Common fixed point theorems of integral type contraction on metric spaces and its applications to system of functional equations

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1 Sarwar et al Fixed Point Theory and Applications :217 DOI 11186/s R E S E A R C H Open Access Common fixed point theorems of integral type contraction on metric spaces and its applications to system of functional equations Muhammad Sarwar 1 Mian Bahadur Zada 1* and İnci M Erhan 2 * Correspondence: mbzmath@gmailcom 1 Department of Mathematics University of Malakand Chakdara DirL Pakistan Full list of author information is available at the end of the article Abstract In this article using the common CLR property common fixed point results for two pairs of weakly compatible mappings satisfying contractive condition of integral type on metric spaces are established Furthermore the existence and uniqueness of common solution for system of functional equations arising in dynamic programming are discussed as an application of a common fixed point theorem presented in this paper MSC: 47H1; 54H25 Keywords: contractive mappings of integral type; weakly compatible mappings; common fixed point; system of functional equations; dynamic programming; common EA property; common CLRproperty 1 Introduction and preliminaries Throughout this paper we assume that R + =[+ opt stands for sup or inf Z and Y are Banach spaces S Z is the state space D Y is the decision space BS denotesthe Banach space of all bounded real-valued functions on S with norm w = sup wx : x S for any w BS and u v : S D R; a i : S D S; H i : S D R RAlso and = ϕ : ϕ : R + R + is Lebesgue integrable with finite integral such that ε ϕt dt >foreachε > = ψ : ψ : R + R + is upper semi-continuous on R + \ ψ = and ψt<tforeacht > 215 Sarwar et al This article is distributed under the terms of the Creative Commons Attribution 4 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made

2 Sarwar et al Fixed Point Theory and Applications :217 Page 2 of 15 Fixed point theory is one of the most fruitful and applicable topics of nonlinear analysis which is widely used not only in other mathematical theories but also in many practical problems of natural sciences and engineering The Banach contraction mapping principle [1] is indeed the most popular result of metric fixed point theory This principle has many application in several domains such as differential equations functional equations integral equations economics wild life and several others Branciari [2] gave an integral version of the Banach contraction principles and proved fixed point theorem for a single-valued contractive mapping of integral type in metric space Afterwards many researchers [3 18] extended the result of Branciari and obtained fixed point and common fixed point theorems for various contractive conditions of integral type on different spaces In particular Liu et al [9] studied fixed point theorems satisfying a contractive condition of integral type and applied their results for the existence and uniqueness of a solution to the following functional equation: f x=opt y D ux y+h1 x y f1 a1 x y x Z 11 Further Liu et al [1] established common fixed point theorems satisfying contractive condition of integral type and applied their results for the existence and uniqueness of common solution to the following system of functional equations: f x=opt y D ux y+h1 x y f1 a1 x y x S gx=opt y D vx y+h2 x y f2 a2 x y 12 x S where x and y signify the state and decision vectors respectively a 1 and a 2 represent the transformations of the process f 1 x andf 2 x denote the optimal return functions with the initial state x The aim of this contribution is to study the existence and uniqueness of common solution for the system of functional equations arising in dynamic programming with the help of common fixed point results satisfying the contractive conditions of integral type in metric space Now we recollect some known definitions and results from the literature which are helpful in the proof of our main results Definition 11 A coincidence point of a pair of self-mapping K L : X X is a point x X for which Kx = Lx A common fixed point of a pair of self-mapping K L : X X is a point x X for which Kx = Lx = x Jungck [19] initiated the concept of weakly compatible maps to study common fixed point theorems Definition 12 [19] A pair of self-mapping K L : X X is weakly compatible if they commute at their coincidence points that is if there exists a point x X such that KLx = LKx whenever Kx = Lx In the study of common fixed points of weakly compatible mappings we often require the assumption of completeness of the space or subspace or continuity of mappings in-

3 Sarwar et al Fixed Point Theory and Applications :217 Page 3 of 15 volved besides some contractive condition Aamri and El Moutawakil [2]introducedthe notion of EA property which requires only the closedness of the subspace and Liu et al [21]extendedtheEA property to common the EApropertyasfollows Definition 13 Let X d be a metric space and K L M N : X X be four self-maps The pairs K M andl N satisfy the common EA property if there exist two sequences x n and y n in X such that lim n Kx n = lim n Mx n = lim n Ly n = lim n Ny n = t X Sintunavarat and Kumam [22] introduced the notion of the CLRpropertywhichnever requires any condition on closedness of the space or subspace and Imdad et al [23]introduced the common CLR propertywhich is an extension of the CLRproperty Definition 14 Let X dbeametricspaceandk L M N : X X be four self maps The pairs K M andl N satisfy the common limit range property with respect to mappings M and NdenotedbyCLR MN ifthereexisttwosequencesx n and y n in X such that lim n Kx n = lim n Mx n = lim n Ly n = lim n Ny n = t MX NX Finally we will need the following results Lemma 11 [9] Let ϕ and r n n N be a non-negative sequence with lim n r n = a Then rn lim n ϕt dt = a ϕt dt Lemma 12 [24] Let E be a set and p q : E R be mappings If opt y E py and opt y E qy are bounded then opty E py opt y E qy sup py qy y E 2 Common fixed point theorems In this section we study common fixed point theorems for weakly compatible mappings using the common CLR and common EA properties Theorem 21 Let X d be a metric space and K L N M : X X be four self-mappings satisfying the following conditions: 1 the pairs K N and L M share CLR NM property; 2 dkxly where ϕ ψ and 1 xy ϕt dt ψ ϕt dt x y X

4 Sarwar et al Fixed Point Theory and Applications :217 Page 4 of 15 1 x y= max dnx My dnx Kx dmy Ly dkx My+dLy Nx dkx NxdLy My 1+dNx My dkx MydLy Nx 1+dNx My 1+dNx Ly+dMy Kx dnx Kx 1+dNx Kx +dmy Ly If the pairs K N and L M are weakly compatible then K L M and N have a unique common fixed point in X Proof Assume that the pairs K N andl M sharetheclr NM property then there exist two sequences x n and y n in X such that lim Kx n = lim Nx n = lim Ly n = lim My n = z for some z MX NX 21 n n n n Since z NX there exists a point u X such that Nu = zthus21becomes lim n Kx n = lim n Nx n = lim n Ly n = lim n My n = z = Nu 22 Now we claim that Ku = NuToprovetheclaimletKu NuThenonputtingx = u and y = y n in condition 2 of Theorem 21wehave dkulyn 1 uy n ϕt dt ψ ϕt dt 23 where 1 u y n = max dnu My n dnu Ku dmy n Ly n 1 [ dku Myn +dly n Nu ] dku NudLy n My n 2 1+dNu My n dku My n dly n Nu dnu Ku 1+dNu Ly n+dmy n Ku 1+dNu My n 1+dNu Ku+dMy n Ly n Taking the upper limit as n in equations 24and23 respectively we have lim 1u y n =max dz Ku 1 [ ] dku z dz Ku = dku z n 2 and dkuz ϕt dt = lim sup n dkulyn lim sup ψ n = ψ < dkuz dkuz ϕt dt 1 uy n ϕt dt ψ lim sup n ϕt dt ϕt dt 1 uy n ϕt dt 24

5 Sarwar et al Fixed Point Theory and Applications :217 Page 5 of 15 which is a contradiction thus Ku = Nu and hence Ku = Nu = z 25 Similarly since z MX so there exists a point v X such that Mv = zthus21becomes lim Kx n = lim Nx n = lim Ly n = lim My n = z = Mv 26 n n n n Now we claim that Lv = MvTosupporttheclaimletLv MvThenonputtingx = x n and y = v in condition 2 of Theorem 21onecanget Lv = Mv = z 27 Therefore from 25 and27 one can write Ku = Nu = Lv = Mv = z 28 Next we show that z is a common fixed point of K L M andn Tothisaimsincethe pairs K NandL M are weakly compatible then using 28wehave Ku = Nu NKu = KNu Kz = Nz 29 and Lv = Mv MLv = LMv Lz = Mz 21 We will show next that Kz = z OtherwiseifKz z using condition 2 of Theorem 21 with x = z and y = vwehave dkzlv 1 zv ϕt dt ψ ϕt dt where 1 z v= max dnz Mv dnz Kz dmv Lv 1 [ ] dkz NzdLv Mv dkz Mv+dLv Nz 2 1+dNz Mv dkz MvdLv Nz dnz Kz 1+dNz Mv 1+dNz Lv+dMv Kz 1+dNz Kz +dmv Lv In the light of 28and29 we get 1 z v=max dkz z 1 [ ] dkz zdz Kz dkz z+dz Kz 2 1+dKz z = dkz z

6 Sarwar et al Fixed Point Theory and Applications :217 Page 6 of 15 and dkzz dkzz ϕt dt ψ ϕt dt < dkzz ϕt dt which is a contradiction Thus Kz = z and from 29 we can write Kz = Nz = z 211 Similarly setting x = u y = z in condition 2 of Theorem 21 and using one can get Lz = Mz = z 212 Therefore from 211and212 it followsthat Kz = Lz = Mz = Nz = z 213 that is z is a common fixed point of K L MandN Finally we prove the uniqueness of the common fixed point of K L MandNAssume that z 1 and z 2 are two distinct common fixed points of K L M andn Thenreplacingx by z 1 and y by z 2 in condition 2 of Theorem 21wehave dz1 z 2 dkz1 Lz 2 1 z 1 z 2 ϕt dt = ϕt dt ψ ϕt dt where 1 z 1 z 2 = max dnz 1 Mz 2 dnz 1 Kz 1 dmz 2 Lz 2 1 [ dkz1 Mz 2 +dlz 2 Nz 1 ] dkz 1 Nz 1 dlz 2 Mz dNz 1 Mz 2 dkz 1 Mz 2 dlz 2 Nz 1 dnz 1 Kz 1 1+dNz 1 Lz 2 +dmz 2 Kz 1 1+dNz 1 Mz 2 1+dNz 1 Kz 1 +dmz 2 Lz 2 = max dz 1 z = dz 1 z 2 [ dz1 z 2 +dz 2 z 1 ] dz 1 z 2 dz 2 z 1 1+dz 1 z 2 so that dz1 z 2 dz1 z 2 ϕt dt ψ ϕt dt < dz1 z 2 ϕt dt which is a contradiction and thus z 1 = z 2 HenceK L M andn have a unique common fixed point in X From Theorem 21 we easily deduce the following corollaries

7 Sarwar et al Fixed Point Theory and Applications :217 Page 7 of 15 Corollary 21 Let X d be a metric space and K N M : X X be three self-mappings satisfying the following conditions: 1 the pairs K N and K M share CKR NM property; 2 dkxky 1 xy ϕt dt ψ ϕt dt x y X where ϕ ψ and 1 x y= max dnx My dnx Kx dmy Ky dkx My+dKy Nx dkx NxdKy My 1+dNx My dkx MydKy Nx 1+dNx My 1+dNx Ky+dMy Kx dnx Kx 1+dNx Kx +dmy Ky If the pairs K N and K M are weakly compatible then K M and N have a unique common fixed point in X Corollary 22 Let X d be a metric space and K M : X X be two self-mappings satisfying the following conditions: 1 the pair K M satisfies the CLR M property; 2 dkxky 1 xy ϕt dt ψ ϕt dt x y X where ϕ ψ and 1 x y= max dmx My dmx Kx dmy Ky dkx My+dKy Mx dkx MxdKy My 1+dMx My dkx MydKy Mx 1+dMx My 1+dMx Ky+dMy Kx dmx Kx 1+dMx Kx +dmy Ky If the pair K M is weakly compatible then K and M have a unique common fixed point in X In a similar way to Theorem 21 the following result can be concluded and proved Theorem 22 Let X d be a metric space and K L N M : X X be four self-mappings satisfying the following conditions: 1 the pairs K N and L M share CLR NM property; 2 dkxly 2 xy ϕt dt ψ ϕt dt x y X

8 Sarwar et al Fixed Point Theory and Applications :217 Page 8 of 15 where ϕ ψ and 2 x y= max dnx My dnx Kx dmy Ly dkx My+dLy Nx dkx NxdLy My 1+dKx Ly dkx MydLy Nx 1+dKx Ly 1+dNx Ly+dMy Kx dnx Kx 1+dNx Kx +dmy Ly If the pairs K N and L M are weakly compatible then K L M and N have a unique common fixed point in X Obviously the CLR MN property implies the common property EAbuttheconverse is not true in general So replacing the CLR MN propertybythecommonpropertyeain Theorem 21and Theorem22 we get thefollowingresults theproofsofwhichcaneasily be done by following the lines of the proof of Theorem 21 becausetheea property together with the closedness property of a suitable subspace gives rise to the closed range property Corollary 23 Let X d be a metric space and K L N M : X X be four self-mappings satisfying the following conditions: 1 the pairs K N and L M share common EA property such that MX or NX is closed subspace of X; 2 dkxly 1 xy ϕt dt ψ ϕt dt x y X where ϕ ψ and 1 x y= max dnx My dnx Kx dmy Ly dkx My+dLy Nx dkx NxdLy My 1+dNx My dkx MydLy Nx 1+dNx My 1+dNx Ly+dMy Kx dnx Kx 1+dNx Kx +dmy Ly If the pairs K N and L M are weakly compatible then K L M and N have a unique common fixed point in X Corollary 24 Let X d be a metric space and K L N M : X X be four self-mappings satisfying the following conditions: 1 the pairs K N and L M share common EA property such that MX or NX is closed subspace of X; 2 dkxly 2 xy ϕt dt ψ ϕt dt x y X

9 Sarwar et al Fixed Point Theory and Applications :217 Page 9 of 15 where ϕ ψ and 2 x y= max dnx My dnx Kx dmy Ly dkx My+dLy Nx dkx NxdLy My 1+dKx Ly dkx MydLy Nx 1+dKx Ly 1+dNx Ly+dMy Kx dnx Kx 1+dNx Kx +dmy Ly If the pairs K N and L M are weakly compatible then K L M and N have a unique common fixed point in X One can obtain further consequences from Theorem 22 and Corollaries 23 and 24 in a similar way to Theorem 21 Remark 21 Theorem 21 and Corollary 23 are still valid if we replace 1 x yby 3 x y= max dnx My dnx Kx dmy Ly dkx My+dLy Nx dkx NxdLy My min 1+dNx My dkx MydLy Nx 1+dNx My 1+dNx Ly+dMy Kx dnx Kx 1+dNx Kx +dmy Ly Similarly Theorem 22 and Corollary 24 are still valid if we replace 1 x yby 4 x y= max dnx My dnx Kx dmy Ly dkx My+dLy Nx dkx NxdLy My min 1+dKx Ly dkx MydLy Nx 1+dKx Ly 1+dNx Ly+dMy Kx dnx Kx 1+dNx Kx +dmy Ly Finally by choosing K = L and N and M as identity mappings we conclude some fixed point theorems for integral type contraction from our main Theorem 21 which can be listed as follows Corollary 25 Let X d be a metric space and K : X X be a self-mapping satisfying the condition dkxky 1 xy ϕt dt ψ ϕt dt x y X where ϕ ψ and 1 x y= max dx y dx Kx dy Ky dkx y+dky x dkx xdky y 1+dx y dkx ydky x 1+dx y for all x y X Then K has a unique fixed point in X 1+dx Ky+dy Kx dx Kx 1+dx Kx +dy Ky

10 Sarwar et al Fixed Point Theory and Applications :217 Page 1 of 15 Corollary 26 Let X d be a metric space and K : X X be a self-mapping satisfying the condition dkxky 2 xy ϕt dt ψ ϕt dt x y X where ϕ ψ and 2 x y= max dx y dx Kx dy Ky dkx y+dky x dkx xdky y 1+dKx y dkx ydky x 1+dKx y for all x y X Then K has a unique fixed point in X 1+dx Ky+dy Kx dx Kx 1+dx Kx +dy Ky Remark 22 Notice that several fixed point theorems such as the celebrated Banach fixed point theorem fixed point theorems for Kannan Chatterjee and Reich type mappings and others can be deduced as particular cases of Corollary 25 To illustrate Theorem 21 we construct the following example Example 21 Let X =2beametricspacewithmetricdx y= x y wherex y X and K L M N be self-maps of Xdefinedby Kx = Mx = 1 ifx 1] 1 if x 1 2 ; Lx = 6 1 ifx 1] 1 2 if x 1 2 and Nx = 1 ifx 1] 1 if x ifx 1] 1 3 if x 1 2 First we verify condition 1 of Theorem 21 Tothisaimletx n = n n+1 n 1 and y n = 1 n+1 n 1 be two sequences in XThen Thus n lim Kx n = lim K n n lim Ly n = lim L n n lim My n = lim M n n lim Nx n = lim N n n n +1 1 n +1 1 n +1 n n +1 =1; =1; =1; =1 lim Kx n = lim Nx n = lim Ly n = lim My n =1 MX NX n n n n That is K NandK M satisfies the common CLR MN property

11 Sarwar et al Fixed Point Theory and Applications :217 Page 11 of 15 Next to verify condition 2 of Theorem 21 let us define ϕ : R + R + by ϕt=2t and ψ : R + R + by ψt= t 5 If x y 1] Then Kx = Ly = My = Nx =1and dkxly 1 xy ϕt dt ==ψ ϕt dt where 1 x y= If x y 1 2 Then Kx = 1 6 Ly = 1 8 My = 1 2 Nx = 1 3 and dkxly 1 24 ϕt dt = 2tdt= t = Also 1 1 x y=max = Thus we obtain 1 xy 3 8 ψ ϕt dt = ψ 2tdt = ψ t = 32 > Hence from the above two cases it follows that dkxly 1 xy ϕt dt ψ ϕt dt x y X dkxly ϕt dt Therefore from Theorem 21 K L M andn have a unique common fixed point which is x =1 3 Applications to existence theorems for functional equations arising in dynamic programming In this section an attempt is made to find the existence and uniqueness of a common solution for a system of functional equations arising in dynamic programming through the help of Theorem 21 Consider the system f 1 x=opt y D ux y+h1 x y f1 a1 x y x S f 2 x=opt y D ux y+h2 x y f2 a2 x y x S f 3 x=opt y D vx y+h3 x y f3 a3 x y x S f 4 x=opt y D vx y+h4 x y f4 a4 x y x S 31 where x and y signify the state and decision vectors respectively a 1 a 2 a 3 anda 4 represent the transformations of the process f 1 x f 2 x f 3 x and f 4 x denote the optimal return functions with the initial state x

12 Sarwar et al Fixed Point Theory and Applications :217 Page 12 of 15 Let K L M N : BS BS be the mappings defined by Khx=opt y D ux y+h1 x y h a1 x y Lhx=opt y D ux y+h2 x y h a2 x y Mhx=opt y D vx y+h3 x y h a3 x y 32 Nhx=opt y D vx y+h4 x y h a4 x y where x h S BS Theorem 31 Let K L M N : BS BS given by 32 be mappings for which the following conditions hold: 1 u v and H i are bounded for i =1234; 2 the pairs K N and L M share CLR NM property; 3 for some h BS KNh = NKh whenever Kh = Nh and LMh = MLh whenever Lh = Mh; 4 for all x y h w S D BS BS H1 xyha 1 xy H 2 xywa 2 xy 1 hw ϕt dt ψ ϕt dt where 1 h w= max Nh Mw Nh Kh Mw Lw 1 [ ] Kh Nh Lw Mw Kh Mw + Lw Nh 2 1+ Nh Mw Kh Mw Lw Nh 1+ Nh Mw 1+ Nh Lw + Mw Kh Nh Kh 1+ Nh Kh + Mw Lw Then the system of functional equations 31 has a unique common solution in BS Proof Since u vandh i are bounded for i =1234thereexistsM >suchthat sup ux y vx y Hi x y t :x y t S D R M 33 Thus by and Lemma 12 K L M N are self-mappings in BS Let x h w S BS BS Suppose that opt y D = inf y D Thenusing32wecanfind y z D such that Khx>ux y+h 1 x y h a1 x y δ; 34 Lwx>ux z+h 2 x z w a2 x z δ; 35 Khx ux z+h 1 x z h a1 x z ; 36 Lwx ux y+h 2 x y w a2 x y ; 37 where x h S BS

13 Sarwar et al Fixed Point Theory and Applications :217 Page 13 of 15 Next with the help of 34and37 we have Khx Lwx>H 1 x y h a1 x y H 2 x y w a2 x y δ max H1 x y h a1 x y H 2 x y w a2 x y H1 x z h a1 x z H 2 x z w a2 x z δ Analogously with the help of 35and36 we have Khx Lwx<H 1 x z h a1 x z H 2 x z w a2 x z + δ max H1 x y h a1 x y H 2 x y w a2 x y H 1 x z h a1 x z H 2 x z w a2 x z + δ So we can write Khx Lwx < max H1 x y h a1 x y H 2 x y w a2 x y H1 x z h a1 x z H 2 x z w a2 x z + δ = max H1 x y h a1 x y H 2 x y w a2 x y + δ H1 x z h a1 x z H 2 x z w a2 x z + δ Khx Lwx < max A B + δ C D + δ 38 where A = H 1 x y ha 1 x y B = H 2 x y wa 2 x y C = H 1 x z ha 1 x z and D = H 2 x z wa 2 x z Similarly one can obtain 38 if opt y D = sup y D Nowusing38 we have Khx Lwx ϕt dt max A B +δ C D +δ = max = max A B +δ A B C D ϕt dt ϕt dt + ϕt dt ϕt dt + A B = max ϕt dt + max A B +δ A B C D +δ A B +δ A B C D +δ C D ϕt dt C D ϕt dt ϕt dt ϕt dt C D +δ C D ϕt dt ϕt dt and by condition 4 of Theorem 31weget Kh Lw 1 hw A B +δ ϕt dt ψ ϕt dt + max ϕt dt A B where x h w S BS BS C D +δ C D ϕt dt 39

14 Sarwar et al Fixed Point Theory and Applications :217 Page 14 of 15 In the light of 33 Theorem 1234 in [25] andϕ foreachε >wecanfindδ Msatisfying ϕt dt ε C [ 3M]withmC δ 31 C where mc denotes the Lebesgue measure of CThus39becomes Kh Lw 1 hw ϕt dt ψ ϕt dt + ε h w BS Taking the limit as ε + weget Kh Lw 1 hw ϕt dt ψ ϕt dt h w BS Thus all the conditions of Theorem 21 are satisfied Hence the mappings K L M N have a unique common fixed point in BS that is the system of functional equations 31has a unique common solution Competing interests The authors declare that they have no competing interests regarding this manuscript Authors contributions All authors read and approved the final version Author details 1 Department of Mathematics University of Malakand Chakdara DirL Pakistan 2 Department of Mathematics Atılım University Ankara Turkey Acknowledgements The authors are grateful to the editor and anonymous reviewers for their careful reviews valuable comments and remarks to improve this paper Received: 16 September 215 Accepted: 1 November 215 References 1 Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux equations integrales Fundam Math Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type Int J Math Math Sci Aliouche A: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type J Math Anal Appl Altun I Türkoglu D: Some fixed point theorems for weakly compatible mappings satisfying an implicit relation TaiwanJMath Altun I Türkoglu D Rhoades BE: Fixed points of weakly compatible maps satisfying a general contractive of integral type Fixed Point Theory Appl 27 ArticleID Djoudi A Aliouche A: Common fixed point theorems of Greguš type for weakly compatible mappings satisfying contractive conditions of integral type J Math Anal Appl Djoudi A Merghadi F: Common fixed point theorems for maps under a contractive condition of integral type J Math Anal Appl Jachymski J: Remarks on contractive conditions of integral type Nonlinear Anal Liu Z Li X Kang SM Cho SY: Fixed point theorems for mappings satisfying contractive conditions of integral type and applications Fixed Point Theory Appl 21 Article ID Liu Z Zou X Kang SM Ume JS: Common fixed points for a pair of mappings satisfying contractive conditions of integral type J Inequal Appl 214 ArticleID Murthy PP Kumar S Tas K: Common fixed points of self maps satisfying an integral type contractive condition in fuzzy metric spaces Math Commun Sintunavarat W Kumam P: Gregus-type common fixed point theorems for tangential multi-valued mappings of integral type in metric spaces Int J Math Math Sci 211 Article ID Alsulami HH Karapınar E O Regan D Shahi P: Fixed points of generalized contractive mappings of integral type Fixed Point Theory Appl 214ArticleID

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