Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 846315 6 pageshttpdxdoiorg1011552013846315

Research ArticleCommon Fixed Points for JH-Operators andOccasionally Weakly g-Biased Pairs under Relaxed Condition onProbabilistic Metric Space

Arvind Bhatt1 and Harish Chandra2

1 Department of Applied Science (Mathematics) Bipin Tripathi Kumaun Institute of Technology Dwarahat (Almora)Uttarakhand Technical University Dehradun Uttarakhand 263653 India

2Department of Mathematics and DST-CIMS Banaras Hindu University Varanasi 221005 India

Correspondence should be addressed to Arvind Bhatt arvindbhu 6junerediffmailcom

Received 14 May 2013 Revised 2 July 2013 Accepted 3 July 2013

Academic Editor C Cuevas

Copyright copy 2013 A Bhatt and H Chandra This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We obtain some fixed point theorems for JH-operators and occasionally weakly g-biasedmaps on a setX together with the function119865 119883 times119883 rarr Δ without using the triangle inequality and without using the symmetric condition Our results extend the results ofBhatt et al (2010)

1 Introduction

Fixed point theory in probabilistic metric spaces can beconsidered as a part of probabilistic analysis which is a oneof the emerging areas of interdisciplinary mathematicalresearch with many diverse applications The theory ofprobabilistic metric spaces was introduced by Menger [1] inconnection with some measurements in Physics Over theyears the theory has found several important applications inthe investigation of physical quantities in quantum particlephysics and string theory as studied by El Naschie [2 3]The area of probabilistic metric spaces is also of fundamentalimportance in probabilistic functional analysis The firsteffort in this direction was made by Sehgal [4] who inhis doctoral dissertation initiated the study of contractionmapping theorems in probabilistic metric spaces Since thenSehgal and Bharucha-Reid [5] obtained a generalization ofBanach Contraction Principle on a complete Menger spacewhich is an important step in the development of fixed pointtheorems in Mengar space

Sessa [6] initiated the tradition of improving commuta-tivity in fixed point theorems by introducing the notion ofweakly commuting maps in metric spaces Jungck [7] soonenlarged this concept to compatible maps The notion ofcompatible mappings in aMengar space has been introduced

by Mishra [8] After this Jungck [9] gave the concept ofweakly compatible maps Aamri and El Moutawakil [10]introduced the (EA) property and thus generalized the con-cept of noncompatible mapsThe results obtained in themet-ric fixed point theory by using the notion of non-compatiblemaps or the (EA) property are very interesting Al-Thagafiand Shahzad [11] defined the concept of occasionally weaklycompatible mappings which is more general than the conceptof weakly compatible maps Bhatt et al [12] have givenapplication of occasionally weakly compatible mappings indynamical system Pathak and Hussain [13] defined the con-cept P-operators Hussain et al [14] gave the concept of JH-operators and occasionally weakly g-biased

In this paper we obtain some fixed point theorems for JH-operators and occasionally weakly biased pairs under relaxedcondition on 119865 Our results extend the results of Bhatt et al[12]

We begin with the following basic definitions of conceptsrelating to probabilistic metric spaces for ready reference andalso for the sake of completeness

Definition 1 (see [15]) A real valued function 119891 on the setof real numbers is called a distribution function if it isnondecreasing left continuous with inf

119905isin119877119891(119905) = 0 and

sup119905isin119877

119891(119905) = 1

2 Journal of Function Spaces and Applications

The Heaviside function 119867 is a distribution functiondefined by

119867(119905) = 0 if 119905 le 0

1 if 119905 gt 0(1)

Definition 2 Let 119883 be a nonempty set and let Δ denote theset of all distribution functions defined on119883 119865 is a mappingfrom119883 times 119883 into Δ satisfying the following condition

119865119909119910

(119905) = 1 lArrrArr 119909 = 119910 (2)where 119865 119883times119883 rarr Δ defined by 119865

119909119910(119905) = 119867(119905 minus119889(119909 119910)) for

all 119909 119910 isin 119883 and 119889 is a function 119889 119883times119883 rarr [0infin) such that119889(119909 119910) = 0 if and only if 119909 = 119910 forall119909 119910 isin 119883 (symmetric andtriangle conditions are not required) A topology 120591(119889) on 119883

is given by119880 isin 120591(119889) if and only if for each 119909 isin 119880 119861(119909 120598) sub 119880

for some 120598 gt 0 where 119861(119909 120598) = 119910 isin 119883 119889(119909 119910) lt 120598

Definition 3 (see [16 17]) Let119883 be a non-empty set A point119909 in 119883 is called a coincidence point of 119891 and 119892 if and onlyif 119891119909 = 119892119909 In this case 119908 = 119891119909 = 119892119909 is called a point ofcoincidence of 119891 and 119892

Let 119862(119891 119892) and 119875119862(119891 119892) denote the sets of coincidencepoints and points of coincidence respectively of the pair(119891 119892) For a space (119883 119865) satisfying (2) and 119860 sube 119883 thediameter of 119860 is defined by

120575 (119860) = sup min 119865119909119910

(119905) 119865119910119909

(119905) 119909 119910 isin 119860 (3)

Here we extend the concept of JH-operators and occa-sionally weakly g-biased pairs and the space (119883 119865) satisfyingcondition (2)

Definition 4 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Two self-maps 119891 and 119892 of a space (119883 119865) are called JH-operators if andonly if there is a point 119908 = 119891119909 = 119892119909 in 119875119862(119891 119892) such that

119865119908119909

(119905) ge 120575 (119875119862 (119891 119892)) 119865119909119908

(119905) ge 120575 (119875119862 (119891 119892))

(4)Definition 5 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Two self-maps 119891 and 119892 of a space (119883 119865) are called weakly g-biased ifand only 119865

119892119891119909119892119909(119905) ge 119865

119891119892119909119891119909(119905) whenever 119891119909 = 119892119909

Definition 6 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Two self-maps 119891 and 119892 of a space (119883 119865) are called occasionally weaklyg-biased if and only if there exists some 119909 isin 119883 such that119891119909 = 119892119909 and 119865

119892119891119909119892119909(119905) ge 119865

119891119892119909119891119909(119905)

Example 7 Let 119883 = [0 +infin) and 119865119909119910

(119905) = 119867(119905 minus 119889(119909 119910))where

119889 (119909 119910) = 119890119909minus119910

minus 1 if 119909 ge 119910

119890119910minus119909

otherwise(5)

Define 119891 119892 119883 rarr 119883 by

119891 (119909) = 2119909 119892 (119909) = 21199092 if 119909 = 0

119891 (119909) = 119892 (119909) =1

2 if 119909 = 0

(6)

In this example 119862(119891 119892) = 0 1 and 119875119862(119891 119892) = 12 2Here 119865

119892119891(1)119892(1)(119905) = 119867(119905 minus 119889(119892119891(1) 119892(1))) = 119867(119905 minus 119889(119892(2)

2)) = 119867(119905minus119889(8 2)) = 119867(119905minus1198906+1) For 119905 = 119890

6minus1119867(119905minus119890

6+1) =

119867(0) = 0 Similarly 119865119891119892(1)119891(1)

(119905) = 119867(119905 minus 119889(119891119892(1) 119891(1))) =

119867(119905minus119889(119891(2) 2)) = 119867(119905minus119889(4 2)) = 119867(119905minus1198902+1) For 119905 = 119890

6minus1

119867(119905 minus 1198902+ 1) = 119867(119890

6minus 1minus 119890

2+ 1) = 119867(119890

6minus 1198902) = 1 Therefore

119865119892119891(1)119892(1)

(119905) le 119865119891119892(1)119891(1)

(119905) Now we can easily show that119865119892119891(0)119892(0)

(119905) ge 119865119891119892(0)119891(0)

(119905)Therefore an occasionally weaklycompatible and a nontrivial weakly g-biased pair (119891 119892) areoccasionally weakly g-biased pairs but the converse does nothold

2 Section II

We note that every symmetric (semimetric) space (119883 119889) [18]can be realized as a probabilistic semi-metric space by taking119865 119883 times 119883 rarr Δ defined by 119865

119909119910(119905) = 119867(119905 minus 119889(119909 119910)) for all 119909

119910 in 119883 So probabilistic semi-metric spaces provide a widerframework than that of the symmetric spaces and are bettersuited in many situations In this paper we have relaxed thesymmetric condition from probabilistic semimetric spaceIn this section we prove some fixed point theorems for apair of JH-operator on space (119883 119865) without imposing therestriction of the triangle inequality or symmetry on119865 In thissection we also prove some fixed point theorems for a pair ofOccasionally weakly biased on space (119883 119865)without imposingthe restriction of the triangle inequality and symmetry onlyon point of coincidence and image of point of coincidence

Theorem 8 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Suppose 119891and 119892 are JH-operators on119883 satisfying the following condition

119865119891119909119891119910

(119905)

ge 119865119892119909119892119910

(119905

119886) +min 119865

119891119909119892119909(119905

119887) 119865119891119910119892119910

(119905

119887)

+min 119865119892119909119892119910

(119905

119888) 119865119892119909119891119909

(119905

119888) 119865119892119910119891119910

(119905

119888)

(7)

for all 119909 119910 isin 119883 with 119891(119909) = 119891119910 and 119905 gt 0 where 0 lt 119886 lt 10 lt 119887 lt 1 and 0 lt 119888 lt 1Then119891 and119892 have a unique commonfixed point

Proof We claim that 119891 and 119892 have a unique point ofcoincidence 119908 = 119891119909 = 119892119909 If possible suppose there isanother point of coincidence119891119910 = 119892119910 = 119908

1and119908

1= 119908Then

119865119891119909119891119910

(119905) = 1 forall119905 gt 0 So from (7) we get

119865119891119909119891119910

(119905)

ge 119865119891119909119891119910

(119905

119886) + 1

+min 119865119891119909119891119910

(119905

119888) 119865119891119909119891119909

(119905

119888) 119865119891119910119891119910

(119905

119888)

= 119865119891119909119891119910

(119905

119886) + 1 + 119865

119891119909119891119910(119905

119888) gt 1

(8)

Journal of Function Spaces and Applications 3

This is a contradiction which implies that 119865119891119909119891119910

(119905) = 1Hence we get 119891119909 = 119891119910 = 119908 = 119911 Therefore there existsa unique element 119908 in 119883 such that 119908 = 119891119909 = 119892119909 Thus120575(119875119862(119891 119892)) = 1 implies that119865

119909119908= 1 and hence 119909 is a unique

common fixed point of 119891 and 119892

Let a function 120601 be defined by 120601 [0 1] rarr [0 1] sati-sfying condition 120601(119902) gt 119902 for all 0 le 119902 lt 1

Theorem 9 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying the condition (2) If 119891and 119892 are occasionally weakly g-biased on 119883 suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (9)

for some point of coincidence 119908 of 119891 119892 and

119865119891119909119891119910

(119905)

ge 120601 [min 119865119892119909119892119910

(119905) 119865119892119909119891119910

(119905) 119865119892119910119891119909

(119905) 119865119892119910119891119910

(119905)]

(10)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

Proof Since 119891 and 119892 are occasionally weakly biased thereexists some 119906 isin 119883 such that 119891119906 = 119892119906 = 119908 and 119865

119892119891119906119892119906(119905) ge

119865119891119892119906119891119906

(119905)We claim that119891119906 is the unique commonfixed pointof 119891 and 119892 For if 119891119891119906 = 119891119906 then from (9) and (10) we get

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119892119891119906119892119906

(119905) 119865119892119891119906119891119906

(119905) 119865119892119906119891119891119906

(119905) 119865119892119906119891119906

(119905)]

= 120601 [min 119865119892119891119906119892119906

(119905) 119865119892119891119906119892119906

(119905) 119865119891119906119891119891119906

(119905) 119865119891119906119891119906

(119905)]

times (since 119891119906 = 119892119906)

(11)

Because 119891 and 119892 are occasionally weakly biased hence

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119891119892119906119891119906

(119905) 119865119891119892119906119891119906

(119905) 119865119891119906119891119891119906

(119905) 1]

= 120601 [min 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 119865119891119906119891119891119906

(119905) 1]

times (since 119891119906 = 119892119906)

(12)

by using condition (9)

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 1]

= 120601 [min 119865119891119891119906119891119906

(119905)]

(13)

Since 120601 [0 1] rarr [0 1] satisfying the condition 120601(119902) gt 119902for all 0 le 119902 lt 1 Therefore 119865

119891119891119906119891119906(119905) gt 119865

119891119891119906119891119906(119905) which is

a contradiction Therefore 119891119891119906 = 119891119906 = 119891119892119906 Hence 119865119892119891119906119892119906

(119905) ge 119865119891119892119906119891119906

= 1 which further implies that 119892119891119906 = 119892119906 =

119891119906 = 119891119891119906 Thus 119891119906 is a common fixed point of 119891 and 119892For uniqueness suppose that 119906 V isin 119883 such that 119891119906 =

119892119906 = 119906 and 119891V = 119892V = V and 119906 = V Then (10) gives

119865119906V (119905) = 119865

119891119906119891V (119905)

ge 120601 [min 119865119892119906119892V (119905) 119865119892119906119891V (119905) 119865119892V119891119906 (119905) 119865119892V119891V (119905)]

= 120601 [min 119865119906V (119905) 119865V119906 (119905) 1]

= 120601 [min 119865119906V (119905) 119865V119906 (119905)]

(14)

Let 120572 = min119865119906V(119905) 119865V119906(119905) gt 0 Then we get 119865

119906V(119905) ge

120601(120572) gt 120572 Similarly we get 119865V119906(119905) ge 120601(120572) gt 120572 Somin119865

119906V(119905) 119865V119906(119905) ge 120601(120572) gt 120572 This is a contradictionTherefore 119906 = V Therefore the common fixed point of 119891and 119892 is unique

Example 10 Letting119883 = [0 1] and 120601 119883 rarr 119883 defined as

120601 (119902) =1 + 119902

2 then 120601 (119902) gt 119902 0 le 119902 lt 1 (15)

and 119865119909119910

(119905) = 119867(119905 minus 119889(119909 119910)) where

119889 (119909 119910) = 119890119909minus119910

minus 1 if 119909 ge 119910

119890119910minus119909

otherwise(16)

Define119891 119892 119883 rarr 119883 by119891(119909) = 2119909 and 119892(119909) = 21199092 if 119909 = 0 1

One has 119891(119909) = 119892(119909) = 12 if 119909 = 0 and 119891(119909) = 119892(119909) = 1

if 119909 = 1In this example we observe that 119862(119891 119892) = 0 1

where (119891 119892) are occasionally weakly g-biased pairs and11986511198911

(119905) = 11986511989111

(119905) Now for 119905 = 11989012 120601[min119865

119892(12)119892(1)(119905)

119865119892(12)119891(1)

(119905) 119865119892(1)119891(12)

(119905) 119865119892(1)119891(1)

(119905)] lt 119865119891(12)119891(1)

(119905)Example 10 is the unique common fixed point of 119891 and 119892

Corollary 11 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) If 119891 and119892 are occasionally weakly g-biased on 119883 suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (17)

whenever 119908 is point of coincidence of 119891 119892 and

119865119891119909119891119910

(119905) ge 120601 [119865119892119909119892119910

(119905)] (18)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

Theproof of the following theorem can be easily obtainedby replacing condition (10) by condition (20) the proof ofTheorem 9

Theorem 12 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) If 119891 and119892 are occasionally weakly g-biased on 119883 Suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (19)

4 Journal of Function Spaces and Applications

whenever 119908 is point of coincidence of 119891 119892 and

119865119891119909119891119910

(119905) gt min 119865119892119909119892119910

(119905) 119865119892119909119891119910

(119905) 119865119892119910119891119909

(119905) 119865119892119910119891119910

(119905)

(20)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

3 Section III

In this section we prove several fixed point theorems for fourself-mappings on (119883 119865) where 119865 119883 times 119883 rarr Δ satisfyingcondition (2) We begin with the following theorem

Theorem 13 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (21)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

gt min 119865119878119909119879119910

(119905) 119865119878119909119891119909

(119905) 119865119879119910119892119910

(119905) 119865119878119909119892119910

119865119879119910119891119909

(119905)

(22)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 isin 119883 such that119891119909 = 119878119909 and 119892119910 = 119879119910 Suppose that 119865

119891119909119892119910(119905) = 1 for all 119905 gt 0

Then from (22)

119865119891119909119892119910

(119905)

gt min 119865119891119909119892119910

(119905) 119865119891119909119891119909

(119905) 119865119892119910119892119910

(119905)

119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)

gt 119865119891119909119892119910

(119905)

(23)

This is a contradiction Hence 119865119891119909119892119910

(119905) = 1 for all 119905 gt 0 Thisimplies that 119891119909 = 119892119910 So 119891119909 = 119878119909 = 119892119910 = 119879119910 Moreover ifthere is another point 119911 such that 119891119911 = 119878119911 then using (22) itfollows that119891119911 = 119878119911 = 119892119910 = 119879119910 or119891119909 = 119891119911 and119908 = 119891119909 = 119878119909

is the unique point of coincidence of 119891 and 119878Thus 120575(119875119862(119891 119878)) = 1 This implies that 119865

119909119891119909(119905) = 1 and

hence 119909 = 119908 is a unique common fixed point of 119891 and 119878Similarly 119910 = 119911 is a unique fixed point of 119892 and 119879 Suppose119908 = 119911 Using (21) and (22) we get

119865119908119911

(119905) = 119865119891119908119892119911

(119905) gt min 119865119908119911

(119905) 119865119911119908

(119905) = 119865119908119911

(119905)

(24)

This is a contradiction Therefore 119908 = 119911 and 119908 is the uniquecommon fixed point of 119891 119892 119878 and 119879

Let the control function 120601 119877+

rarr 119877+ be a continuous

nondecreasing function such that 120601(2119905) ge 2120601(119905) and 120601(1) = 1Let a function 120595 be defined by 120595 [0 1] rarr [0 1] satisfyingthe condition 120595(119902) gt 119902 for all 0 le 119902 lt 1

Theorem 14 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (25)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) (26)

where119872120601(119909 119910)

= min 120601 (119865119878119909119879119910

(119905)) 120601 (119865119878119909119891119909

(119905)) 120601 (119865119892119910119879119910

(119905))

1

2[120601 (119865119878119909119892119910

(119905)) + 120601 (119865119891119909119879119910

(119905))]

(27)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 in 119883 such that119908 = 119891119909 = 119878119909 and 119911 = 119892119910 = 119879119910 We claim that 119891119909 = 119892119910Suppose that 119891119909 = 119892119910 Then from (25) and (26) we get

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) gt 120595 (119865

119891119909119892119910(119905)) (28)

which is a contradiction Therefore 120595(119865119891119909119892119910

(119905)) = 1 whichfurther implies that 119865

119891119909119892119910(119905) = 1 Hence the claim follows

that is 119908 = 119891119909 = 119892119910 = 119911 Now from the repeated use ofcondition (26) we can show that 119891 119892 and 119878 and 119879 have aunique common fixed point

DefineG = 120601 120601 (R+)5rarr R+ such that

if 119906 isin R+ such that 119906 ge 120601 (119906 1 1 119906 119906)

119906 ge 120601 (1 119906 1 119906 119906) or 119906 ge 120601 (1 1 119906 119906 119906)

then 119906 = 1

(1198921)

Theorem 15 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (29)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

ge 120601 (119865119878119909119879119910

(119905) 119865119891119909119878119909

(119905) 119865119892119910119879119910

(119905)

119865119891119909119879119910

(119905) 119865119892119910119878119909

(119905))

(30)

Journal of Function Spaces and Applications 5

for all 119909 119910 isin 119883 then119891 119892 and 119878 and 119879 have a unique commonfixed point

Proof It follows from the given assumptions that there existsa point 119909 isin 119883 such that 119891119909 = 119878119909 and there exists anotherpoint 119910 isin 119883 for which 119892119910 = 119879119910 Suppose that 119891119909 = 119892119910 Thenfrom (30) we have

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)) (31)

Since119891119909 and 119892119910 are points of coincidence of 119891 119878 and 119892 119879respectively hence from (30) we get

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119891119909119892119910

(119905)) (32)

Therefore from (1198921) we get 119865

119891119909119892119910(119905) = 1 This shows that

119891119909 = 119892119910 Suppose that there exists another point 119911 such that119891119911 = 119878119911 Then using (30) one obtains 119891119911 = 119878119911 = 119892119910 =

119879119910 = 119891119909 = 119878119909 Hence 119908 = 119891119909 = 119891119911 is the unique pointof coincidence of 119891 and 119878 120575(119875119862(119891 119878)) = 1 This implies that119865119909119891119909

(119905) = 1 and hence 119909 = 119908 is a unique common fixedpoint of 119891 and 119878 Similarly there exists a unique point V isin 119883

such that V = 119892119911 = 119879V It then follows that V = 119908 and 119908 is acommon fixed point of 119891 119892 119878 and 119879 and 119908 is unique

4 Application to Dynamic Programming

Throughout in this section we assume that 119883 and 119884 areBanach spaces 119878 sub 119883 is a state space and119863 sub 119884 is a decisionspace We denote by 119861(119878) the set of all bounded real valuedfunctions defined on 119878

As suggested by Bellman and Lee [19] the basic form ofthe functional equations arising in dynamic programming is

119891 (119909) = opt119910119867(119909 119910 119891 (119879 (119909 119910))) (33)

where 119909 and 119910 represent the state and decision vectorsrespectively 119879 represents the transformation of the processand 119891(119909) represents the optimal return function with initialstate 119909 (here opt denotes maximum or minimum)

We now study the existence and uniqueness of a commonsolution of the following functional equations arising indynamic programming

120595 (119909) = sup119910isin119863

119867(119909 119910 120595 (119879 (119909 119910))) 119909 isin 119878

119875 (119909) = sup119910isin119863

119865 (119909 119910 119875 (119879 (119909 119910))) 119909 isin 119878

(34)

where 119879 119878 times 119863 rarr 119878119867 and 119865 119878 times 119863 timesR rarr RAs an application of Corollary 11 the existence and

uniqueness of a common solution of the functional equationsarising in dynamic programming can be established whichextends Theorem 18 [12]

Definition 16 Let 119883 be a non-empty set and 119889 a function 119889

119883 times 119883 rarr [0infin) such that

119889 (119909 119910) = 0 iff 119909 = 119910 forall119909 119910 isin 119883 (35)

Corollary 17 Let 119883 be a non-empty set and 119889 119883 times 119883 rarr

[0infin) a function satisfying condition (35) If 119891 and 119892 are JH-operators self-mappings of119883 and

119889 (119891119909 119891119910) le 120601 119889 (119892119909 119892119910) forall119909 119910 isin 119883 (36)

where 120601 R+ rarr R+ a nondecreasing function satisfying thecondition 120601(119905) lt 119905 for each 119905 gt 0 then 119891 and 119892 have a uniquecommon fixed point

Proof The proof of this corollary can be easily obtained

We now present main result of this section

Theorem 18 Suppose that the following conditions (i) (ii)(iii) and (iv) are satisfied

(i) 119867 and 119865 are bounded(ii) |119867(119909 119910 ℎ(119905))minus119867(119909 119910 119896(119905))| le 120601|119892ℎ(119905)minus119892119896(119905)| for all

(119909 119910) isin 119878 times 119863 ℎ 119896 isin 119861(119878) and 119905 isin 119878 where 120601 R+ rarr

R+ is a nondecreasing function satisfying the condition120601(119905) lt 119905 for each 119905 gt 0 and 119891 and 119892 are defined asfollows

119891ℎ (119909) = sup119910isin119863

119867(119909 119910 ℎ (119879 (119909 119910))) 119909 isin 119878 ℎ isin 119861 (119878)

119892119896 (119909) = sup119910isin119863

119865 (119909 119910 119896 (119879 (119909 119910))) 119909 isin 119878 119896 isin 119861 (119878)

(37)

(iii) If there is a point 119891119906(119909) = 119892119906(119909) = 119896(119909) for some119906(119909) isin 119861(119904) implies 119889(119896119909 119906119909) le 120575(119875119862(119891 119892))

then 119896(119909) is the unique common solution of (34)

Proof For any ℎ 119896 isin 119861(119878) let

119889 (ℎ 119896) = sup |ℎ (119909) minus 119896 (119909)| 119909 isin 119878 (38)

From conditions (i) (ii) (iii) it follows that 119891 and 119892 are self-mappings of 119861(119878)

Letting ℎ1 ℎ2be any two points of 119861(119878) 119909 isin 119878 and 120578 any

positive number then there exist 1199101 1199102isin 119863 such that

119891ℎ1(119909) lt 119867 (119909 119910

1 ℎ1(1199091)) + 120578 (39)

119891ℎ2(119909) lt 119867 (119909 119910

2 ℎ2(1199092)) + 120578 (40)

119891ℎ1(119909) ge 119867 (119909 119910

2 ℎ1(1199092)) (41)

119891ℎ2(119909) ge 119867 (119909 119910

1 ℎ2(1199091)) (42)

Subtracting (42) from (39) and using (ii) we have

119891ℎ1(119909) minus 119891ℎ

2(119909)

lt 119867 (119909 1199101 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091)) + 120578

le1003816100381610038161003816119867 (119909 119910

1 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091))1003816100381610038161003816 + 120578

le 120601 (1003816100381610038161003816119892ℎ1 (1199091) minus 119892119896

2(1199091)1003816100381610038161003816 + 120578)

le 120601 (119889 (119892ℎ1 119892ℎ2) + 120578)

(43)

6 Journal of Function Spaces and Applications

Similarly from (40) and (41) we get

119891ℎ1(119909) minus 119891ℎ

2(119909) gt 120601 (119889 (119892ℎ

1 119892ℎ2)) minus 120578 (44)

Hence1003816100381610038161003816119891ℎ1 (119909) minus 119891ℎ

2(119909)

1003816100381610038161003816 lt 120601 (119889 (119892ℎ1 119892ℎ2)) + 120578 (45)

Since (42) is true for any 119909 isin 119878 and 120578 any positive number

119889 (119891ℎ1 119891ℎ2) le 120601 (119889 (119892ℎ

1 119892ℎ2)) (46)

Therefore from Corollary 17 119896(119909) is the unique commonfixed point of 119891 and 119892 that is 119896(119909) is the unique commonsolution of functional equation (34)

Acknowledgment

Authors are grateful to referee for careful reading of this paperand for valuable comments

References

[1] K Menger ldquoStatistical metricesrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 no12 pp 535ndash537 1942

[2] M S El Naschie ldquoOn the uncertainty of Cantorian geometryand the two-slit experimentrdquo Chaos Solitons amp Fractals vol 9no 3 pp 517ndash529 1998

[3] M S El Naschie ldquoOn the unification of heterotic strings Mtheory and 120576(infin) theoryrdquo Chaos Solitons amp Fractals vol 11 no14 pp 2397ndash2408 2000

[4] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD dissertation] Wayne State UniversityDetroit Mich USA 1966

[5] VM Sehgal and A T Bharucha-Reid ldquoFixed points of contrac-tion mappings on probabilistic metric spacesrdquo MathematicalSystems Theory vol 6 pp 97ndash102 1972

[6] S Sessa ldquoOn a weak commutativity condition of mappingsin fixed point considerationsrdquo Publications de lrsquoInstitut Math-ematique vol 32(46) pp 149ndash153 1982

[7] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[8] S NMishra ldquoCommon fixed points of compatible mappings inPM-spacesrdquo Mathematica Japonica vol 36 no 2 pp 283ndash2891991

[9] G Jungck ldquoCommon fixed points for noncontinuous nonselfmaps on nonmetric spacesrdquo Far East Journal of MathematicalSciences vol 4 no 2 pp 199ndash215 1996

[10] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[11] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[12] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 73 no 1 pp 176ndash182 2010

[13] H K Pathak and N Hussain ldquoCommon fixed points for 119875-operator pair with applicationsrdquoAppliedMathematics andCom-putation vol 217 no 7 pp 3137ndash3143 2010

[14] N Hussain M A Khamsi and A Latif ldquoCommon fixed pointsfor 119869119867-operators and occasionally weakly biased pairs underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2133ndash2140 2011

[15] B Schweizer and A Skalar Statistical Metric Spaces NorthHolland Amsterdam The Netherlands 1983

[16] G Jungck and B E Rhoades ldquoFixed point theorems for occa-sionally weakly compatible mappingsrdquo Fixed Point Theory vol7 no 2 pp 286ndash296 2006

[17] G Jungck and N Hussain ldquoCompatible maps and invariantapproximationsrdquo Journal of Mathematical Analysis and Appli-cations vol 325 no 2 pp 1003ndash1012 2007

[18] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[19] B Bellman and E S Lee ldquoFunctional equation arising indynamic programmingrdquo Aequationes Mathematicae vol 17 pp1ndash18 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

The Heaviside function 119867 is a distribution functiondefined by

119867(119905) = 0 if 119905 le 0

1 if 119905 gt 0(1)

Definition 2 Let 119883 be a nonempty set and let Δ denote theset of all distribution functions defined on119883 119865 is a mappingfrom119883 times 119883 into Δ satisfying the following condition

119865119909119910

(119905) = 1 lArrrArr 119909 = 119910 (2)where 119865 119883times119883 rarr Δ defined by 119865

119909119910(119905) = 119867(119905 minus119889(119909 119910)) for

all 119909 119910 isin 119883 and 119889 is a function 119889 119883times119883 rarr [0infin) such that119889(119909 119910) = 0 if and only if 119909 = 119910 forall119909 119910 isin 119883 (symmetric andtriangle conditions are not required) A topology 120591(119889) on 119883

is given by119880 isin 120591(119889) if and only if for each 119909 isin 119880 119861(119909 120598) sub 119880

for some 120598 gt 0 where 119861(119909 120598) = 119910 isin 119883 119889(119909 119910) lt 120598

Definition 3 (see [16 17]) Let119883 be a non-empty set A point119909 in 119883 is called a coincidence point of 119891 and 119892 if and onlyif 119891119909 = 119892119909 In this case 119908 = 119891119909 = 119892119909 is called a point ofcoincidence of 119891 and 119892

Let 119862(119891 119892) and 119875119862(119891 119892) denote the sets of coincidencepoints and points of coincidence respectively of the pair(119891 119892) For a space (119883 119865) satisfying (2) and 119860 sube 119883 thediameter of 119860 is defined by

120575 (119860) = sup min 119865119909119910

(119905) 119865119910119909

(119905) 119909 119910 isin 119860 (3)

Here we extend the concept of JH-operators and occa-sionally weakly g-biased pairs and the space (119883 119865) satisfyingcondition (2)

Definition 4 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Two self-maps 119891 and 119892 of a space (119883 119865) are called JH-operators if andonly if there is a point 119908 = 119891119909 = 119892119909 in 119875119862(119891 119892) such that

119865119908119909

(119905) ge 120575 (119875119862 (119891 119892)) 119865119909119908

(119905) ge 120575 (119875119862 (119891 119892))

(4)Definition 5 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Two self-maps 119891 and 119892 of a space (119883 119865) are called weakly g-biased ifand only 119865

119892119891119909119892119909(119905) ge 119865

119891119892119909119891119909(119905) whenever 119891119909 = 119892119909

Definition 6 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Two self-maps 119891 and 119892 of a space (119883 119865) are called occasionally weaklyg-biased if and only if there exists some 119909 isin 119883 such that119891119909 = 119892119909 and 119865

119892119891119909119892119909(119905) ge 119865

119891119892119909119891119909(119905)

Example 7 Let 119883 = [0 +infin) and 119865119909119910

(119905) = 119867(119905 minus 119889(119909 119910))where

119889 (119909 119910) = 119890119909minus119910

minus 1 if 119909 ge 119910

119890119910minus119909

otherwise(5)

Define 119891 119892 119883 rarr 119883 by

119891 (119909) = 2119909 119892 (119909) = 21199092 if 119909 = 0

119891 (119909) = 119892 (119909) =1

2 if 119909 = 0

(6)

In this example 119862(119891 119892) = 0 1 and 119875119862(119891 119892) = 12 2Here 119865

119892119891(1)119892(1)(119905) = 119867(119905 minus 119889(119892119891(1) 119892(1))) = 119867(119905 minus 119889(119892(2)

2)) = 119867(119905minus119889(8 2)) = 119867(119905minus1198906+1) For 119905 = 119890

6minus1119867(119905minus119890

6+1) =

119867(0) = 0 Similarly 119865119891119892(1)119891(1)

(119905) = 119867(119905 minus 119889(119891119892(1) 119891(1))) =

119867(119905minus119889(119891(2) 2)) = 119867(119905minus119889(4 2)) = 119867(119905minus1198902+1) For 119905 = 119890

6minus1

119867(119905 minus 1198902+ 1) = 119867(119890

6minus 1minus 119890

2+ 1) = 119867(119890

6minus 1198902) = 1 Therefore

119865119892119891(1)119892(1)

(119905) le 119865119891119892(1)119891(1)

(119905) Now we can easily show that119865119892119891(0)119892(0)

(119905) ge 119865119891119892(0)119891(0)

(119905)Therefore an occasionally weaklycompatible and a nontrivial weakly g-biased pair (119891 119892) areoccasionally weakly g-biased pairs but the converse does nothold

2 Section II

We note that every symmetric (semimetric) space (119883 119889) [18]can be realized as a probabilistic semi-metric space by taking119865 119883 times 119883 rarr Δ defined by 119865

119909119910(119905) = 119867(119905 minus 119889(119909 119910)) for all 119909

119910 in 119883 So probabilistic semi-metric spaces provide a widerframework than that of the symmetric spaces and are bettersuited in many situations In this paper we have relaxed thesymmetric condition from probabilistic semimetric spaceIn this section we prove some fixed point theorems for apair of JH-operator on space (119883 119865) without imposing therestriction of the triangle inequality or symmetry on119865 In thissection we also prove some fixed point theorems for a pair ofOccasionally weakly biased on space (119883 119865)without imposingthe restriction of the triangle inequality and symmetry onlyon point of coincidence and image of point of coincidence

Theorem 8 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) Suppose 119891and 119892 are JH-operators on119883 satisfying the following condition

119865119891119909119891119910

(119905)

ge 119865119892119909119892119910

(119905

119886) +min 119865

119891119909119892119909(119905

119887) 119865119891119910119892119910

(119905

119887)

+min 119865119892119909119892119910

(119905

119888) 119865119892119909119891119909

(119905

119888) 119865119892119910119891119910

(119905

119888)

(7)

for all 119909 119910 isin 119883 with 119891(119909) = 119891119910 and 119905 gt 0 where 0 lt 119886 lt 10 lt 119887 lt 1 and 0 lt 119888 lt 1Then119891 and119892 have a unique commonfixed point

Proof We claim that 119891 and 119892 have a unique point ofcoincidence 119908 = 119891119909 = 119892119909 If possible suppose there isanother point of coincidence119891119910 = 119892119910 = 119908

1and119908

1= 119908Then

119865119891119909119891119910

(119905) = 1 forall119905 gt 0 So from (7) we get

119865119891119909119891119910

(119905)

ge 119865119891119909119891119910

(119905

119886) + 1

+min 119865119891119909119891119910

(119905

119888) 119865119891119909119891119909

(119905

119888) 119865119891119910119891119910

(119905

119888)

= 119865119891119909119891119910

(119905

119886) + 1 + 119865

119891119909119891119910(119905

119888) gt 1

(8)

Journal of Function Spaces and Applications 3

This is a contradiction which implies that 119865119891119909119891119910

(119905) = 1Hence we get 119891119909 = 119891119910 = 119908 = 119911 Therefore there existsa unique element 119908 in 119883 such that 119908 = 119891119909 = 119892119909 Thus120575(119875119862(119891 119892)) = 1 implies that119865

119909119908= 1 and hence 119909 is a unique

common fixed point of 119891 and 119892

Let a function 120601 be defined by 120601 [0 1] rarr [0 1] sati-sfying condition 120601(119902) gt 119902 for all 0 le 119902 lt 1

Theorem 9 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying the condition (2) If 119891and 119892 are occasionally weakly g-biased on 119883 suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (9)

for some point of coincidence 119908 of 119891 119892 and

119865119891119909119891119910

(119905)

ge 120601 [min 119865119892119909119892119910

(119905) 119865119892119909119891119910

(119905) 119865119892119910119891119909

(119905) 119865119892119910119891119910

(119905)]

(10)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

Proof Since 119891 and 119892 are occasionally weakly biased thereexists some 119906 isin 119883 such that 119891119906 = 119892119906 = 119908 and 119865

119892119891119906119892119906(119905) ge

119865119891119892119906119891119906

(119905)We claim that119891119906 is the unique commonfixed pointof 119891 and 119892 For if 119891119891119906 = 119891119906 then from (9) and (10) we get

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119892119891119906119892119906

(119905) 119865119892119891119906119891119906

(119905) 119865119892119906119891119891119906

(119905) 119865119892119906119891119906

(119905)]

= 120601 [min 119865119892119891119906119892119906

(119905) 119865119892119891119906119892119906

(119905) 119865119891119906119891119891119906

(119905) 119865119891119906119891119906

(119905)]

times (since 119891119906 = 119892119906)

(11)

Because 119891 and 119892 are occasionally weakly biased hence

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119891119892119906119891119906

(119905) 119865119891119892119906119891119906

(119905) 119865119891119906119891119891119906

(119905) 1]

= 120601 [min 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 119865119891119906119891119891119906

(119905) 1]

times (since 119891119906 = 119892119906)

(12)

by using condition (9)

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 1]

= 120601 [min 119865119891119891119906119891119906

(119905)]

(13)

Since 120601 [0 1] rarr [0 1] satisfying the condition 120601(119902) gt 119902for all 0 le 119902 lt 1 Therefore 119865

119891119891119906119891119906(119905) gt 119865

119891119891119906119891119906(119905) which is

a contradiction Therefore 119891119891119906 = 119891119906 = 119891119892119906 Hence 119865119892119891119906119892119906

(119905) ge 119865119891119892119906119891119906

= 1 which further implies that 119892119891119906 = 119892119906 =

119891119906 = 119891119891119906 Thus 119891119906 is a common fixed point of 119891 and 119892For uniqueness suppose that 119906 V isin 119883 such that 119891119906 =

119892119906 = 119906 and 119891V = 119892V = V and 119906 = V Then (10) gives

119865119906V (119905) = 119865

119891119906119891V (119905)

ge 120601 [min 119865119892119906119892V (119905) 119865119892119906119891V (119905) 119865119892V119891119906 (119905) 119865119892V119891V (119905)]

= 120601 [min 119865119906V (119905) 119865V119906 (119905) 1]

= 120601 [min 119865119906V (119905) 119865V119906 (119905)]

(14)

Let 120572 = min119865119906V(119905) 119865V119906(119905) gt 0 Then we get 119865

119906V(119905) ge

120601(120572) gt 120572 Similarly we get 119865V119906(119905) ge 120601(120572) gt 120572 Somin119865

119906V(119905) 119865V119906(119905) ge 120601(120572) gt 120572 This is a contradictionTherefore 119906 = V Therefore the common fixed point of 119891and 119892 is unique

Example 10 Letting119883 = [0 1] and 120601 119883 rarr 119883 defined as

120601 (119902) =1 + 119902

2 then 120601 (119902) gt 119902 0 le 119902 lt 1 (15)

and 119865119909119910

(119905) = 119867(119905 minus 119889(119909 119910)) where

119889 (119909 119910) = 119890119909minus119910

minus 1 if 119909 ge 119910

119890119910minus119909

otherwise(16)

Define119891 119892 119883 rarr 119883 by119891(119909) = 2119909 and 119892(119909) = 21199092 if 119909 = 0 1

One has 119891(119909) = 119892(119909) = 12 if 119909 = 0 and 119891(119909) = 119892(119909) = 1

if 119909 = 1In this example we observe that 119862(119891 119892) = 0 1

where (119891 119892) are occasionally weakly g-biased pairs and11986511198911

(119905) = 11986511989111

(119905) Now for 119905 = 11989012 120601[min119865

119892(12)119892(1)(119905)

119865119892(12)119891(1)

(119905) 119865119892(1)119891(12)

(119905) 119865119892(1)119891(1)

(119905)] lt 119865119891(12)119891(1)

(119905)Example 10 is the unique common fixed point of 119891 and 119892

Corollary 11 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) If 119891 and119892 are occasionally weakly g-biased on 119883 suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (17)

whenever 119908 is point of coincidence of 119891 119892 and

119865119891119909119891119910

(119905) ge 120601 [119865119892119909119892119910

(119905)] (18)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

Theproof of the following theorem can be easily obtainedby replacing condition (10) by condition (20) the proof ofTheorem 9

Theorem 12 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) If 119891 and119892 are occasionally weakly g-biased on 119883 Suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (19)

4 Journal of Function Spaces and Applications

whenever 119908 is point of coincidence of 119891 119892 and

119865119891119909119891119910

(119905) gt min 119865119892119909119892119910

(119905) 119865119892119909119891119910

(119905) 119865119892119910119891119909

(119905) 119865119892119910119891119910

(119905)

(20)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

3 Section III

In this section we prove several fixed point theorems for fourself-mappings on (119883 119865) where 119865 119883 times 119883 rarr Δ satisfyingcondition (2) We begin with the following theorem

Theorem 13 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (21)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

gt min 119865119878119909119879119910

(119905) 119865119878119909119891119909

(119905) 119865119879119910119892119910

(119905) 119865119878119909119892119910

119865119879119910119891119909

(119905)

(22)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 isin 119883 such that119891119909 = 119878119909 and 119892119910 = 119879119910 Suppose that 119865

119891119909119892119910(119905) = 1 for all 119905 gt 0

Then from (22)

119865119891119909119892119910

(119905)

gt min 119865119891119909119892119910

(119905) 119865119891119909119891119909

(119905) 119865119892119910119892119910

(119905)

119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)

gt 119865119891119909119892119910

(119905)

(23)

This is a contradiction Hence 119865119891119909119892119910

(119905) = 1 for all 119905 gt 0 Thisimplies that 119891119909 = 119892119910 So 119891119909 = 119878119909 = 119892119910 = 119879119910 Moreover ifthere is another point 119911 such that 119891119911 = 119878119911 then using (22) itfollows that119891119911 = 119878119911 = 119892119910 = 119879119910 or119891119909 = 119891119911 and119908 = 119891119909 = 119878119909

is the unique point of coincidence of 119891 and 119878Thus 120575(119875119862(119891 119878)) = 1 This implies that 119865

119909119891119909(119905) = 1 and

hence 119909 = 119908 is a unique common fixed point of 119891 and 119878Similarly 119910 = 119911 is a unique fixed point of 119892 and 119879 Suppose119908 = 119911 Using (21) and (22) we get

119865119908119911

(119905) = 119865119891119908119892119911

(119905) gt min 119865119908119911

(119905) 119865119911119908

(119905) = 119865119908119911

(119905)

(24)

This is a contradiction Therefore 119908 = 119911 and 119908 is the uniquecommon fixed point of 119891 119892 119878 and 119879

Let the control function 120601 119877+

rarr 119877+ be a continuous

nondecreasing function such that 120601(2119905) ge 2120601(119905) and 120601(1) = 1Let a function 120595 be defined by 120595 [0 1] rarr [0 1] satisfyingthe condition 120595(119902) gt 119902 for all 0 le 119902 lt 1

Theorem 14 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (25)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) (26)

where119872120601(119909 119910)

= min 120601 (119865119878119909119879119910

(119905)) 120601 (119865119878119909119891119909

(119905)) 120601 (119865119892119910119879119910

(119905))

1

2[120601 (119865119878119909119892119910

(119905)) + 120601 (119865119891119909119879119910

(119905))]

(27)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 in 119883 such that119908 = 119891119909 = 119878119909 and 119911 = 119892119910 = 119879119910 We claim that 119891119909 = 119892119910Suppose that 119891119909 = 119892119910 Then from (25) and (26) we get

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) gt 120595 (119865

119891119909119892119910(119905)) (28)

which is a contradiction Therefore 120595(119865119891119909119892119910

(119905)) = 1 whichfurther implies that 119865

119891119909119892119910(119905) = 1 Hence the claim follows

that is 119908 = 119891119909 = 119892119910 = 119911 Now from the repeated use ofcondition (26) we can show that 119891 119892 and 119878 and 119879 have aunique common fixed point

DefineG = 120601 120601 (R+)5rarr R+ such that

if 119906 isin R+ such that 119906 ge 120601 (119906 1 1 119906 119906)

119906 ge 120601 (1 119906 1 119906 119906) or 119906 ge 120601 (1 1 119906 119906 119906)

then 119906 = 1

(1198921)

Theorem 15 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (29)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

ge 120601 (119865119878119909119879119910

(119905) 119865119891119909119878119909

(119905) 119865119892119910119879119910

(119905)

119865119891119909119879119910

(119905) 119865119892119910119878119909

(119905))

(30)

Journal of Function Spaces and Applications 5

for all 119909 119910 isin 119883 then119891 119892 and 119878 and 119879 have a unique commonfixed point

Proof It follows from the given assumptions that there existsa point 119909 isin 119883 such that 119891119909 = 119878119909 and there exists anotherpoint 119910 isin 119883 for which 119892119910 = 119879119910 Suppose that 119891119909 = 119892119910 Thenfrom (30) we have

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)) (31)

Since119891119909 and 119892119910 are points of coincidence of 119891 119878 and 119892 119879respectively hence from (30) we get

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119891119909119892119910

(119905)) (32)

Therefore from (1198921) we get 119865

119891119909119892119910(119905) = 1 This shows that

119891119909 = 119892119910 Suppose that there exists another point 119911 such that119891119911 = 119878119911 Then using (30) one obtains 119891119911 = 119878119911 = 119892119910 =

119879119910 = 119891119909 = 119878119909 Hence 119908 = 119891119909 = 119891119911 is the unique pointof coincidence of 119891 and 119878 120575(119875119862(119891 119878)) = 1 This implies that119865119909119891119909

(119905) = 1 and hence 119909 = 119908 is a unique common fixedpoint of 119891 and 119878 Similarly there exists a unique point V isin 119883

such that V = 119892119911 = 119879V It then follows that V = 119908 and 119908 is acommon fixed point of 119891 119892 119878 and 119879 and 119908 is unique

4 Application to Dynamic Programming

Throughout in this section we assume that 119883 and 119884 areBanach spaces 119878 sub 119883 is a state space and119863 sub 119884 is a decisionspace We denote by 119861(119878) the set of all bounded real valuedfunctions defined on 119878

As suggested by Bellman and Lee [19] the basic form ofthe functional equations arising in dynamic programming is

119891 (119909) = opt119910119867(119909 119910 119891 (119879 (119909 119910))) (33)

where 119909 and 119910 represent the state and decision vectorsrespectively 119879 represents the transformation of the processand 119891(119909) represents the optimal return function with initialstate 119909 (here opt denotes maximum or minimum)

We now study the existence and uniqueness of a commonsolution of the following functional equations arising indynamic programming

120595 (119909) = sup119910isin119863

119867(119909 119910 120595 (119879 (119909 119910))) 119909 isin 119878

119875 (119909) = sup119910isin119863

119865 (119909 119910 119875 (119879 (119909 119910))) 119909 isin 119878

(34)

where 119879 119878 times 119863 rarr 119878119867 and 119865 119878 times 119863 timesR rarr RAs an application of Corollary 11 the existence and

uniqueness of a common solution of the functional equationsarising in dynamic programming can be established whichextends Theorem 18 [12]

Definition 16 Let 119883 be a non-empty set and 119889 a function 119889

119883 times 119883 rarr [0infin) such that

119889 (119909 119910) = 0 iff 119909 = 119910 forall119909 119910 isin 119883 (35)

Corollary 17 Let 119883 be a non-empty set and 119889 119883 times 119883 rarr

[0infin) a function satisfying condition (35) If 119891 and 119892 are JH-operators self-mappings of119883 and

119889 (119891119909 119891119910) le 120601 119889 (119892119909 119892119910) forall119909 119910 isin 119883 (36)

where 120601 R+ rarr R+ a nondecreasing function satisfying thecondition 120601(119905) lt 119905 for each 119905 gt 0 then 119891 and 119892 have a uniquecommon fixed point

Proof The proof of this corollary can be easily obtained

We now present main result of this section

Theorem 18 Suppose that the following conditions (i) (ii)(iii) and (iv) are satisfied

(i) 119867 and 119865 are bounded(ii) |119867(119909 119910 ℎ(119905))minus119867(119909 119910 119896(119905))| le 120601|119892ℎ(119905)minus119892119896(119905)| for all

(119909 119910) isin 119878 times 119863 ℎ 119896 isin 119861(119878) and 119905 isin 119878 where 120601 R+ rarr

R+ is a nondecreasing function satisfying the condition120601(119905) lt 119905 for each 119905 gt 0 and 119891 and 119892 are defined asfollows

119891ℎ (119909) = sup119910isin119863

119867(119909 119910 ℎ (119879 (119909 119910))) 119909 isin 119878 ℎ isin 119861 (119878)

119892119896 (119909) = sup119910isin119863

119865 (119909 119910 119896 (119879 (119909 119910))) 119909 isin 119878 119896 isin 119861 (119878)

(37)

(iii) If there is a point 119891119906(119909) = 119892119906(119909) = 119896(119909) for some119906(119909) isin 119861(119904) implies 119889(119896119909 119906119909) le 120575(119875119862(119891 119892))

then 119896(119909) is the unique common solution of (34)

Proof For any ℎ 119896 isin 119861(119878) let

119889 (ℎ 119896) = sup |ℎ (119909) minus 119896 (119909)| 119909 isin 119878 (38)

From conditions (i) (ii) (iii) it follows that 119891 and 119892 are self-mappings of 119861(119878)

Letting ℎ1 ℎ2be any two points of 119861(119878) 119909 isin 119878 and 120578 any

positive number then there exist 1199101 1199102isin 119863 such that

119891ℎ1(119909) lt 119867 (119909 119910

1 ℎ1(1199091)) + 120578 (39)

119891ℎ2(119909) lt 119867 (119909 119910

2 ℎ2(1199092)) + 120578 (40)

119891ℎ1(119909) ge 119867 (119909 119910

2 ℎ1(1199092)) (41)

119891ℎ2(119909) ge 119867 (119909 119910

1 ℎ2(1199091)) (42)

Subtracting (42) from (39) and using (ii) we have

119891ℎ1(119909) minus 119891ℎ

2(119909)

lt 119867 (119909 1199101 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091)) + 120578

le1003816100381610038161003816119867 (119909 119910

1 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091))1003816100381610038161003816 + 120578

le 120601 (1003816100381610038161003816119892ℎ1 (1199091) minus 119892119896

2(1199091)1003816100381610038161003816 + 120578)

le 120601 (119889 (119892ℎ1 119892ℎ2) + 120578)

(43)

6 Journal of Function Spaces and Applications

Similarly from (40) and (41) we get

119891ℎ1(119909) minus 119891ℎ

2(119909) gt 120601 (119889 (119892ℎ

1 119892ℎ2)) minus 120578 (44)

Hence1003816100381610038161003816119891ℎ1 (119909) minus 119891ℎ

2(119909)

1003816100381610038161003816 lt 120601 (119889 (119892ℎ1 119892ℎ2)) + 120578 (45)

Since (42) is true for any 119909 isin 119878 and 120578 any positive number

119889 (119891ℎ1 119891ℎ2) le 120601 (119889 (119892ℎ

1 119892ℎ2)) (46)

Therefore from Corollary 17 119896(119909) is the unique commonfixed point of 119891 and 119892 that is 119896(119909) is the unique commonsolution of functional equation (34)

Acknowledgment

Authors are grateful to referee for careful reading of this paperand for valuable comments

References

[1] K Menger ldquoStatistical metricesrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 no12 pp 535ndash537 1942

[2] M S El Naschie ldquoOn the uncertainty of Cantorian geometryand the two-slit experimentrdquo Chaos Solitons amp Fractals vol 9no 3 pp 517ndash529 1998

[3] M S El Naschie ldquoOn the unification of heterotic strings Mtheory and 120576(infin) theoryrdquo Chaos Solitons amp Fractals vol 11 no14 pp 2397ndash2408 2000

[4] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD dissertation] Wayne State UniversityDetroit Mich USA 1966

[5] VM Sehgal and A T Bharucha-Reid ldquoFixed points of contrac-tion mappings on probabilistic metric spacesrdquo MathematicalSystems Theory vol 6 pp 97ndash102 1972

[6] S Sessa ldquoOn a weak commutativity condition of mappingsin fixed point considerationsrdquo Publications de lrsquoInstitut Math-ematique vol 32(46) pp 149ndash153 1982

[7] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[8] S NMishra ldquoCommon fixed points of compatible mappings inPM-spacesrdquo Mathematica Japonica vol 36 no 2 pp 283ndash2891991

[9] G Jungck ldquoCommon fixed points for noncontinuous nonselfmaps on nonmetric spacesrdquo Far East Journal of MathematicalSciences vol 4 no 2 pp 199ndash215 1996

[10] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[11] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[12] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 73 no 1 pp 176ndash182 2010

[13] H K Pathak and N Hussain ldquoCommon fixed points for 119875-operator pair with applicationsrdquoAppliedMathematics andCom-putation vol 217 no 7 pp 3137ndash3143 2010

[14] N Hussain M A Khamsi and A Latif ldquoCommon fixed pointsfor 119869119867-operators and occasionally weakly biased pairs underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2133ndash2140 2011

[15] B Schweizer and A Skalar Statistical Metric Spaces NorthHolland Amsterdam The Netherlands 1983

[16] G Jungck and B E Rhoades ldquoFixed point theorems for occa-sionally weakly compatible mappingsrdquo Fixed Point Theory vol7 no 2 pp 286ndash296 2006

[17] G Jungck and N Hussain ldquoCompatible maps and invariantapproximationsrdquo Journal of Mathematical Analysis and Appli-cations vol 325 no 2 pp 1003ndash1012 2007

[18] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[19] B Bellman and E S Lee ldquoFunctional equation arising indynamic programmingrdquo Aequationes Mathematicae vol 17 pp1ndash18 1979

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

This is a contradiction which implies that 119865119891119909119891119910

(119905) = 1Hence we get 119891119909 = 119891119910 = 119908 = 119911 Therefore there existsa unique element 119908 in 119883 such that 119908 = 119891119909 = 119892119909 Thus120575(119875119862(119891 119892)) = 1 implies that119865

119909119908= 1 and hence 119909 is a unique

common fixed point of 119891 and 119892

Let a function 120601 be defined by 120601 [0 1] rarr [0 1] sati-sfying condition 120601(119902) gt 119902 for all 0 le 119902 lt 1

Theorem 9 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying the condition (2) If 119891and 119892 are occasionally weakly g-biased on 119883 suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (9)

for some point of coincidence 119908 of 119891 119892 and

119865119891119909119891119910

(119905)

ge 120601 [min 119865119892119909119892119910

(119905) 119865119892119909119891119910

(119905) 119865119892119910119891119909

(119905) 119865119892119910119891119910

(119905)]

(10)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

Proof Since 119891 and 119892 are occasionally weakly biased thereexists some 119906 isin 119883 such that 119891119906 = 119892119906 = 119908 and 119865

119892119891119906119892119906(119905) ge

119865119891119892119906119891119906

(119905)We claim that119891119906 is the unique commonfixed pointof 119891 and 119892 For if 119891119891119906 = 119891119906 then from (9) and (10) we get

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119892119891119906119892119906

(119905) 119865119892119891119906119891119906

(119905) 119865119892119906119891119891119906

(119905) 119865119892119906119891119906

(119905)]

= 120601 [min 119865119892119891119906119892119906

(119905) 119865119892119891119906119892119906

(119905) 119865119891119906119891119891119906

(119905) 119865119891119906119891119906

(119905)]

times (since 119891119906 = 119892119906)

(11)

Because 119891 and 119892 are occasionally weakly biased hence

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119891119892119906119891119906

(119905) 119865119891119892119906119891119906

(119905) 119865119891119906119891119891119906

(119905) 1]

= 120601 [min 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 119865119891119906119891119891119906

(119905) 1]

times (since 119891119906 = 119892119906)

(12)

by using condition (9)

119865119891119891119906119891119906

(119905)

ge 120601 [min 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 119865119891119891119906119891119906

(119905) 1]

= 120601 [min 119865119891119891119906119891119906

(119905)]

(13)

Since 120601 [0 1] rarr [0 1] satisfying the condition 120601(119902) gt 119902for all 0 le 119902 lt 1 Therefore 119865

119891119891119906119891119906(119905) gt 119865

119891119891119906119891119906(119905) which is

a contradiction Therefore 119891119891119906 = 119891119906 = 119891119892119906 Hence 119865119892119891119906119892119906

(119905) ge 119865119891119892119906119891119906

= 1 which further implies that 119892119891119906 = 119892119906 =

119891119906 = 119891119891119906 Thus 119891119906 is a common fixed point of 119891 and 119892For uniqueness suppose that 119906 V isin 119883 such that 119891119906 =

119892119906 = 119906 and 119891V = 119892V = V and 119906 = V Then (10) gives

119865119906V (119905) = 119865

119891119906119891V (119905)

ge 120601 [min 119865119892119906119892V (119905) 119865119892119906119891V (119905) 119865119892V119891119906 (119905) 119865119892V119891V (119905)]

= 120601 [min 119865119906V (119905) 119865V119906 (119905) 1]

= 120601 [min 119865119906V (119905) 119865V119906 (119905)]

(14)

Let 120572 = min119865119906V(119905) 119865V119906(119905) gt 0 Then we get 119865

119906V(119905) ge

120601(120572) gt 120572 Similarly we get 119865V119906(119905) ge 120601(120572) gt 120572 Somin119865

119906V(119905) 119865V119906(119905) ge 120601(120572) gt 120572 This is a contradictionTherefore 119906 = V Therefore the common fixed point of 119891and 119892 is unique

Example 10 Letting119883 = [0 1] and 120601 119883 rarr 119883 defined as

120601 (119902) =1 + 119902

2 then 120601 (119902) gt 119902 0 le 119902 lt 1 (15)

and 119865119909119910

(119905) = 119867(119905 minus 119889(119909 119910)) where

119889 (119909 119910) = 119890119909minus119910

minus 1 if 119909 ge 119910

119890119910minus119909

otherwise(16)

Define119891 119892 119883 rarr 119883 by119891(119909) = 2119909 and 119892(119909) = 21199092 if 119909 = 0 1

One has 119891(119909) = 119892(119909) = 12 if 119909 = 0 and 119891(119909) = 119892(119909) = 1

if 119909 = 1In this example we observe that 119862(119891 119892) = 0 1

where (119891 119892) are occasionally weakly g-biased pairs and11986511198911

(119905) = 11986511989111

(119905) Now for 119905 = 11989012 120601[min119865

119892(12)119892(1)(119905)

119865119892(12)119891(1)

(119905) 119865119892(1)119891(12)

(119905) 119865119892(1)119891(1)

(119905)] lt 119865119891(12)119891(1)

(119905)Example 10 is the unique common fixed point of 119891 and 119892

Corollary 11 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) If 119891 and119892 are occasionally weakly g-biased on 119883 suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (17)

whenever 119908 is point of coincidence of 119891 119892 and

119865119891119909119891119910

(119905) ge 120601 [119865119892119909119892119910

(119905)] (18)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

Theproof of the following theorem can be easily obtainedby replacing condition (10) by condition (20) the proof ofTheorem 9

Theorem 12 Let 119883 be a non-empty set together with thefunction 119865 119883 times 119883 rarr Δ satisfying condition (2) If 119891 and119892 are occasionally weakly g-biased on 119883 Suppose

119865119891119908119908

(119905) = 119865119908119891119908

(119905) (19)

4 Journal of Function Spaces and Applications

whenever 119908 is point of coincidence of 119891 119892 and

119865119891119909119891119910

(119905) gt min 119865119892119909119892119910

(119905) 119865119892119909119891119910

(119905) 119865119892119910119891119909

(119905) 119865119892119910119891119910

(119905)

(20)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

3 Section III

In this section we prove several fixed point theorems for fourself-mappings on (119883 119865) where 119865 119883 times 119883 rarr Δ satisfyingcondition (2) We begin with the following theorem

Theorem 13 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (21)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

gt min 119865119878119909119879119910

(119905) 119865119878119909119891119909

(119905) 119865119879119910119892119910

(119905) 119865119878119909119892119910

119865119879119910119891119909

(119905)

(22)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 isin 119883 such that119891119909 = 119878119909 and 119892119910 = 119879119910 Suppose that 119865

119891119909119892119910(119905) = 1 for all 119905 gt 0

Then from (22)

119865119891119909119892119910

(119905)

gt min 119865119891119909119892119910

(119905) 119865119891119909119891119909

(119905) 119865119892119910119892119910

(119905)

119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)

gt 119865119891119909119892119910

(119905)

(23)

This is a contradiction Hence 119865119891119909119892119910

(119905) = 1 for all 119905 gt 0 Thisimplies that 119891119909 = 119892119910 So 119891119909 = 119878119909 = 119892119910 = 119879119910 Moreover ifthere is another point 119911 such that 119891119911 = 119878119911 then using (22) itfollows that119891119911 = 119878119911 = 119892119910 = 119879119910 or119891119909 = 119891119911 and119908 = 119891119909 = 119878119909

is the unique point of coincidence of 119891 and 119878Thus 120575(119875119862(119891 119878)) = 1 This implies that 119865

119909119891119909(119905) = 1 and

hence 119909 = 119908 is a unique common fixed point of 119891 and 119878Similarly 119910 = 119911 is a unique fixed point of 119892 and 119879 Suppose119908 = 119911 Using (21) and (22) we get

119865119908119911

(119905) = 119865119891119908119892119911

(119905) gt min 119865119908119911

(119905) 119865119911119908

(119905) = 119865119908119911

(119905)

(24)

This is a contradiction Therefore 119908 = 119911 and 119908 is the uniquecommon fixed point of 119891 119892 119878 and 119879

Let the control function 120601 119877+

rarr 119877+ be a continuous

nondecreasing function such that 120601(2119905) ge 2120601(119905) and 120601(1) = 1Let a function 120595 be defined by 120595 [0 1] rarr [0 1] satisfyingthe condition 120595(119902) gt 119902 for all 0 le 119902 lt 1

Theorem 14 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (25)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) (26)

where119872120601(119909 119910)

= min 120601 (119865119878119909119879119910

(119905)) 120601 (119865119878119909119891119909

(119905)) 120601 (119865119892119910119879119910

(119905))

1

2[120601 (119865119878119909119892119910

(119905)) + 120601 (119865119891119909119879119910

(119905))]

(27)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 in 119883 such that119908 = 119891119909 = 119878119909 and 119911 = 119892119910 = 119879119910 We claim that 119891119909 = 119892119910Suppose that 119891119909 = 119892119910 Then from (25) and (26) we get

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) gt 120595 (119865

119891119909119892119910(119905)) (28)

which is a contradiction Therefore 120595(119865119891119909119892119910

(119905)) = 1 whichfurther implies that 119865

119891119909119892119910(119905) = 1 Hence the claim follows

that is 119908 = 119891119909 = 119892119910 = 119911 Now from the repeated use ofcondition (26) we can show that 119891 119892 and 119878 and 119879 have aunique common fixed point

DefineG = 120601 120601 (R+)5rarr R+ such that

if 119906 isin R+ such that 119906 ge 120601 (119906 1 1 119906 119906)

119906 ge 120601 (1 119906 1 119906 119906) or 119906 ge 120601 (1 1 119906 119906 119906)

then 119906 = 1

(1198921)

Theorem 15 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (29)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

ge 120601 (119865119878119909119879119910

(119905) 119865119891119909119878119909

(119905) 119865119892119910119879119910

(119905)

119865119891119909119879119910

(119905) 119865119892119910119878119909

(119905))

(30)

Journal of Function Spaces and Applications 5

for all 119909 119910 isin 119883 then119891 119892 and 119878 and 119879 have a unique commonfixed point

Proof It follows from the given assumptions that there existsa point 119909 isin 119883 such that 119891119909 = 119878119909 and there exists anotherpoint 119910 isin 119883 for which 119892119910 = 119879119910 Suppose that 119891119909 = 119892119910 Thenfrom (30) we have

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)) (31)

Since119891119909 and 119892119910 are points of coincidence of 119891 119878 and 119892 119879respectively hence from (30) we get

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119891119909119892119910

(119905)) (32)

Therefore from (1198921) we get 119865

119891119909119892119910(119905) = 1 This shows that

119891119909 = 119892119910 Suppose that there exists another point 119911 such that119891119911 = 119878119911 Then using (30) one obtains 119891119911 = 119878119911 = 119892119910 =

119879119910 = 119891119909 = 119878119909 Hence 119908 = 119891119909 = 119891119911 is the unique pointof coincidence of 119891 and 119878 120575(119875119862(119891 119878)) = 1 This implies that119865119909119891119909

(119905) = 1 and hence 119909 = 119908 is a unique common fixedpoint of 119891 and 119878 Similarly there exists a unique point V isin 119883

such that V = 119892119911 = 119879V It then follows that V = 119908 and 119908 is acommon fixed point of 119891 119892 119878 and 119879 and 119908 is unique

4 Application to Dynamic Programming

Throughout in this section we assume that 119883 and 119884 areBanach spaces 119878 sub 119883 is a state space and119863 sub 119884 is a decisionspace We denote by 119861(119878) the set of all bounded real valuedfunctions defined on 119878

As suggested by Bellman and Lee [19] the basic form ofthe functional equations arising in dynamic programming is

119891 (119909) = opt119910119867(119909 119910 119891 (119879 (119909 119910))) (33)

where 119909 and 119910 represent the state and decision vectorsrespectively 119879 represents the transformation of the processand 119891(119909) represents the optimal return function with initialstate 119909 (here opt denotes maximum or minimum)

We now study the existence and uniqueness of a commonsolution of the following functional equations arising indynamic programming

120595 (119909) = sup119910isin119863

119867(119909 119910 120595 (119879 (119909 119910))) 119909 isin 119878

119875 (119909) = sup119910isin119863

119865 (119909 119910 119875 (119879 (119909 119910))) 119909 isin 119878

(34)

where 119879 119878 times 119863 rarr 119878119867 and 119865 119878 times 119863 timesR rarr RAs an application of Corollary 11 the existence and

uniqueness of a common solution of the functional equationsarising in dynamic programming can be established whichextends Theorem 18 [12]

Definition 16 Let 119883 be a non-empty set and 119889 a function 119889

119883 times 119883 rarr [0infin) such that

119889 (119909 119910) = 0 iff 119909 = 119910 forall119909 119910 isin 119883 (35)

Corollary 17 Let 119883 be a non-empty set and 119889 119883 times 119883 rarr

[0infin) a function satisfying condition (35) If 119891 and 119892 are JH-operators self-mappings of119883 and

119889 (119891119909 119891119910) le 120601 119889 (119892119909 119892119910) forall119909 119910 isin 119883 (36)

where 120601 R+ rarr R+ a nondecreasing function satisfying thecondition 120601(119905) lt 119905 for each 119905 gt 0 then 119891 and 119892 have a uniquecommon fixed point

Proof The proof of this corollary can be easily obtained

We now present main result of this section

Theorem 18 Suppose that the following conditions (i) (ii)(iii) and (iv) are satisfied

(i) 119867 and 119865 are bounded(ii) |119867(119909 119910 ℎ(119905))minus119867(119909 119910 119896(119905))| le 120601|119892ℎ(119905)minus119892119896(119905)| for all

(119909 119910) isin 119878 times 119863 ℎ 119896 isin 119861(119878) and 119905 isin 119878 where 120601 R+ rarr

R+ is a nondecreasing function satisfying the condition120601(119905) lt 119905 for each 119905 gt 0 and 119891 and 119892 are defined asfollows

119891ℎ (119909) = sup119910isin119863

119867(119909 119910 ℎ (119879 (119909 119910))) 119909 isin 119878 ℎ isin 119861 (119878)

119892119896 (119909) = sup119910isin119863

119865 (119909 119910 119896 (119879 (119909 119910))) 119909 isin 119878 119896 isin 119861 (119878)

(37)

(iii) If there is a point 119891119906(119909) = 119892119906(119909) = 119896(119909) for some119906(119909) isin 119861(119904) implies 119889(119896119909 119906119909) le 120575(119875119862(119891 119892))

then 119896(119909) is the unique common solution of (34)

Proof For any ℎ 119896 isin 119861(119878) let

119889 (ℎ 119896) = sup |ℎ (119909) minus 119896 (119909)| 119909 isin 119878 (38)

From conditions (i) (ii) (iii) it follows that 119891 and 119892 are self-mappings of 119861(119878)

Letting ℎ1 ℎ2be any two points of 119861(119878) 119909 isin 119878 and 120578 any

positive number then there exist 1199101 1199102isin 119863 such that

119891ℎ1(119909) lt 119867 (119909 119910

1 ℎ1(1199091)) + 120578 (39)

119891ℎ2(119909) lt 119867 (119909 119910

2 ℎ2(1199092)) + 120578 (40)

119891ℎ1(119909) ge 119867 (119909 119910

2 ℎ1(1199092)) (41)

119891ℎ2(119909) ge 119867 (119909 119910

1 ℎ2(1199091)) (42)

Subtracting (42) from (39) and using (ii) we have

119891ℎ1(119909) minus 119891ℎ

2(119909)

lt 119867 (119909 1199101 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091)) + 120578

le1003816100381610038161003816119867 (119909 119910

1 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091))1003816100381610038161003816 + 120578

le 120601 (1003816100381610038161003816119892ℎ1 (1199091) minus 119892119896

2(1199091)1003816100381610038161003816 + 120578)

le 120601 (119889 (119892ℎ1 119892ℎ2) + 120578)

(43)

6 Journal of Function Spaces and Applications

Similarly from (40) and (41) we get

119891ℎ1(119909) minus 119891ℎ

2(119909) gt 120601 (119889 (119892ℎ

1 119892ℎ2)) minus 120578 (44)

Hence1003816100381610038161003816119891ℎ1 (119909) minus 119891ℎ

2(119909)

1003816100381610038161003816 lt 120601 (119889 (119892ℎ1 119892ℎ2)) + 120578 (45)

Since (42) is true for any 119909 isin 119878 and 120578 any positive number

119889 (119891ℎ1 119891ℎ2) le 120601 (119889 (119892ℎ

1 119892ℎ2)) (46)

Therefore from Corollary 17 119896(119909) is the unique commonfixed point of 119891 and 119892 that is 119896(119909) is the unique commonsolution of functional equation (34)

Acknowledgment

Authors are grateful to referee for careful reading of this paperand for valuable comments

References

[1] K Menger ldquoStatistical metricesrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 no12 pp 535ndash537 1942

[2] M S El Naschie ldquoOn the uncertainty of Cantorian geometryand the two-slit experimentrdquo Chaos Solitons amp Fractals vol 9no 3 pp 517ndash529 1998

[3] M S El Naschie ldquoOn the unification of heterotic strings Mtheory and 120576(infin) theoryrdquo Chaos Solitons amp Fractals vol 11 no14 pp 2397ndash2408 2000

[4] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD dissertation] Wayne State UniversityDetroit Mich USA 1966

[5] VM Sehgal and A T Bharucha-Reid ldquoFixed points of contrac-tion mappings on probabilistic metric spacesrdquo MathematicalSystems Theory vol 6 pp 97ndash102 1972

[6] S Sessa ldquoOn a weak commutativity condition of mappingsin fixed point considerationsrdquo Publications de lrsquoInstitut Math-ematique vol 32(46) pp 149ndash153 1982

[7] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[8] S NMishra ldquoCommon fixed points of compatible mappings inPM-spacesrdquo Mathematica Japonica vol 36 no 2 pp 283ndash2891991

[9] G Jungck ldquoCommon fixed points for noncontinuous nonselfmaps on nonmetric spacesrdquo Far East Journal of MathematicalSciences vol 4 no 2 pp 199ndash215 1996

[10] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[11] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[12] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 73 no 1 pp 176ndash182 2010

[13] H K Pathak and N Hussain ldquoCommon fixed points for 119875-operator pair with applicationsrdquoAppliedMathematics andCom-putation vol 217 no 7 pp 3137ndash3143 2010

[14] N Hussain M A Khamsi and A Latif ldquoCommon fixed pointsfor 119869119867-operators and occasionally weakly biased pairs underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2133ndash2140 2011

[15] B Schweizer and A Skalar Statistical Metric Spaces NorthHolland Amsterdam The Netherlands 1983

[16] G Jungck and B E Rhoades ldquoFixed point theorems for occa-sionally weakly compatible mappingsrdquo Fixed Point Theory vol7 no 2 pp 286ndash296 2006

[17] G Jungck and N Hussain ldquoCompatible maps and invariantapproximationsrdquo Journal of Mathematical Analysis and Appli-cations vol 325 no 2 pp 1003ndash1012 2007

[18] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[19] B Bellman and E S Lee ldquoFunctional equation arising indynamic programmingrdquo Aequationes Mathematicae vol 17 pp1ndash18 1979

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Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

whenever 119908 is point of coincidence of 119891 119892 and

119865119891119909119891119910

(119905) gt min 119865119892119909119892119910

(119905) 119865119892119909119891119910

(119905) 119865119892119910119891119909

(119905) 119865119892119910119891119910

(119905)

(20)

for some 119909 119910 isin 119883 and 119905 gt 0 Then 119891 and 119892 have a uniquecommon fixed point

3 Section III

In this section we prove several fixed point theorems for fourself-mappings on (119883 119865) where 119865 119883 times 119883 rarr Δ satisfyingcondition (2) We begin with the following theorem

Theorem 13 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (21)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

gt min 119865119878119909119879119910

(119905) 119865119878119909119891119909

(119905) 119865119879119910119892119910

(119905) 119865119878119909119892119910

119865119879119910119891119909

(119905)

(22)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 isin 119883 such that119891119909 = 119878119909 and 119892119910 = 119879119910 Suppose that 119865

119891119909119892119910(119905) = 1 for all 119905 gt 0

Then from (22)

119865119891119909119892119910

(119905)

gt min 119865119891119909119892119910

(119905) 119865119891119909119891119909

(119905) 119865119892119910119892119910

(119905)

119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)

gt 119865119891119909119892119910

(119905)

(23)

This is a contradiction Hence 119865119891119909119892119910

(119905) = 1 for all 119905 gt 0 Thisimplies that 119891119909 = 119892119910 So 119891119909 = 119878119909 = 119892119910 = 119879119910 Moreover ifthere is another point 119911 such that 119891119911 = 119878119911 then using (22) itfollows that119891119911 = 119878119911 = 119892119910 = 119879119910 or119891119909 = 119891119911 and119908 = 119891119909 = 119878119909

is the unique point of coincidence of 119891 and 119878Thus 120575(119875119862(119891 119878)) = 1 This implies that 119865

119909119891119909(119905) = 1 and

hence 119909 = 119908 is a unique common fixed point of 119891 and 119878Similarly 119910 = 119911 is a unique fixed point of 119892 and 119879 Suppose119908 = 119911 Using (21) and (22) we get

119865119908119911

(119905) = 119865119891119908119892119911

(119905) gt min 119865119908119911

(119905) 119865119911119908

(119905) = 119865119908119911

(119905)

(24)

This is a contradiction Therefore 119908 = 119911 and 119908 is the uniquecommon fixed point of 119891 119892 119878 and 119879

Let the control function 120601 119877+

rarr 119877+ be a continuous

nondecreasing function such that 120601(2119905) ge 2120601(119905) and 120601(1) = 1Let a function 120595 be defined by 120595 [0 1] rarr [0 1] satisfyingthe condition 120595(119902) gt 119902 for all 0 le 119902 lt 1

Theorem 14 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (25)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) (26)

where119872120601(119909 119910)

= min 120601 (119865119878119909119879119910

(119905)) 120601 (119865119878119909119891119909

(119905)) 120601 (119865119892119910119879119910

(119905))

1

2[120601 (119865119878119909119892119910

(119905)) + 120601 (119865119891119909119879119910

(119905))]

(27)

for each 119909 119910 isin 119883 for which 119891119909 = 119892119910 then 119891 119892 119878 and 119879 havea unique common fixed point

Proof By hypothesis there exist points 119909 119910 in 119883 such that119908 = 119891119909 = 119878119909 and 119911 = 119892119910 = 119879119910 We claim that 119891119909 = 119892119910Suppose that 119891119909 = 119892119910 Then from (25) and (26) we get

120595 (119865119891119909119892119910

(119905)) ge 120595 (119872120601(119909 119910)) gt 120595 (119865

119891119909119892119910(119905)) (28)

which is a contradiction Therefore 120595(119865119891119909119892119910

(119905)) = 1 whichfurther implies that 119865

119891119909119892119910(119905) = 1 Hence the claim follows

that is 119908 = 119891119909 = 119892119910 = 119911 Now from the repeated use ofcondition (26) we can show that 119891 119892 and 119878 and 119879 have aunique common fixed point

DefineG = 120601 120601 (R+)5rarr R+ such that

if 119906 isin R+ such that 119906 ge 120601 (119906 1 1 119906 119906)

119906 ge 120601 (1 119906 1 119906 119906) or 119906 ge 120601 (1 1 119906 119906 119906)

then 119906 = 1

(1198921)

Theorem 15 Let 119883 be a non-empty set and 119865 119883 times 119883 rarr Δ

satisfying condition (2) Suppose that 119891 119892 119878 and 119879 are self-mappings of119883 and that the pairs 119891 119878 and 119892 119879 are each JH-operators on 119883 If

119865119911119908

(119905) = 119865119908119911

(119905) (29)

whenever119908 and 119911 are points of coincidence of 119891 119878 and 119892 119879respectively and

119865119891119909119892119910

(119905)

ge 120601 (119865119878119909119879119910

(119905) 119865119891119909119878119909

(119905) 119865119892119910119879119910

(119905)

119865119891119909119879119910

(119905) 119865119892119910119878119909

(119905))

(30)

Journal of Function Spaces and Applications 5

for all 119909 119910 isin 119883 then119891 119892 and 119878 and 119879 have a unique commonfixed point

Proof It follows from the given assumptions that there existsa point 119909 isin 119883 such that 119891119909 = 119878119909 and there exists anotherpoint 119910 isin 119883 for which 119892119910 = 119879119910 Suppose that 119891119909 = 119892119910 Thenfrom (30) we have

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)) (31)

Since119891119909 and 119892119910 are points of coincidence of 119891 119878 and 119892 119879respectively hence from (30) we get

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119891119909119892119910

(119905)) (32)

Therefore from (1198921) we get 119865

119891119909119892119910(119905) = 1 This shows that

119891119909 = 119892119910 Suppose that there exists another point 119911 such that119891119911 = 119878119911 Then using (30) one obtains 119891119911 = 119878119911 = 119892119910 =

119879119910 = 119891119909 = 119878119909 Hence 119908 = 119891119909 = 119891119911 is the unique pointof coincidence of 119891 and 119878 120575(119875119862(119891 119878)) = 1 This implies that119865119909119891119909

(119905) = 1 and hence 119909 = 119908 is a unique common fixedpoint of 119891 and 119878 Similarly there exists a unique point V isin 119883

such that V = 119892119911 = 119879V It then follows that V = 119908 and 119908 is acommon fixed point of 119891 119892 119878 and 119879 and 119908 is unique

4 Application to Dynamic Programming

Throughout in this section we assume that 119883 and 119884 areBanach spaces 119878 sub 119883 is a state space and119863 sub 119884 is a decisionspace We denote by 119861(119878) the set of all bounded real valuedfunctions defined on 119878

As suggested by Bellman and Lee [19] the basic form ofthe functional equations arising in dynamic programming is

119891 (119909) = opt119910119867(119909 119910 119891 (119879 (119909 119910))) (33)

where 119909 and 119910 represent the state and decision vectorsrespectively 119879 represents the transformation of the processand 119891(119909) represents the optimal return function with initialstate 119909 (here opt denotes maximum or minimum)

We now study the existence and uniqueness of a commonsolution of the following functional equations arising indynamic programming

120595 (119909) = sup119910isin119863

119867(119909 119910 120595 (119879 (119909 119910))) 119909 isin 119878

119875 (119909) = sup119910isin119863

119865 (119909 119910 119875 (119879 (119909 119910))) 119909 isin 119878

(34)

where 119879 119878 times 119863 rarr 119878119867 and 119865 119878 times 119863 timesR rarr RAs an application of Corollary 11 the existence and

uniqueness of a common solution of the functional equationsarising in dynamic programming can be established whichextends Theorem 18 [12]

Definition 16 Let 119883 be a non-empty set and 119889 a function 119889

119883 times 119883 rarr [0infin) such that

119889 (119909 119910) = 0 iff 119909 = 119910 forall119909 119910 isin 119883 (35)

Corollary 17 Let 119883 be a non-empty set and 119889 119883 times 119883 rarr

[0infin) a function satisfying condition (35) If 119891 and 119892 are JH-operators self-mappings of119883 and

119889 (119891119909 119891119910) le 120601 119889 (119892119909 119892119910) forall119909 119910 isin 119883 (36)

where 120601 R+ rarr R+ a nondecreasing function satisfying thecondition 120601(119905) lt 119905 for each 119905 gt 0 then 119891 and 119892 have a uniquecommon fixed point

Proof The proof of this corollary can be easily obtained

We now present main result of this section

Theorem 18 Suppose that the following conditions (i) (ii)(iii) and (iv) are satisfied

(i) 119867 and 119865 are bounded(ii) |119867(119909 119910 ℎ(119905))minus119867(119909 119910 119896(119905))| le 120601|119892ℎ(119905)minus119892119896(119905)| for all

(119909 119910) isin 119878 times 119863 ℎ 119896 isin 119861(119878) and 119905 isin 119878 where 120601 R+ rarr

R+ is a nondecreasing function satisfying the condition120601(119905) lt 119905 for each 119905 gt 0 and 119891 and 119892 are defined asfollows

119891ℎ (119909) = sup119910isin119863

119867(119909 119910 ℎ (119879 (119909 119910))) 119909 isin 119878 ℎ isin 119861 (119878)

119892119896 (119909) = sup119910isin119863

119865 (119909 119910 119896 (119879 (119909 119910))) 119909 isin 119878 119896 isin 119861 (119878)

(37)

(iii) If there is a point 119891119906(119909) = 119892119906(119909) = 119896(119909) for some119906(119909) isin 119861(119904) implies 119889(119896119909 119906119909) le 120575(119875119862(119891 119892))

then 119896(119909) is the unique common solution of (34)

Proof For any ℎ 119896 isin 119861(119878) let

119889 (ℎ 119896) = sup |ℎ (119909) minus 119896 (119909)| 119909 isin 119878 (38)

From conditions (i) (ii) (iii) it follows that 119891 and 119892 are self-mappings of 119861(119878)

Letting ℎ1 ℎ2be any two points of 119861(119878) 119909 isin 119878 and 120578 any

positive number then there exist 1199101 1199102isin 119863 such that

119891ℎ1(119909) lt 119867 (119909 119910

1 ℎ1(1199091)) + 120578 (39)

119891ℎ2(119909) lt 119867 (119909 119910

2 ℎ2(1199092)) + 120578 (40)

119891ℎ1(119909) ge 119867 (119909 119910

2 ℎ1(1199092)) (41)

119891ℎ2(119909) ge 119867 (119909 119910

1 ℎ2(1199091)) (42)

Subtracting (42) from (39) and using (ii) we have

119891ℎ1(119909) minus 119891ℎ

2(119909)

lt 119867 (119909 1199101 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091)) + 120578

le1003816100381610038161003816119867 (119909 119910

1 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091))1003816100381610038161003816 + 120578

le 120601 (1003816100381610038161003816119892ℎ1 (1199091) minus 119892119896

2(1199091)1003816100381610038161003816 + 120578)

le 120601 (119889 (119892ℎ1 119892ℎ2) + 120578)

(43)

6 Journal of Function Spaces and Applications

Similarly from (40) and (41) we get

119891ℎ1(119909) minus 119891ℎ

2(119909) gt 120601 (119889 (119892ℎ

1 119892ℎ2)) minus 120578 (44)

Hence1003816100381610038161003816119891ℎ1 (119909) minus 119891ℎ

2(119909)

1003816100381610038161003816 lt 120601 (119889 (119892ℎ1 119892ℎ2)) + 120578 (45)

Since (42) is true for any 119909 isin 119878 and 120578 any positive number

119889 (119891ℎ1 119891ℎ2) le 120601 (119889 (119892ℎ

1 119892ℎ2)) (46)

Therefore from Corollary 17 119896(119909) is the unique commonfixed point of 119891 and 119892 that is 119896(119909) is the unique commonsolution of functional equation (34)

Acknowledgment

Authors are grateful to referee for careful reading of this paperand for valuable comments

References

[1] K Menger ldquoStatistical metricesrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 no12 pp 535ndash537 1942

[2] M S El Naschie ldquoOn the uncertainty of Cantorian geometryand the two-slit experimentrdquo Chaos Solitons amp Fractals vol 9no 3 pp 517ndash529 1998

[3] M S El Naschie ldquoOn the unification of heterotic strings Mtheory and 120576(infin) theoryrdquo Chaos Solitons amp Fractals vol 11 no14 pp 2397ndash2408 2000

[4] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD dissertation] Wayne State UniversityDetroit Mich USA 1966

[5] VM Sehgal and A T Bharucha-Reid ldquoFixed points of contrac-tion mappings on probabilistic metric spacesrdquo MathematicalSystems Theory vol 6 pp 97ndash102 1972

[6] S Sessa ldquoOn a weak commutativity condition of mappingsin fixed point considerationsrdquo Publications de lrsquoInstitut Math-ematique vol 32(46) pp 149ndash153 1982

[7] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[8] S NMishra ldquoCommon fixed points of compatible mappings inPM-spacesrdquo Mathematica Japonica vol 36 no 2 pp 283ndash2891991

[9] G Jungck ldquoCommon fixed points for noncontinuous nonselfmaps on nonmetric spacesrdquo Far East Journal of MathematicalSciences vol 4 no 2 pp 199ndash215 1996

[10] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[11] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[12] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 73 no 1 pp 176ndash182 2010

[13] H K Pathak and N Hussain ldquoCommon fixed points for 119875-operator pair with applicationsrdquoAppliedMathematics andCom-putation vol 217 no 7 pp 3137ndash3143 2010

[14] N Hussain M A Khamsi and A Latif ldquoCommon fixed pointsfor 119869119867-operators and occasionally weakly biased pairs underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2133ndash2140 2011

[15] B Schweizer and A Skalar Statistical Metric Spaces NorthHolland Amsterdam The Netherlands 1983

[16] G Jungck and B E Rhoades ldquoFixed point theorems for occa-sionally weakly compatible mappingsrdquo Fixed Point Theory vol7 no 2 pp 286ndash296 2006

[17] G Jungck and N Hussain ldquoCompatible maps and invariantapproximationsrdquo Journal of Mathematical Analysis and Appli-cations vol 325 no 2 pp 1003ndash1012 2007

[18] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[19] B Bellman and E S Lee ldquoFunctional equation arising indynamic programmingrdquo Aequationes Mathematicae vol 17 pp1ndash18 1979

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 Function Spaces and Applications 5

for all 119909 119910 isin 119883 then119891 119892 and 119878 and 119879 have a unique commonfixed point

Proof It follows from the given assumptions that there existsa point 119909 isin 119883 such that 119891119909 = 119878119909 and there exists anotherpoint 119910 isin 119883 for which 119892119910 = 119879119910 Suppose that 119891119909 = 119892119910 Thenfrom (30) we have

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119892119910119891119909

(119905)) (31)

Since119891119909 and 119892119910 are points of coincidence of 119891 119878 and 119892 119879respectively hence from (30) we get

119865119891119909119892119910

(119905) ge 120601 (119865119891119909119892119910

(119905) 0 0 119865119891119909119892119910

(119905) 119865119891119909119892119910

(119905)) (32)

Therefore from (1198921) we get 119865

119891119909119892119910(119905) = 1 This shows that

119891119909 = 119892119910 Suppose that there exists another point 119911 such that119891119911 = 119878119911 Then using (30) one obtains 119891119911 = 119878119911 = 119892119910 =

119879119910 = 119891119909 = 119878119909 Hence 119908 = 119891119909 = 119891119911 is the unique pointof coincidence of 119891 and 119878 120575(119875119862(119891 119878)) = 1 This implies that119865119909119891119909

(119905) = 1 and hence 119909 = 119908 is a unique common fixedpoint of 119891 and 119878 Similarly there exists a unique point V isin 119883

such that V = 119892119911 = 119879V It then follows that V = 119908 and 119908 is acommon fixed point of 119891 119892 119878 and 119879 and 119908 is unique

4 Application to Dynamic Programming

Throughout in this section we assume that 119883 and 119884 areBanach spaces 119878 sub 119883 is a state space and119863 sub 119884 is a decisionspace We denote by 119861(119878) the set of all bounded real valuedfunctions defined on 119878

As suggested by Bellman and Lee [19] the basic form ofthe functional equations arising in dynamic programming is

119891 (119909) = opt119910119867(119909 119910 119891 (119879 (119909 119910))) (33)

where 119909 and 119910 represent the state and decision vectorsrespectively 119879 represents the transformation of the processand 119891(119909) represents the optimal return function with initialstate 119909 (here opt denotes maximum or minimum)

We now study the existence and uniqueness of a commonsolution of the following functional equations arising indynamic programming

120595 (119909) = sup119910isin119863

119867(119909 119910 120595 (119879 (119909 119910))) 119909 isin 119878

119875 (119909) = sup119910isin119863

119865 (119909 119910 119875 (119879 (119909 119910))) 119909 isin 119878

(34)

where 119879 119878 times 119863 rarr 119878119867 and 119865 119878 times 119863 timesR rarr RAs an application of Corollary 11 the existence and

uniqueness of a common solution of the functional equationsarising in dynamic programming can be established whichextends Theorem 18 [12]

Definition 16 Let 119883 be a non-empty set and 119889 a function 119889

119883 times 119883 rarr [0infin) such that

119889 (119909 119910) = 0 iff 119909 = 119910 forall119909 119910 isin 119883 (35)

Corollary 17 Let 119883 be a non-empty set and 119889 119883 times 119883 rarr

[0infin) a function satisfying condition (35) If 119891 and 119892 are JH-operators self-mappings of119883 and

119889 (119891119909 119891119910) le 120601 119889 (119892119909 119892119910) forall119909 119910 isin 119883 (36)

where 120601 R+ rarr R+ a nondecreasing function satisfying thecondition 120601(119905) lt 119905 for each 119905 gt 0 then 119891 and 119892 have a uniquecommon fixed point

Proof The proof of this corollary can be easily obtained

We now present main result of this section

Theorem 18 Suppose that the following conditions (i) (ii)(iii) and (iv) are satisfied

(i) 119867 and 119865 are bounded(ii) |119867(119909 119910 ℎ(119905))minus119867(119909 119910 119896(119905))| le 120601|119892ℎ(119905)minus119892119896(119905)| for all

(119909 119910) isin 119878 times 119863 ℎ 119896 isin 119861(119878) and 119905 isin 119878 where 120601 R+ rarr

R+ is a nondecreasing function satisfying the condition120601(119905) lt 119905 for each 119905 gt 0 and 119891 and 119892 are defined asfollows

119891ℎ (119909) = sup119910isin119863

119867(119909 119910 ℎ (119879 (119909 119910))) 119909 isin 119878 ℎ isin 119861 (119878)

119892119896 (119909) = sup119910isin119863

119865 (119909 119910 119896 (119879 (119909 119910))) 119909 isin 119878 119896 isin 119861 (119878)

(37)

(iii) If there is a point 119891119906(119909) = 119892119906(119909) = 119896(119909) for some119906(119909) isin 119861(119904) implies 119889(119896119909 119906119909) le 120575(119875119862(119891 119892))

then 119896(119909) is the unique common solution of (34)

Proof For any ℎ 119896 isin 119861(119878) let

119889 (ℎ 119896) = sup |ℎ (119909) minus 119896 (119909)| 119909 isin 119878 (38)

From conditions (i) (ii) (iii) it follows that 119891 and 119892 are self-mappings of 119861(119878)

Letting ℎ1 ℎ2be any two points of 119861(119878) 119909 isin 119878 and 120578 any

positive number then there exist 1199101 1199102isin 119863 such that

119891ℎ1(119909) lt 119867 (119909 119910

1 ℎ1(1199091)) + 120578 (39)

119891ℎ2(119909) lt 119867 (119909 119910

2 ℎ2(1199092)) + 120578 (40)

119891ℎ1(119909) ge 119867 (119909 119910

2 ℎ1(1199092)) (41)

119891ℎ2(119909) ge 119867 (119909 119910

1 ℎ2(1199091)) (42)

Subtracting (42) from (39) and using (ii) we have

119891ℎ1(119909) minus 119891ℎ

2(119909)

lt 119867 (119909 1199101 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091)) + 120578

le1003816100381610038161003816119867 (119909 119910

1 ℎ1(1199091)) minus 119867 (119909 119910

1 ℎ2(1199091))1003816100381610038161003816 + 120578

le 120601 (1003816100381610038161003816119892ℎ1 (1199091) minus 119892119896

2(1199091)1003816100381610038161003816 + 120578)

le 120601 (119889 (119892ℎ1 119892ℎ2) + 120578)

(43)

6 Journal of Function Spaces and Applications

Similarly from (40) and (41) we get

119891ℎ1(119909) minus 119891ℎ

2(119909) gt 120601 (119889 (119892ℎ

1 119892ℎ2)) minus 120578 (44)

Hence1003816100381610038161003816119891ℎ1 (119909) minus 119891ℎ

2(119909)

1003816100381610038161003816 lt 120601 (119889 (119892ℎ1 119892ℎ2)) + 120578 (45)

Since (42) is true for any 119909 isin 119878 and 120578 any positive number

119889 (119891ℎ1 119891ℎ2) le 120601 (119889 (119892ℎ

1 119892ℎ2)) (46)

Therefore from Corollary 17 119896(119909) is the unique commonfixed point of 119891 and 119892 that is 119896(119909) is the unique commonsolution of functional equation (34)

Acknowledgment

Authors are grateful to referee for careful reading of this paperand for valuable comments

References

[1] K Menger ldquoStatistical metricesrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 no12 pp 535ndash537 1942

[2] M S El Naschie ldquoOn the uncertainty of Cantorian geometryand the two-slit experimentrdquo Chaos Solitons amp Fractals vol 9no 3 pp 517ndash529 1998

[3] M S El Naschie ldquoOn the unification of heterotic strings Mtheory and 120576(infin) theoryrdquo Chaos Solitons amp Fractals vol 11 no14 pp 2397ndash2408 2000

[4] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD dissertation] Wayne State UniversityDetroit Mich USA 1966

[5] VM Sehgal and A T Bharucha-Reid ldquoFixed points of contrac-tion mappings on probabilistic metric spacesrdquo MathematicalSystems Theory vol 6 pp 97ndash102 1972

[6] S Sessa ldquoOn a weak commutativity condition of mappingsin fixed point considerationsrdquo Publications de lrsquoInstitut Math-ematique vol 32(46) pp 149ndash153 1982

[7] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[8] S NMishra ldquoCommon fixed points of compatible mappings inPM-spacesrdquo Mathematica Japonica vol 36 no 2 pp 283ndash2891991

[9] G Jungck ldquoCommon fixed points for noncontinuous nonselfmaps on nonmetric spacesrdquo Far East Journal of MathematicalSciences vol 4 no 2 pp 199ndash215 1996

[10] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[11] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[12] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 73 no 1 pp 176ndash182 2010

[13] H K Pathak and N Hussain ldquoCommon fixed points for 119875-operator pair with applicationsrdquoAppliedMathematics andCom-putation vol 217 no 7 pp 3137ndash3143 2010

[14] N Hussain M A Khamsi and A Latif ldquoCommon fixed pointsfor 119869119867-operators and occasionally weakly biased pairs underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2133ndash2140 2011

[15] B Schweizer and A Skalar Statistical Metric Spaces NorthHolland Amsterdam The Netherlands 1983

[16] G Jungck and B E Rhoades ldquoFixed point theorems for occa-sionally weakly compatible mappingsrdquo Fixed Point Theory vol7 no 2 pp 286ndash296 2006

[17] G Jungck and N Hussain ldquoCompatible maps and invariantapproximationsrdquo Journal of Mathematical Analysis and Appli-cations vol 325 no 2 pp 1003ndash1012 2007

[18] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[19] B Bellman and E S Lee ldquoFunctional equation arising indynamic programmingrdquo Aequationes Mathematicae vol 17 pp1ndash18 1979

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

6 Journal of Function Spaces and Applications

Similarly from (40) and (41) we get

119891ℎ1(119909) minus 119891ℎ

2(119909) gt 120601 (119889 (119892ℎ

1 119892ℎ2)) minus 120578 (44)

Hence1003816100381610038161003816119891ℎ1 (119909) minus 119891ℎ

2(119909)

1003816100381610038161003816 lt 120601 (119889 (119892ℎ1 119892ℎ2)) + 120578 (45)

Since (42) is true for any 119909 isin 119878 and 120578 any positive number

119889 (119891ℎ1 119891ℎ2) le 120601 (119889 (119892ℎ

1 119892ℎ2)) (46)

Therefore from Corollary 17 119896(119909) is the unique commonfixed point of 119891 and 119892 that is 119896(119909) is the unique commonsolution of functional equation (34)

Acknowledgment

Authors are grateful to referee for careful reading of this paperand for valuable comments

References

[1] K Menger ldquoStatistical metricesrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 no12 pp 535ndash537 1942

[2] M S El Naschie ldquoOn the uncertainty of Cantorian geometryand the two-slit experimentrdquo Chaos Solitons amp Fractals vol 9no 3 pp 517ndash529 1998

[3] M S El Naschie ldquoOn the unification of heterotic strings Mtheory and 120576(infin) theoryrdquo Chaos Solitons amp Fractals vol 11 no14 pp 2397ndash2408 2000

[4] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD dissertation] Wayne State UniversityDetroit Mich USA 1966

[5] VM Sehgal and A T Bharucha-Reid ldquoFixed points of contrac-tion mappings on probabilistic metric spacesrdquo MathematicalSystems Theory vol 6 pp 97ndash102 1972

[6] S Sessa ldquoOn a weak commutativity condition of mappingsin fixed point considerationsrdquo Publications de lrsquoInstitut Math-ematique vol 32(46) pp 149ndash153 1982

[7] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[8] S NMishra ldquoCommon fixed points of compatible mappings inPM-spacesrdquo Mathematica Japonica vol 36 no 2 pp 283ndash2891991

[9] G Jungck ldquoCommon fixed points for noncontinuous nonselfmaps on nonmetric spacesrdquo Far East Journal of MathematicalSciences vol 4 no 2 pp 199ndash215 1996

[10] M Aamri and D El Moutawakil ldquoSome new common fixedpoint theorems under strict contractive conditionsrdquo Journal ofMathematical Analysis and Applications vol 270 no 1 pp 181ndash188 2002

[11] M A Al-Thagafi andN Shahzad ldquoGeneralized 119868-nonexpansiveselfmaps and invariant approximationsrdquo Acta MathematicaSinica vol 24 no 5 pp 867ndash876 2008

[12] A Bhatt H Chandra and D R Sahu ldquoCommon fixed pointtheorems for occasionally weakly compatible mappings underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 73 no 1 pp 176ndash182 2010

[13] H K Pathak and N Hussain ldquoCommon fixed points for 119875-operator pair with applicationsrdquoAppliedMathematics andCom-putation vol 217 no 7 pp 3137ndash3143 2010

[14] N Hussain M A Khamsi and A Latif ldquoCommon fixed pointsfor 119869119867-operators and occasionally weakly biased pairs underrelaxed conditionsrdquo Nonlinear Analysis Theory Methods ampApplications vol 74 no 6 pp 2133ndash2140 2011

[15] B Schweizer and A Skalar Statistical Metric Spaces NorthHolland Amsterdam The Netherlands 1983

[16] G Jungck and B E Rhoades ldquoFixed point theorems for occa-sionally weakly compatible mappingsrdquo Fixed Point Theory vol7 no 2 pp 286ndash296 2006

[17] G Jungck and N Hussain ldquoCompatible maps and invariantapproximationsrdquo Journal of Mathematical Analysis and Appli-cations vol 325 no 2 pp 1003ndash1012 2007

[18] W A Wilson ldquoOn semi-metric spacesrdquo American Journal ofMathematics vol 53 no 2 pp 361ndash373 1931

[19] B Bellman and E S Lee ldquoFunctional equation arising indynamic programmingrdquo Aequationes Mathematicae vol 17 pp1ndash18 1979

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