

[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012

[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012

[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Discrete Dynamics in Nature and Society

Example 2 Consider the fractional differential equation

(11986354minus 119910) (119905) sdot (11986314minus 119910) (119905) minus 11199052 (11986314minus 119910) (119905)+ 119905 intinfin119905

(119904 minus 119905)minus120572 119910 (119904) 119889119904 = 0 119905 gt 0 (60)

In (60) 120572 = 14 119901(119905) = 11199052 119902(119905) = 119905 and 119891(119909) = ℎ(119909) = 119909Proceeding the same process as Example 1 we see that

(19) holds Taking 1199050 = 1 1198961 = 1198962 = 1 119903(119905) = 1 It is clear thatconditions (1198601) minus (1198603) hold Furthermore taking 119867(119905 119904) =(119905 minus 119904)14 it meets1198671015840119904(119905 119904) = minus(14)(119905 minus 119904) lt 0 for (119905 119904) isin D0D0 fl (119905 119904) 119905 gt 119904 ge 1199050

Since

lim sup119905997888rarrinfin

1119867 (119905 1199050)sdot int1199051199050

119867(119905 119904) [[119903 (119904) 119902 (119904) minus(1199031015840 (119904))2411989611198962Γ (1 minus 120572) 119903 (119904)]]119889119904

= lim sup119905997888rarrinfin

1(119905 minus 1199050)14 int119905

1199050

119904 (119905 minus 119904)14 119889119904 = lim sup119905997888rarrinfin

sdot 1(119905 minus 1199050)14 [45 (119905 minus 1199050)54 1199050 + 1645 (119905 minus 1199050)94]

= lim sup119905997888rarrinfin

(45 (119905 minus 1199050) 1199050 + 1645 (119905 minus 1199050)2) = infin

(61)

which shows that (42) holds byTheorem 3 every solution of(12) is oscillatory

5 Conclusion

In the paper by using the generalized Riccati transformationand inequality technique we study a class of 2120572 + 1 orderfractional differential equations in the form (12) which con-tains the damping term and has not been studied beforeTheoscillation criteria of (12) are obtained and some examples aregiven to reinforce our results

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper

Authorsrsquo Contributions

Hui Liu carried out the main results and completed thecorresponding proof Run Xu participated in the proofand helped in completing Section 4 All authors read andapproved the final manuscript

Acknowledgments

This research is supported by National Science Foundation ofChina (11671227)

References

[1] G W Leibniz Mathematische Schriften Grorg Olms Verlags-buchhandlung Hildesheim Germany 1962

[2] K B Oldham and J SpanierThe Fractional Calculus AcademicPress San Diego USA 1974

[3] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fraction Differential Equations toMethods of Their Solution and some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego CA USA 1999

[4] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations ElsevierScience BV Amsterdam Netherlands 2006

[5] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calculus to viscoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[6] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Goron andBreach Science Publishers Yverdon Switzerland 1993

[8] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[9] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fraction differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 pp 401ndash418 2002

[10] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011

[11] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[12] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheoryMethodsampApplicationsA vol 72 no 3-4 pp 1768ndash17772010

[13] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[14] S R Grace R P Agarwal P J Y Wong and A Zafer ldquoOnthe oscillation of fractional differential equationsrdquo FractionalCalculus and Applied Analysis vol 15 no 2 pp 222ndash231 2012

[15] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 33 18pages 2012

[16] D-X Chen ldquoOscillatory behavior of a class of fractionaldifferential equations with dampingrdquo ldquoPolitehnicardquo University ofBucharest Scientific Bulletin Series A Applied Mathematics andPhysics vol 75 no 1 pp 107ndash118 2013

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 9

[17] Z Han Y Zhao Y Sun and C Zhang ldquoOscillation for a classof fractional differential equationrdquoDiscrete Dynamics in Natureand Society vol 2013 Article ID 390282 6 pages 2013

[18] R Xu ldquoOscillation criteria for nonlinear fractional differentialequationsrdquo Journal of AppliedMathematics vol 2013 Article ID971357 7 pages 2013

[19] Q Huizeng and B Zheng ldquoOscillation of a class of fractionaldifferential equations with damping termrdquoThe Scientific WorldJournal vol 2013 Article ID 685621 9 pages 2013

[20] J O Alzabut and T Abdeljawad ldquoSufficient conditions for theoscillation of nonlinear fractional difference equationsrdquo Journalof Fractional Calculus and Applications vol 5 no 1 pp 177ndash1872014

[21] B Abdalla K Abodayeh T Abdeljawad and J Alzabut ldquoNewoscillation criteria for forced nonlinear fractional differenceequationsrdquo Vietnam Journal of Mathematics vol 45 no 4 pp609ndash618 2017

[22] Z Bai and R Xu ldquoThe asymptotic behavior of solutions for aclass of nonlinear fractional difference equations with dampingtermrdquoDiscreteDynamics inNature and Society vol 2018ArticleID 5232147 11 pages 2018

[23] B Abdalla and T Abdeljawad ldquoOn the oscillation of Hadamardfractional differential equationsrdquo Advances in Difference Equa-tions vol 2018 no 409 2018

[24] J Alzabut T Abdeljawad and H Alrabaiah ldquoOscillationcriteria for forced and damped nabla fractional differenceequationsrdquo Journal of Computational Analysis and Applicationsvol 24 no 8 pp 1387ndash1394 2018

[25] B Abdalla J Alzabut and T Abdeljawad ldquoOn the oscillationof higher order fractional difference equations with mixednonlinearitiesrdquo Hacettepe Journal of Mathematics and Statisticsvol 47 no 2 pp 207ndash217 2018

[26] W Wusheng ldquoAnalytic invariant curves of a nonlinear secondorder difference equationrdquo Acta Mathematica Scientia vol 29no 2 pp 415ndash426 2009

[27] H Liu and F Meng ldquoSome new nonlinear integral inequalitieswith weakly singular kernel and their applications to FDEsrdquoJournal of Inequalities and Applications vol 2015 no 209 2015

[28] R Xu andFMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 no 1 pp 1ndash162016

[29] R Xu ldquoSome new nonlinear weakly singular integral inequali-ties and their applicationsrdquo Journal ofMathematical Inequalitiesvol 11 no 4 pp 1007ndash1018 2017

[30] Y Guan Z Zhao and X Lin ldquoOn the existence of positive solu-tions and negative solutions of singular fractional differentialequations via global bifurcation techniquesrdquo Boundary ValueProblems vol 2016 no 1 article 141 2016

[31] X Wang and R Xu ldquoThe existence and uniqueness of solutionsand Lyapunov-type inequality for CFR fractional differentialequationsrdquo Journal of Function Spaces vol 2018 Article ID5875108 7 pages 2018

[32] D Zan and R Xu ldquoThe existence results of solutions for systemof fractional differential equations with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Society vol 2018Article ID 8534820 8 pages 2018

[33] R Xu and Y Zhang ldquoGeneralized Gronwall fractional summa-tion inequalities and their applicationsrdquo Journal of Inequalitiesand Applications vol 2015 no 242 2015

[34] M Fanwei ldquoA new approach for solving fractional partialdifferential equationrdquo Journal of Applied Mathematics pp 1ndash112013

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Mathematical PhysicsAdvances in

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International Journal of

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

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Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

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Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom