research article weak contraction condition...
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Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 967045 5 pageshttpdxdoiorg1011552013967045
Research ArticleWeak Contraction Condition Involving Cubic Terms of119889(119909 119910) under the Fixed Point Consideration
Penumurthy Parvateesam Murthy and K N V V Vara Prasad
Department of Pure and Applied Mathematics Guru Ghasidas Vishwavidyalaya Bilaspur Chhattisgrah 495009 India
Correspondence should be addressed to K N V V Vara Prasad kvaraprasad71gmailcom
Received 31 January 2013 Accepted 14 April 2013
Academic Editor Krassimir T Atanassov
Copyright copy 2013 P P Murthy and K N V V Vara PrasadThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
A fixed point theorem is presented for single-valued map with using generalized 120593-weak contractive condition involving variouscombinations of 119889(119909 119910) on a complete metric space Our result is an extension as well as a generalization of Alber and Guerre-Delabriere (1997) in particular It also generalizes the results of Rhoades (2001) Choudhury and Dutta (2000) and Dutta andChoudhury (2008)
1 Introduction
Let (119883 119889) be a metric space A map 119879 119883 rarr 119883 is acontraction if for each 119909 119910 isin 119883 there exists a constant119896 isin (0 1) such that 119889(119879119909 119879119910) le 119896 119889(119909 119910)
A map 119879 119883 rarr 119883 is a 120593-weak contraction if for each119909 119910 isin 119864 there exists a function 120593 [0infin) rarr [0infin) 120593(119905) gt0 for all 119905 gt 0 and 120593(0) = 0 such that 119889(119879119909 119879119910) le 119889(119909 119910) minus
120593(119889(119909 119910))In [1] Alber and Guerre-Delabriere introduced the con-
cept of weak contraction in Hilbert spaces Rhoades [2] hasshown that the result whichAlber andGuerre-Delabriere hadproved in [2] is also valid in complete metric spaces
In this paper we introduced generalized 120593-weak contrac-tive condition involving various combinations of 119889(119909 119910) Ourresult is an extension as well as a generalization of Alber andGuerre-Delabriere [1] and Rhoades [2] in particular It alsogeneralizes the results of [3 4]
Now We state the result of Rhoades as follows
Theorem 1 (see [2 Theorem 2]) Let (119883 119889) be a completemetric space and let 119879 be a 120593-weak contraction on 119883 120593
[0 +infin) rarr [0 +infin)which is a continuous and nondecreasingfunction with 120593(119905) gt 0 for all 119905 isin (0infin) and 120593(0) = 0 then 119879
has a unique fixed point in119883If one takes 120593(119905) = (1 minus 119896)119905 where 0 lt 119896 lt 1 then weak
contraction reduces to contraction mapping
In this paper a new type of inequality is introduced withcubic terms involving 119901(ge0) called a ldquogeneralized 120593-weakcontractive condition with cubic terms involving 119901rdquo
Let (119883 119889) be ametric space and119879 a self-map of119883 satisfyingthe following condition
[1 + 119901119889 (119909 119910)] 1198892(119879119909 119879119910)
le 119901max 1
2[1198892(119909 119879119909) 119889 (119910 119879119910) + 119889 (119909 119879119909) 119889
2(119910 119879119910)]
119889 (119909 119879119909) 119889 (119909 119879119910) 119889 (119910 119879119909)
119889 (119909 119879119910) 119889 (119910 119879119909) 119889 (119910 119879119910)
+ 119898 (119909 119910) minus 120593 (119898 (119909 119910))
(1)
where
119898(119909 119910)
=max1198892 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910) 119889 (119909 119879119910) 119889 (119910 119879119909)
1
2[119889 (119909 119879119909) 119889 (119909 119879119910) + 119889 (119910 119879119909) 119889 (119910 119879119910)]
(2)
2 Journal of Mathematics
119901 ge 0 is a real number is and 120593 [0infin) rarr [0infin) is acontinuous function with 120593(119905) = 0 hArr 119905 = 0 and 120593(119905) gt 0 foreach 119905 gt 0
2 Main Result
Lemma 2 Let 119879 be a self map of a metric space 119883 satisfying(1) For any sequence 119909
119899 in 119883 defined by 119909
119899+1= 119879119909119899 119899 ge 0
Then the sequence 119909119899 is Cauchy in119883
Proof Let 1199090
isin 119883 be an arbitrary point Constructing thesequence 119909
119899 follows
119909119899+1
= 119879119909119899 119899 isin 119873
0 (3)
If 119909119899= 119909119899+1
for some 119899 then trivially 119879 has a fixed point Weassume 119909
119899+1= 119909119899 for all 119899 isin 119873
0 We write 120572
119899= 119889(119909
119899 119909119899+1
)First we prove that 120572
119899 is a nonincreasing sequence and
converges to 0Case 1 If 119899 is even taking 119909 = 119909
2119899and 119910 = 119909
2119899+1in (1) we get
[1 + 119901 119889 (1199092119899 1199092119899+1
)] 1198892(1198791199092119899 1198791199092119899+1
)
le 119901max 1
2[1198892(1199092119899 1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)
+119889 (1199092119899 1198791199092119899) 1198892(1199092119899+1
1198791199092119899+1
)]
119889 (1199092119899 1198791199092119899) 119889 (1199092119899 1198791199092119899+1
)
times 119889 (1199092119899+1
1198791199092119899) 119889 (119909
2119899 1198791199092119899+1
)
times 119889 (1199092119899+1
1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)
+ 119898 (1199092119899 1199092119899+1
) minus 120593 (119898 (1199092119899 1199092119899+1
))
(4)
where
119898(1199092119899 1199092119899+1
)
= max 1198892 (1199092119899 1199092119899+1
) 119889 (1199092119899 1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)
119889 (1199092119899 1198791199092119899+1
) 119889 (1199092119899+1
1198791199092119899)
1
2[119889 (1199092119899 1198791199092119899) 119889 (1199092119899 1198791199092119899+1
)
+119889 (1199092119899+1
1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)]
(5)
By using (3) we get
[1 + 119901 119889 (1199092119899 1199092119899+1
)] 1198892(1199092119899+1
1199092119899+2
)
le 119901max 1
2[1198892(1199092119899 1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)
+119889 (1199092119899 1199092119899+1
) 1198892(1199092119899+1
1199092119899+2
)]
119889 (1199092119899 1199092119899+1
) 119889 (1199092119899 1199092119899+2
)
times 119889 (1199092119899+1
1199092119899+1
) 119889 (1199092119899 1199092119899+2
)
times 119889 (1199092119899+1
1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)
+ 119898 (1199092119899 1199092119899+1
) minus 120593 (119898 (1199092119899 1199092119899+1
))
(6)
where
119898(1199092119899 1199092119899+1
)
= max 1198892 (1199092119899 1199092119899+1
) 119889 (1199092119899 1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)
119889 (1199092119899 1199092119899+2
) 119889 (1199092119899+1
1199092119899+1
)
1
2[119889 (1199092119899 1199092119899+1
) 119889 (1199092119899 1199092119899+2
)
+119889 (1199092119899+1
1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)]
(7)
Now consider 1205722119899
= 119889(1199092119899 1199092119899+1
) then we have
[1 + 1199011205722119899] 1205722
2119899+1
le 119901max 1
2[1205722
21198991205722119899+1
+ 12057221198991205722
2119899+1] 0 0
+ 119898 (1199092119899 1199092119899+1
) minus 120593 (119898 (1199092119899 1199092119899+1
))
(8)
where119898(1199092119899 1199092119899+1
) = max12057222119899 12057221198991205722119899+1
0 (12)[1205722119899119889(1199092119899
1199092119899+2
) + 0]By triangular inequality and using property of 120593 we get
119889 (1199092119899 1199092119899+2
) le 119889 (1199092119899 1199092119899+1
) + 119889 (1199092119899+1
1199092119899+2
)
= 1205722119899
+ 1205722119899+1
(9)
Then
119898(1199092119899 1199092119899+1
) le 1198981015840(119909 119910)
= max 12057222119899 1205722119899+1
1205722119899 0
1
21205722119899
(1205722119899
+ 1205722119899+1
) 0
(10)
If 1205722119899
lt 1205722119899+1
then (8) reduces to 1199011205722
2119899+1le 119901120572
2
2119899+1minus
120593(1205722
2119899+1) a contradiction Thus 120572
2
2119899+1le 1205722
2119899implies that
1205722119899+1
le 1205722119899
In a similar way if 119899 is odd we can obtain 1205722119899+2
lt 1205722119899+1
It follows that the sequence 120572
119899 is decreasing
Let lim119899rarrinfin
120572119899= 119903 for some 119903 ge 0
Journal of Mathematics 3
Suppose 119903 gt 0 then from inequality (1) we have
[1 + 119901 119889 (119909119899 119909119899+1
)] 1198892(119879119909119899 119879119909119899+1
)
le 119901max 1
2[1198892(119909119899 119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
+119889 (119909119899 119879119909119899) 1198892(119909119899+1
119879119909119899+1
)]
119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899+1
)
times 119889 (119909119899+1
119879119909119899) 119889 (119909
119899 119879119909119899+1
)
times119889 (119909119899+1
119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
+ 119898 (119909119899 119909119899+1
) minus 120593 (119898 (119909119899 119909119899+1
))
(11)
where119898(119909119899 119909119899+1
)
= max 1198892 (119909119899 119909119899+1
) 119889 (119909119899 119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
119889 (119909119899 119879119909119899+1
) 119889 (119909119899+1
119879119909119899)
1
2[119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899+1
)
+119889 (119909119899+1
119879119909119899) 119889 (119909119899+1
119879119909119899+1
)]
(12)
By using (3) we get
[1 + 119901119889 (119909119899 119909119899+1
)] 1198892(119909119899+1
119909119899+2
)
le 119901max 1
2[1198892(119909119899 119909119899+1
) 119889 (119909119899+1
119909119899+2
)
+119889 (119909119899 119909119899+1
) 1198892(119909119899+1
119909119899+2
)]
119889 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
)
119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
) 119889 (119909119899+1
119909119899+2
)
+ 119898 (119909119899 119909119899+1
) minus 120593 (119898 (119909119899 119909119899+1
))
(13)
where119898(119909119899 119909119899+1
)
= max 1198892 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+1
) 119889 (119909119899+1
119909119899+2
)
119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
)
1
2[119889 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+2
)
+119889 (119909119899+1
119909119899+1
) 119889 (119909119899+1
119909119899+2
)]
(14)
Using triangular inequality and property of 120593 and takinglimits 119899 rarr infin we get
[1 + 119901119903] 1199032le 1199011199033+ 1199032minus 120593 (119903
2) (15)
Then 120593(1199032) le 0 since 119903 is positive then by the property of 120593
we get 119903 = 0 We conclude that
lim119899rarrinfin
120572119899= lim119899rarrinfin
119889 (119909119899 119909119899+1
) = 119903 = 0 (16)
Now we show that 119909119899 is a Cauchy sequence Suppose
we assume that 119909119899 is not a Cauchy sequence then there is
120576 gt 0 for which we can find two sequences of positive integers119898(119896) and 119899(119896) such that for all positive integers 119896 119899(119896) gt
119898(119896) gt 119896
119889 (119909119898(119896)
119909119899(119896)
) ge 120598 119889 (119909119898(119896)
119909119899(119896)minus1
) lt 120598 (17)
Now
120598 le 119889 (119909119898(119896)
119909119899(119896)
) le 119889 (119909119898(119896)
119909119899(119896)minus1
) + 119889 (119909119899(119896)minus1
119909119899(119896)
)
(18)
Letting 119896 rarr infin we get
lim119896rarrinfin
119889 (119909119898(119896)
119909119899(119896)
) = 120598 (19)
Now from the triangular inequality we have1003816100381610038161003816119889 (119909119899(119896)
119909119898(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816 le 119889 (119909
119898(119896) 119909119898(119896)+1
)
(20)
Taking limits as 119896 rarr infin and using (16) and (19) we have
lim119896rarrinfin
119889 (119909119899(119896)
119909119898(119896)+1
) = 120598 (21)
Again from the triangular inequality we have1003816100381610038161003816119889 (119909119898(119896)
119909119899(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816 le 119889 (119909
119899(119896) 119909119899(119896)+1
)
(22)
Taking limits as 119896 rarr infin and using (16) and (19) we have
lim119896rarrinfin
119889 (119909119898(119896)
119909119899(119896)+1
) = 120598 (23)
Again by using triangular inequality we have1003816100381610038161003816119889 (119909119898(119896)+1
119909119899(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816
le 119889 (119909119898(119896)
119909119898(119896)+1
) + 119889 (119909119899(119896)
119909119899(119896)+1
)
(24)
Taking limit 119896 rarr infin in the above inequality and using (16)and (19) we have
lim119896rarrinfin
119889 (119909119899(119896)+1
119909119898(119896)+1
) = 120598 (25)
Again putting 119909 = 119909119898(119896)
and 119910 = 119909119899(119896)
in (1) we get
[1 + 119901119889 (119909119898(119896)
119909119899(119896)
)] 1198892(119879119909119898(119896)
119879119909119899(119896)
)
le 119901 1
2[1198892(119909119898(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
+119889 (119909119898(119896)
119879119909119898(119896)
) 1198892(119909119899(119896)
119879119909119899(119896)
)]
119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
)
119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
+ 119898 (119909119898(119896)
119909119899(119896)
) minus 120593 (119898 (119909119898(119896)
119909119899(119896)
))
(26)
4 Journal of Mathematics
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
)
1
2[119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119898(119896)
119879119909119899(119896)
)
+119889 (119909119899(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)]
(27)
Using (3) then we obtain
[1 + 119901119889 (119909119898(119896)
119909119899(119896)
)] 1198892(119909119898(119896)+1
119909119899(119896)+1
)
le 119901 1
2[1198892(119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+119889 (119909119898(119896)
119909119898(119896)+1
) 1198892(119909119899(119896)
119909119899(119896)+1
)]
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+ 119898 (119909119898(119896)
119909119899(119896)
) minus 120593 (119898 (119909119898(119896)
119909119899(119896)
))
(28)
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
119889 (119909119898(119896)
119909119899(119896)+1
) 119889 (119909119899(119896)
119909119898(119896)+1
)
1
2[119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
+119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)]
(29)
Letting 119896 rarr infin and using (16)ndash(25) we get
[1 + 119901120598] 1205982
le 119901max 1
2[0 + 0] 0 0 + 120598
2minus 120593 (120598
2)
= 1205982minus 120593 (120598
2)
(30)
a contradiction Thus 119909119899 is a Cauchy in119883
Theorem 3 Let 119879 be a self-map of a complete metric space 119883satisfying (1) Then 119879 has a unique fixed point in119883
Proof From Lemma 2 the sequence 119909119899 is a Cauchy in 119883
Since (119883 119889) is a complete metric space then there exists apoint 119911 isin 119883 such that
lim119899rarrinfin
119909119899= 119911 (31)
Now we prove that 119911 is a fixed point of 119879Taking 119909 = 119909
119899and 119910 = 119911 in (1) we have
[1 + 119901119889 (119909119899 119911)] 119889
2(119879119909119899 119879119911)
le 119901max 1
2[1198892(119909119899 119879119909119899) 119889 (119911 119879119911)
+119889 (119909119899 119879119909119899) 1198892(119911 119879119911)]
119889 (119909119899 119879119909119899) 119889 (119909119899 119879119911) 119889 (119911 119879119909
119899)
119889 (119909119899 119879119911) 119889 (119911 119879119909
119899) 119889 (119911 119879119911)
+ 119898 (119909119899 119911) minus 120593 (119898 (119909
119899 119911))
(32)
where
119898(119909119899 119911)
= max 1198892 (119909119899 119911)
119889 (119909119899 119879119909119899) 119889 (119911 119879119911) 119889 (119909
119899 119879119911) 119889 (119911 119879119909
119899)
1
2[119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899)
+119889 (119911 119879119909119899) 119889 (119911 119879119911)]
(33)
Using (31) and (3) we get
[1 + 119901119889 (119911 119911)] 1198892(119911 119879119911)
le 119901max 1
2[1198892(119911 119911) 119889 (119911 119879119911) + 119889 (119911 119911) 119889
2(119911 119879119911)]
119889 (119911 119911) 119889 (119911 119879119911) 119889 (119911 119911)
119889 (119911 119879119911) 119889 (119911 119911) 119889 (119911 119879119911)
+ 119898 (119911 119911) minus 120593 (119898 (119911 119911))
(34)
Hence 1198892(119911 119879119911) le 0 rArr 119879119911 = 119911Then 119879 has a fixed point in119883To prove the uniqueness of the fixed point we assume that
1199111and 1199112are two fixed points of 119879 Taking 119909 = 119911
1and 119910 = 119911
2
in (1) we easily get 119889(1199111 1199112) = 0 wich implies that 119911
1= 1199112
Therefore 119879 has a unique fixed point in119883
Corollary 4 Let 119879 be a self-map of a complete metric space119883satisfying the condition
1198892(119879119909 119879119910) le 119898 (119909 119910) minus 120593 (119898 (119909 119910)) (35)
Journal of Mathematics 5
where
119898(119909 119910)
= max 1198892 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)
119889 (119909 119879119910) 119889 (119910 119879119909)
1
2[119889 (119909 119879119909) 119889 (119909 119879119910) + 119889 (119910 119879119909) 119889 (119910 119879119910)]
(36)
for all 119909 119910 isin 119883 and 120593 [0infin) rarr [0infin) is a continuousfunction with 120593(119905) = 0 hArr 119905 = 0 and 120593(119905) gt 0 for each 119905 gt 0Then 119879 has a unique fixed point in119883
Proof 119901 = 0 in Theorem 3 we have the result
Now we give an example to support our result
Example 5 Let119883 = 0 1 2 and let 119889 be the usual metric on119883 Let 119879 119883 rarr 119883 be defined by 1198790 = 1198791 = 0 and 1198792 = 1And define 120593 [0infin) rarr [0infin) by 120593(119905) = 1199052 For any valueof 119901 gt 0 and 119909 119910 isin 119883 then it is easy to verify that inequality(1) holds HenceTheorem 3 holds well
Acknowledgment
Penumurthy Parvateesam Murthy is thankful to the Uni-versity Grants Commission New Delhi India for financialassistance throughMajor Reserch Project File no 42-322013(SR)
References
[1] Y I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications I Gohberg and Y Lybich Eds vol98 of Operator Theory Advances and Applications pp 7ndash22Birkhauser Basel Switzerland 1997
[2] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications A Theoryand Methods vol 47 no 4 pp 2683ndash2693 2001
[3] B S Choudhury and P N Dutta ldquoA unified fixed point resultin metric spaces involving a two variable functionrdquo Filomat no14 pp 43ndash48 2000
[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008
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2 Journal of Mathematics
119901 ge 0 is a real number is and 120593 [0infin) rarr [0infin) is acontinuous function with 120593(119905) = 0 hArr 119905 = 0 and 120593(119905) gt 0 foreach 119905 gt 0
2 Main Result
Lemma 2 Let 119879 be a self map of a metric space 119883 satisfying(1) For any sequence 119909
119899 in 119883 defined by 119909
119899+1= 119879119909119899 119899 ge 0
Then the sequence 119909119899 is Cauchy in119883
Proof Let 1199090
isin 119883 be an arbitrary point Constructing thesequence 119909
119899 follows
119909119899+1
= 119879119909119899 119899 isin 119873
0 (3)
If 119909119899= 119909119899+1
for some 119899 then trivially 119879 has a fixed point Weassume 119909
119899+1= 119909119899 for all 119899 isin 119873
0 We write 120572
119899= 119889(119909
119899 119909119899+1
)First we prove that 120572
119899 is a nonincreasing sequence and
converges to 0Case 1 If 119899 is even taking 119909 = 119909
2119899and 119910 = 119909
2119899+1in (1) we get
[1 + 119901 119889 (1199092119899 1199092119899+1
)] 1198892(1198791199092119899 1198791199092119899+1
)
le 119901max 1
2[1198892(1199092119899 1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)
+119889 (1199092119899 1198791199092119899) 1198892(1199092119899+1
1198791199092119899+1
)]
119889 (1199092119899 1198791199092119899) 119889 (1199092119899 1198791199092119899+1
)
times 119889 (1199092119899+1
1198791199092119899) 119889 (119909
2119899 1198791199092119899+1
)
times 119889 (1199092119899+1
1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)
+ 119898 (1199092119899 1199092119899+1
) minus 120593 (119898 (1199092119899 1199092119899+1
))
(4)
where
119898(1199092119899 1199092119899+1
)
= max 1198892 (1199092119899 1199092119899+1
) 119889 (1199092119899 1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)
119889 (1199092119899 1198791199092119899+1
) 119889 (1199092119899+1
1198791199092119899)
1
2[119889 (1199092119899 1198791199092119899) 119889 (1199092119899 1198791199092119899+1
)
+119889 (1199092119899+1
1198791199092119899) 119889 (1199092119899+1
1198791199092119899+1
)]
(5)
By using (3) we get
[1 + 119901 119889 (1199092119899 1199092119899+1
)] 1198892(1199092119899+1
1199092119899+2
)
le 119901max 1
2[1198892(1199092119899 1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)
+119889 (1199092119899 1199092119899+1
) 1198892(1199092119899+1
1199092119899+2
)]
119889 (1199092119899 1199092119899+1
) 119889 (1199092119899 1199092119899+2
)
times 119889 (1199092119899+1
1199092119899+1
) 119889 (1199092119899 1199092119899+2
)
times 119889 (1199092119899+1
1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)
+ 119898 (1199092119899 1199092119899+1
) minus 120593 (119898 (1199092119899 1199092119899+1
))
(6)
where
119898(1199092119899 1199092119899+1
)
= max 1198892 (1199092119899 1199092119899+1
) 119889 (1199092119899 1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)
119889 (1199092119899 1199092119899+2
) 119889 (1199092119899+1
1199092119899+1
)
1
2[119889 (1199092119899 1199092119899+1
) 119889 (1199092119899 1199092119899+2
)
+119889 (1199092119899+1
1199092119899+1
) 119889 (1199092119899+1
1199092119899+2
)]
(7)
Now consider 1205722119899
= 119889(1199092119899 1199092119899+1
) then we have
[1 + 1199011205722119899] 1205722
2119899+1
le 119901max 1
2[1205722
21198991205722119899+1
+ 12057221198991205722
2119899+1] 0 0
+ 119898 (1199092119899 1199092119899+1
) minus 120593 (119898 (1199092119899 1199092119899+1
))
(8)
where119898(1199092119899 1199092119899+1
) = max12057222119899 12057221198991205722119899+1
0 (12)[1205722119899119889(1199092119899
1199092119899+2
) + 0]By triangular inequality and using property of 120593 we get
119889 (1199092119899 1199092119899+2
) le 119889 (1199092119899 1199092119899+1
) + 119889 (1199092119899+1
1199092119899+2
)
= 1205722119899
+ 1205722119899+1
(9)
Then
119898(1199092119899 1199092119899+1
) le 1198981015840(119909 119910)
= max 12057222119899 1205722119899+1
1205722119899 0
1
21205722119899
(1205722119899
+ 1205722119899+1
) 0
(10)
If 1205722119899
lt 1205722119899+1
then (8) reduces to 1199011205722
2119899+1le 119901120572
2
2119899+1minus
120593(1205722
2119899+1) a contradiction Thus 120572
2
2119899+1le 1205722
2119899implies that
1205722119899+1
le 1205722119899
In a similar way if 119899 is odd we can obtain 1205722119899+2
lt 1205722119899+1
It follows that the sequence 120572
119899 is decreasing
Let lim119899rarrinfin
120572119899= 119903 for some 119903 ge 0
Journal of Mathematics 3
Suppose 119903 gt 0 then from inequality (1) we have
[1 + 119901 119889 (119909119899 119909119899+1
)] 1198892(119879119909119899 119879119909119899+1
)
le 119901max 1
2[1198892(119909119899 119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
+119889 (119909119899 119879119909119899) 1198892(119909119899+1
119879119909119899+1
)]
119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899+1
)
times 119889 (119909119899+1
119879119909119899) 119889 (119909
119899 119879119909119899+1
)
times119889 (119909119899+1
119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
+ 119898 (119909119899 119909119899+1
) minus 120593 (119898 (119909119899 119909119899+1
))
(11)
where119898(119909119899 119909119899+1
)
= max 1198892 (119909119899 119909119899+1
) 119889 (119909119899 119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
119889 (119909119899 119879119909119899+1
) 119889 (119909119899+1
119879119909119899)
1
2[119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899+1
)
+119889 (119909119899+1
119879119909119899) 119889 (119909119899+1
119879119909119899+1
)]
(12)
By using (3) we get
[1 + 119901119889 (119909119899 119909119899+1
)] 1198892(119909119899+1
119909119899+2
)
le 119901max 1
2[1198892(119909119899 119909119899+1
) 119889 (119909119899+1
119909119899+2
)
+119889 (119909119899 119909119899+1
) 1198892(119909119899+1
119909119899+2
)]
119889 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
)
119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
) 119889 (119909119899+1
119909119899+2
)
+ 119898 (119909119899 119909119899+1
) minus 120593 (119898 (119909119899 119909119899+1
))
(13)
where119898(119909119899 119909119899+1
)
= max 1198892 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+1
) 119889 (119909119899+1
119909119899+2
)
119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
)
1
2[119889 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+2
)
+119889 (119909119899+1
119909119899+1
) 119889 (119909119899+1
119909119899+2
)]
(14)
Using triangular inequality and property of 120593 and takinglimits 119899 rarr infin we get
[1 + 119901119903] 1199032le 1199011199033+ 1199032minus 120593 (119903
2) (15)
Then 120593(1199032) le 0 since 119903 is positive then by the property of 120593
we get 119903 = 0 We conclude that
lim119899rarrinfin
120572119899= lim119899rarrinfin
119889 (119909119899 119909119899+1
) = 119903 = 0 (16)
Now we show that 119909119899 is a Cauchy sequence Suppose
we assume that 119909119899 is not a Cauchy sequence then there is
120576 gt 0 for which we can find two sequences of positive integers119898(119896) and 119899(119896) such that for all positive integers 119896 119899(119896) gt
119898(119896) gt 119896
119889 (119909119898(119896)
119909119899(119896)
) ge 120598 119889 (119909119898(119896)
119909119899(119896)minus1
) lt 120598 (17)
Now
120598 le 119889 (119909119898(119896)
119909119899(119896)
) le 119889 (119909119898(119896)
119909119899(119896)minus1
) + 119889 (119909119899(119896)minus1
119909119899(119896)
)
(18)
Letting 119896 rarr infin we get
lim119896rarrinfin
119889 (119909119898(119896)
119909119899(119896)
) = 120598 (19)
Now from the triangular inequality we have1003816100381610038161003816119889 (119909119899(119896)
119909119898(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816 le 119889 (119909
119898(119896) 119909119898(119896)+1
)
(20)
Taking limits as 119896 rarr infin and using (16) and (19) we have
lim119896rarrinfin
119889 (119909119899(119896)
119909119898(119896)+1
) = 120598 (21)
Again from the triangular inequality we have1003816100381610038161003816119889 (119909119898(119896)
119909119899(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816 le 119889 (119909
119899(119896) 119909119899(119896)+1
)
(22)
Taking limits as 119896 rarr infin and using (16) and (19) we have
lim119896rarrinfin
119889 (119909119898(119896)
119909119899(119896)+1
) = 120598 (23)
Again by using triangular inequality we have1003816100381610038161003816119889 (119909119898(119896)+1
119909119899(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816
le 119889 (119909119898(119896)
119909119898(119896)+1
) + 119889 (119909119899(119896)
119909119899(119896)+1
)
(24)
Taking limit 119896 rarr infin in the above inequality and using (16)and (19) we have
lim119896rarrinfin
119889 (119909119899(119896)+1
119909119898(119896)+1
) = 120598 (25)
Again putting 119909 = 119909119898(119896)
and 119910 = 119909119899(119896)
in (1) we get
[1 + 119901119889 (119909119898(119896)
119909119899(119896)
)] 1198892(119879119909119898(119896)
119879119909119899(119896)
)
le 119901 1
2[1198892(119909119898(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
+119889 (119909119898(119896)
119879119909119898(119896)
) 1198892(119909119899(119896)
119879119909119899(119896)
)]
119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
)
119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
+ 119898 (119909119898(119896)
119909119899(119896)
) minus 120593 (119898 (119909119898(119896)
119909119899(119896)
))
(26)
4 Journal of Mathematics
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
)
1
2[119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119898(119896)
119879119909119899(119896)
)
+119889 (119909119899(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)]
(27)
Using (3) then we obtain
[1 + 119901119889 (119909119898(119896)
119909119899(119896)
)] 1198892(119909119898(119896)+1
119909119899(119896)+1
)
le 119901 1
2[1198892(119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+119889 (119909119898(119896)
119909119898(119896)+1
) 1198892(119909119899(119896)
119909119899(119896)+1
)]
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+ 119898 (119909119898(119896)
119909119899(119896)
) minus 120593 (119898 (119909119898(119896)
119909119899(119896)
))
(28)
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
119889 (119909119898(119896)
119909119899(119896)+1
) 119889 (119909119899(119896)
119909119898(119896)+1
)
1
2[119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
+119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)]
(29)
Letting 119896 rarr infin and using (16)ndash(25) we get
[1 + 119901120598] 1205982
le 119901max 1
2[0 + 0] 0 0 + 120598
2minus 120593 (120598
2)
= 1205982minus 120593 (120598
2)
(30)
a contradiction Thus 119909119899 is a Cauchy in119883
Theorem 3 Let 119879 be a self-map of a complete metric space 119883satisfying (1) Then 119879 has a unique fixed point in119883
Proof From Lemma 2 the sequence 119909119899 is a Cauchy in 119883
Since (119883 119889) is a complete metric space then there exists apoint 119911 isin 119883 such that
lim119899rarrinfin
119909119899= 119911 (31)
Now we prove that 119911 is a fixed point of 119879Taking 119909 = 119909
119899and 119910 = 119911 in (1) we have
[1 + 119901119889 (119909119899 119911)] 119889
2(119879119909119899 119879119911)
le 119901max 1
2[1198892(119909119899 119879119909119899) 119889 (119911 119879119911)
+119889 (119909119899 119879119909119899) 1198892(119911 119879119911)]
119889 (119909119899 119879119909119899) 119889 (119909119899 119879119911) 119889 (119911 119879119909
119899)
119889 (119909119899 119879119911) 119889 (119911 119879119909
119899) 119889 (119911 119879119911)
+ 119898 (119909119899 119911) minus 120593 (119898 (119909
119899 119911))
(32)
where
119898(119909119899 119911)
= max 1198892 (119909119899 119911)
119889 (119909119899 119879119909119899) 119889 (119911 119879119911) 119889 (119909
119899 119879119911) 119889 (119911 119879119909
119899)
1
2[119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899)
+119889 (119911 119879119909119899) 119889 (119911 119879119911)]
(33)
Using (31) and (3) we get
[1 + 119901119889 (119911 119911)] 1198892(119911 119879119911)
le 119901max 1
2[1198892(119911 119911) 119889 (119911 119879119911) + 119889 (119911 119911) 119889
2(119911 119879119911)]
119889 (119911 119911) 119889 (119911 119879119911) 119889 (119911 119911)
119889 (119911 119879119911) 119889 (119911 119911) 119889 (119911 119879119911)
+ 119898 (119911 119911) minus 120593 (119898 (119911 119911))
(34)
Hence 1198892(119911 119879119911) le 0 rArr 119879119911 = 119911Then 119879 has a fixed point in119883To prove the uniqueness of the fixed point we assume that
1199111and 1199112are two fixed points of 119879 Taking 119909 = 119911
1and 119910 = 119911
2
in (1) we easily get 119889(1199111 1199112) = 0 wich implies that 119911
1= 1199112
Therefore 119879 has a unique fixed point in119883
Corollary 4 Let 119879 be a self-map of a complete metric space119883satisfying the condition
1198892(119879119909 119879119910) le 119898 (119909 119910) minus 120593 (119898 (119909 119910)) (35)
Journal of Mathematics 5
where
119898(119909 119910)
= max 1198892 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)
119889 (119909 119879119910) 119889 (119910 119879119909)
1
2[119889 (119909 119879119909) 119889 (119909 119879119910) + 119889 (119910 119879119909) 119889 (119910 119879119910)]
(36)
for all 119909 119910 isin 119883 and 120593 [0infin) rarr [0infin) is a continuousfunction with 120593(119905) = 0 hArr 119905 = 0 and 120593(119905) gt 0 for each 119905 gt 0Then 119879 has a unique fixed point in119883
Proof 119901 = 0 in Theorem 3 we have the result
Now we give an example to support our result
Example 5 Let119883 = 0 1 2 and let 119889 be the usual metric on119883 Let 119879 119883 rarr 119883 be defined by 1198790 = 1198791 = 0 and 1198792 = 1And define 120593 [0infin) rarr [0infin) by 120593(119905) = 1199052 For any valueof 119901 gt 0 and 119909 119910 isin 119883 then it is easy to verify that inequality(1) holds HenceTheorem 3 holds well
Acknowledgment
Penumurthy Parvateesam Murthy is thankful to the Uni-versity Grants Commission New Delhi India for financialassistance throughMajor Reserch Project File no 42-322013(SR)
References
[1] Y I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications I Gohberg and Y Lybich Eds vol98 of Operator Theory Advances and Applications pp 7ndash22Birkhauser Basel Switzerland 1997
[2] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications A Theoryand Methods vol 47 no 4 pp 2683ndash2693 2001
[3] B S Choudhury and P N Dutta ldquoA unified fixed point resultin metric spaces involving a two variable functionrdquo Filomat no14 pp 43ndash48 2000
[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 3
Suppose 119903 gt 0 then from inequality (1) we have
[1 + 119901 119889 (119909119899 119909119899+1
)] 1198892(119879119909119899 119879119909119899+1
)
le 119901max 1
2[1198892(119909119899 119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
+119889 (119909119899 119879119909119899) 1198892(119909119899+1
119879119909119899+1
)]
119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899+1
)
times 119889 (119909119899+1
119879119909119899) 119889 (119909
119899 119879119909119899+1
)
times119889 (119909119899+1
119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
+ 119898 (119909119899 119909119899+1
) minus 120593 (119898 (119909119899 119909119899+1
))
(11)
where119898(119909119899 119909119899+1
)
= max 1198892 (119909119899 119909119899+1
) 119889 (119909119899 119879119909119899) 119889 (119909119899+1
119879119909119899+1
)
119889 (119909119899 119879119909119899+1
) 119889 (119909119899+1
119879119909119899)
1
2[119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899+1
)
+119889 (119909119899+1
119879119909119899) 119889 (119909119899+1
119879119909119899+1
)]
(12)
By using (3) we get
[1 + 119901119889 (119909119899 119909119899+1
)] 1198892(119909119899+1
119909119899+2
)
le 119901max 1
2[1198892(119909119899 119909119899+1
) 119889 (119909119899+1
119909119899+2
)
+119889 (119909119899 119909119899+1
) 1198892(119909119899+1
119909119899+2
)]
119889 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
)
119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
) 119889 (119909119899+1
119909119899+2
)
+ 119898 (119909119899 119909119899+1
) minus 120593 (119898 (119909119899 119909119899+1
))
(13)
where119898(119909119899 119909119899+1
)
= max 1198892 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+1
) 119889 (119909119899+1
119909119899+2
)
119889 (119909119899 119909119899+2
) 119889 (119909119899+1
119909119899+1
)
1
2[119889 (119909119899 119909119899+1
) 119889 (119909119899 119909119899+2
)
+119889 (119909119899+1
119909119899+1
) 119889 (119909119899+1
119909119899+2
)]
(14)
Using triangular inequality and property of 120593 and takinglimits 119899 rarr infin we get
[1 + 119901119903] 1199032le 1199011199033+ 1199032minus 120593 (119903
2) (15)
Then 120593(1199032) le 0 since 119903 is positive then by the property of 120593
we get 119903 = 0 We conclude that
lim119899rarrinfin
120572119899= lim119899rarrinfin
119889 (119909119899 119909119899+1
) = 119903 = 0 (16)
Now we show that 119909119899 is a Cauchy sequence Suppose
we assume that 119909119899 is not a Cauchy sequence then there is
120576 gt 0 for which we can find two sequences of positive integers119898(119896) and 119899(119896) such that for all positive integers 119896 119899(119896) gt
119898(119896) gt 119896
119889 (119909119898(119896)
119909119899(119896)
) ge 120598 119889 (119909119898(119896)
119909119899(119896)minus1
) lt 120598 (17)
Now
120598 le 119889 (119909119898(119896)
119909119899(119896)
) le 119889 (119909119898(119896)
119909119899(119896)minus1
) + 119889 (119909119899(119896)minus1
119909119899(119896)
)
(18)
Letting 119896 rarr infin we get
lim119896rarrinfin
119889 (119909119898(119896)
119909119899(119896)
) = 120598 (19)
Now from the triangular inequality we have1003816100381610038161003816119889 (119909119899(119896)
119909119898(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816 le 119889 (119909
119898(119896) 119909119898(119896)+1
)
(20)
Taking limits as 119896 rarr infin and using (16) and (19) we have
lim119896rarrinfin
119889 (119909119899(119896)
119909119898(119896)+1
) = 120598 (21)
Again from the triangular inequality we have1003816100381610038161003816119889 (119909119898(119896)
119909119899(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816 le 119889 (119909
119899(119896) 119909119899(119896)+1
)
(22)
Taking limits as 119896 rarr infin and using (16) and (19) we have
lim119896rarrinfin
119889 (119909119898(119896)
119909119899(119896)+1
) = 120598 (23)
Again by using triangular inequality we have1003816100381610038161003816119889 (119909119898(119896)+1
119909119899(119896)+1
) minus 119889 (119909119898(119896)
119909119899(119896)
)1003816100381610038161003816
le 119889 (119909119898(119896)
119909119898(119896)+1
) + 119889 (119909119899(119896)
119909119899(119896)+1
)
(24)
Taking limit 119896 rarr infin in the above inequality and using (16)and (19) we have
lim119896rarrinfin
119889 (119909119899(119896)+1
119909119898(119896)+1
) = 120598 (25)
Again putting 119909 = 119909119898(119896)
and 119910 = 119909119899(119896)
in (1) we get
[1 + 119901119889 (119909119898(119896)
119909119899(119896)
)] 1198892(119879119909119898(119896)
119879119909119899(119896)
)
le 119901 1
2[1198892(119909119898(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
+119889 (119909119898(119896)
119879119909119898(119896)
) 1198892(119909119899(119896)
119879119909119899(119896)
)]
119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
)
119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
+ 119898 (119909119898(119896)
119909119899(119896)
) minus 120593 (119898 (119909119898(119896)
119909119899(119896)
))
(26)
4 Journal of Mathematics
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
)
1
2[119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119898(119896)
119879119909119899(119896)
)
+119889 (119909119899(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)]
(27)
Using (3) then we obtain
[1 + 119901119889 (119909119898(119896)
119909119899(119896)
)] 1198892(119909119898(119896)+1
119909119899(119896)+1
)
le 119901 1
2[1198892(119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+119889 (119909119898(119896)
119909119898(119896)+1
) 1198892(119909119899(119896)
119909119899(119896)+1
)]
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+ 119898 (119909119898(119896)
119909119899(119896)
) minus 120593 (119898 (119909119898(119896)
119909119899(119896)
))
(28)
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
119889 (119909119898(119896)
119909119899(119896)+1
) 119889 (119909119899(119896)
119909119898(119896)+1
)
1
2[119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
+119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)]
(29)
Letting 119896 rarr infin and using (16)ndash(25) we get
[1 + 119901120598] 1205982
le 119901max 1
2[0 + 0] 0 0 + 120598
2minus 120593 (120598
2)
= 1205982minus 120593 (120598
2)
(30)
a contradiction Thus 119909119899 is a Cauchy in119883
Theorem 3 Let 119879 be a self-map of a complete metric space 119883satisfying (1) Then 119879 has a unique fixed point in119883
Proof From Lemma 2 the sequence 119909119899 is a Cauchy in 119883
Since (119883 119889) is a complete metric space then there exists apoint 119911 isin 119883 such that
lim119899rarrinfin
119909119899= 119911 (31)
Now we prove that 119911 is a fixed point of 119879Taking 119909 = 119909
119899and 119910 = 119911 in (1) we have
[1 + 119901119889 (119909119899 119911)] 119889
2(119879119909119899 119879119911)
le 119901max 1
2[1198892(119909119899 119879119909119899) 119889 (119911 119879119911)
+119889 (119909119899 119879119909119899) 1198892(119911 119879119911)]
119889 (119909119899 119879119909119899) 119889 (119909119899 119879119911) 119889 (119911 119879119909
119899)
119889 (119909119899 119879119911) 119889 (119911 119879119909
119899) 119889 (119911 119879119911)
+ 119898 (119909119899 119911) minus 120593 (119898 (119909
119899 119911))
(32)
where
119898(119909119899 119911)
= max 1198892 (119909119899 119911)
119889 (119909119899 119879119909119899) 119889 (119911 119879119911) 119889 (119909
119899 119879119911) 119889 (119911 119879119909
119899)
1
2[119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899)
+119889 (119911 119879119909119899) 119889 (119911 119879119911)]
(33)
Using (31) and (3) we get
[1 + 119901119889 (119911 119911)] 1198892(119911 119879119911)
le 119901max 1
2[1198892(119911 119911) 119889 (119911 119879119911) + 119889 (119911 119911) 119889
2(119911 119879119911)]
119889 (119911 119911) 119889 (119911 119879119911) 119889 (119911 119911)
119889 (119911 119879119911) 119889 (119911 119911) 119889 (119911 119879119911)
+ 119898 (119911 119911) minus 120593 (119898 (119911 119911))
(34)
Hence 1198892(119911 119879119911) le 0 rArr 119879119911 = 119911Then 119879 has a fixed point in119883To prove the uniqueness of the fixed point we assume that
1199111and 1199112are two fixed points of 119879 Taking 119909 = 119911
1and 119910 = 119911
2
in (1) we easily get 119889(1199111 1199112) = 0 wich implies that 119911
1= 1199112
Therefore 119879 has a unique fixed point in119883
Corollary 4 Let 119879 be a self-map of a complete metric space119883satisfying the condition
1198892(119879119909 119879119910) le 119898 (119909 119910) minus 120593 (119898 (119909 119910)) (35)
Journal of Mathematics 5
where
119898(119909 119910)
= max 1198892 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)
119889 (119909 119879119910) 119889 (119910 119879119909)
1
2[119889 (119909 119879119909) 119889 (119909 119879119910) + 119889 (119910 119879119909) 119889 (119910 119879119910)]
(36)
for all 119909 119910 isin 119883 and 120593 [0infin) rarr [0infin) is a continuousfunction with 120593(119905) = 0 hArr 119905 = 0 and 120593(119905) gt 0 for each 119905 gt 0Then 119879 has a unique fixed point in119883
Proof 119901 = 0 in Theorem 3 we have the result
Now we give an example to support our result
Example 5 Let119883 = 0 1 2 and let 119889 be the usual metric on119883 Let 119879 119883 rarr 119883 be defined by 1198790 = 1198791 = 0 and 1198792 = 1And define 120593 [0infin) rarr [0infin) by 120593(119905) = 1199052 For any valueof 119901 gt 0 and 119909 119910 isin 119883 then it is easy to verify that inequality(1) holds HenceTheorem 3 holds well
Acknowledgment
Penumurthy Parvateesam Murthy is thankful to the Uni-versity Grants Commission New Delhi India for financialassistance throughMajor Reserch Project File no 42-322013(SR)
References
[1] Y I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications I Gohberg and Y Lybich Eds vol98 of Operator Theory Advances and Applications pp 7ndash22Birkhauser Basel Switzerland 1997
[2] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications A Theoryand Methods vol 47 no 4 pp 2683ndash2693 2001
[3] B S Choudhury and P N Dutta ldquoA unified fixed point resultin metric spaces involving a two variable functionrdquo Filomat no14 pp 43ndash48 2000
[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Mathematics
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)
119889 (119909119898(119896)
119879119909119899(119896)
) 119889 (119909119899(119896)
119879119909119898(119896)
)
1
2[119889 (119909119898(119896)
119879119909119898(119896)
) 119889 (119909119898(119896)
119879119909119899(119896)
)
+119889 (119909119899(119896)
119879119909119898(119896)
) 119889 (119909119899(119896)
119879119909119899(119896)
)]
(27)
Using (3) then we obtain
[1 + 119901119889 (119909119898(119896)
119909119899(119896)
)] 1198892(119909119898(119896)+1
119909119899(119896)+1
)
le 119901 1
2[1198892(119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+119889 (119909119898(119896)
119909119898(119896)+1
) 1198892(119909119899(119896)
119909119899(119896)+1
)]
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
times 119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
+ 119898 (119909119898(119896)
119909119899(119896)
) minus 120593 (119898 (119909119898(119896)
119909119899(119896)
))
(28)
where
119898(119909119898(119896)
119909119899(119896)
)
= max 1198892 (119909119898(119896)
119909119899(119896)
)
119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)
119889 (119909119898(119896)
119909119899(119896)+1
) 119889 (119909119899(119896)
119909119898(119896)+1
)
1
2[119889 (119909119898(119896)
119909119898(119896)+1
) 119889 (119909119898(119896)
119909119899(119896)+1
)
+119889 (119909119899(119896)
119909119898(119896)+1
) 119889 (119909119899(119896)
119909119899(119896)+1
)]
(29)
Letting 119896 rarr infin and using (16)ndash(25) we get
[1 + 119901120598] 1205982
le 119901max 1
2[0 + 0] 0 0 + 120598
2minus 120593 (120598
2)
= 1205982minus 120593 (120598
2)
(30)
a contradiction Thus 119909119899 is a Cauchy in119883
Theorem 3 Let 119879 be a self-map of a complete metric space 119883satisfying (1) Then 119879 has a unique fixed point in119883
Proof From Lemma 2 the sequence 119909119899 is a Cauchy in 119883
Since (119883 119889) is a complete metric space then there exists apoint 119911 isin 119883 such that
lim119899rarrinfin
119909119899= 119911 (31)
Now we prove that 119911 is a fixed point of 119879Taking 119909 = 119909
119899and 119910 = 119911 in (1) we have
[1 + 119901119889 (119909119899 119911)] 119889
2(119879119909119899 119879119911)
le 119901max 1
2[1198892(119909119899 119879119909119899) 119889 (119911 119879119911)
+119889 (119909119899 119879119909119899) 1198892(119911 119879119911)]
119889 (119909119899 119879119909119899) 119889 (119909119899 119879119911) 119889 (119911 119879119909
119899)
119889 (119909119899 119879119911) 119889 (119911 119879119909
119899) 119889 (119911 119879119911)
+ 119898 (119909119899 119911) minus 120593 (119898 (119909
119899 119911))
(32)
where
119898(119909119899 119911)
= max 1198892 (119909119899 119911)
119889 (119909119899 119879119909119899) 119889 (119911 119879119911) 119889 (119909
119899 119879119911) 119889 (119911 119879119909
119899)
1
2[119889 (119909119899 119879119909119899) 119889 (119909119899 119879119909119899)
+119889 (119911 119879119909119899) 119889 (119911 119879119911)]
(33)
Using (31) and (3) we get
[1 + 119901119889 (119911 119911)] 1198892(119911 119879119911)
le 119901max 1
2[1198892(119911 119911) 119889 (119911 119879119911) + 119889 (119911 119911) 119889
2(119911 119879119911)]
119889 (119911 119911) 119889 (119911 119879119911) 119889 (119911 119911)
119889 (119911 119879119911) 119889 (119911 119911) 119889 (119911 119879119911)
+ 119898 (119911 119911) minus 120593 (119898 (119911 119911))
(34)
Hence 1198892(119911 119879119911) le 0 rArr 119879119911 = 119911Then 119879 has a fixed point in119883To prove the uniqueness of the fixed point we assume that
1199111and 1199112are two fixed points of 119879 Taking 119909 = 119911
1and 119910 = 119911
2
in (1) we easily get 119889(1199111 1199112) = 0 wich implies that 119911
1= 1199112
Therefore 119879 has a unique fixed point in119883
Corollary 4 Let 119879 be a self-map of a complete metric space119883satisfying the condition
1198892(119879119909 119879119910) le 119898 (119909 119910) minus 120593 (119898 (119909 119910)) (35)
Journal of Mathematics 5
where
119898(119909 119910)
= max 1198892 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)
119889 (119909 119879119910) 119889 (119910 119879119909)
1
2[119889 (119909 119879119909) 119889 (119909 119879119910) + 119889 (119910 119879119909) 119889 (119910 119879119910)]
(36)
for all 119909 119910 isin 119883 and 120593 [0infin) rarr [0infin) is a continuousfunction with 120593(119905) = 0 hArr 119905 = 0 and 120593(119905) gt 0 for each 119905 gt 0Then 119879 has a unique fixed point in119883
Proof 119901 = 0 in Theorem 3 we have the result
Now we give an example to support our result
Example 5 Let119883 = 0 1 2 and let 119889 be the usual metric on119883 Let 119879 119883 rarr 119883 be defined by 1198790 = 1198791 = 0 and 1198792 = 1And define 120593 [0infin) rarr [0infin) by 120593(119905) = 1199052 For any valueof 119901 gt 0 and 119909 119910 isin 119883 then it is easy to verify that inequality(1) holds HenceTheorem 3 holds well
Acknowledgment
Penumurthy Parvateesam Murthy is thankful to the Uni-versity Grants Commission New Delhi India for financialassistance throughMajor Reserch Project File no 42-322013(SR)
References
[1] Y I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications I Gohberg and Y Lybich Eds vol98 of Operator Theory Advances and Applications pp 7ndash22Birkhauser Basel Switzerland 1997
[2] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications A Theoryand Methods vol 47 no 4 pp 2683ndash2693 2001
[3] B S Choudhury and P N Dutta ldquoA unified fixed point resultin metric spaces involving a two variable functionrdquo Filomat no14 pp 43ndash48 2000
[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 5
where
119898(119909 119910)
= max 1198892 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)
119889 (119909 119879119910) 119889 (119910 119879119909)
1
2[119889 (119909 119879119909) 119889 (119909 119879119910) + 119889 (119910 119879119909) 119889 (119910 119879119910)]
(36)
for all 119909 119910 isin 119883 and 120593 [0infin) rarr [0infin) is a continuousfunction with 120593(119905) = 0 hArr 119905 = 0 and 120593(119905) gt 0 for each 119905 gt 0Then 119879 has a unique fixed point in119883
Proof 119901 = 0 in Theorem 3 we have the result
Now we give an example to support our result
Example 5 Let119883 = 0 1 2 and let 119889 be the usual metric on119883 Let 119879 119883 rarr 119883 be defined by 1198790 = 1198791 = 0 and 1198792 = 1And define 120593 [0infin) rarr [0infin) by 120593(119905) = 1199052 For any valueof 119901 gt 0 and 119909 119910 isin 119883 then it is easy to verify that inequality(1) holds HenceTheorem 3 holds well
Acknowledgment
Penumurthy Parvateesam Murthy is thankful to the Uni-versity Grants Commission New Delhi India for financialassistance throughMajor Reserch Project File no 42-322013(SR)
References
[1] Y I Alber and S Guerre-Delabriere ldquoPrinciple of weaklycontractive maps in Hilbert spacesrdquo in New Results in OperatorTheory and its Applications I Gohberg and Y Lybich Eds vol98 of Operator Theory Advances and Applications pp 7ndash22Birkhauser Basel Switzerland 1997
[2] B E Rhoades ldquoSome theorems on weakly contractive mapsrdquoNonlinear Analysis Theory Methods amp Applications A Theoryand Methods vol 47 no 4 pp 2683ndash2693 2001
[3] B S Choudhury and P N Dutta ldquoA unified fixed point resultin metric spaces involving a two variable functionrdquo Filomat no14 pp 43ndash48 2000
[4] P N Dutta and B S Choudhury ldquoA generalisation of con-traction principle in metric spacesrdquo Fixed Point Theory andApplications vol 2008 Article ID 406368 8 pages 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of