On locally contractive fuzzy set-valued mappings

Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 R E S E A R C H Open Access On locally contractive fuzzy set-valued mappings Jamshaid Ahmad1, Akbar Azam1and Salvador Romaguera2* *Correspondence: [email protected].es 2Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, 46022, Spain Full list of author information is available at the end of the article Abstract We prove the existence of common fuzzy fixed points for a sequence of locally contractive fuzzy mappings satisfying generalized Banach type contraction conditions in a complete metric space by using iterations. Our main result generalizes and unifies several well-known fixed-point theorems for multivalued maps. Illustrative examples are also given. MSC: 46S40; 47H10; 54H25 Keywords: fixed point; fuzzy mapping; contractive mapping; locally contractive 1 Introduction The Banach contraction theorem and its subsequent generalizations play a fundamental role in the field of fixed point theory. In particular, Heilpern introduced in []thenotion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework. Subsequently several other authors [–] have studied and established the existence of fixed points of fuzzy mappings. The aim of this paper is to prove a common fixed-point theorem for a sequence of fuzzy mappings in the context of metric spaces without the assumption of linearity. Our results generalize and unify several typical theorems of the literature. 2Preliminaries Given a metric space (X,d), denote by CB(X) the family of all nonempty closed bounded subsets of (X,d). As usual, for ζ∈Xand A∈CB(X), we define d(ζ,A)=inf a∈Ad(ζ,a). Then the Hausdorff metric Hon CB(X)inducedbydis defined as H(A,B)=maxsup a∈A d(a,B),sup b∈B d(A,b), for all A,B∈CB(X). Afuzzysetin(X,d) is a function with domain Xand values in I= [,]. IXdenotes the collection of all fuzzy sets in X.IfAis a fuzzy set and ζ∈X, then the function value A(ζ) is called the grade of membership of ζin A.Theα-level set of a fuzzy set Ais denoted by ©2014 Ahmad et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 2 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 Aα, and it is defined as follows: Aα=ζ:A(ζ)≥αif α∈(,], A=closureofζ:A(ζ)> . According to Heilpern [], a fuzzy set Ain a metric linear space (X,d)issaidtobe an approximate quantity if Aαis compact and convex in X,foreachα∈(, ], and supζ∈XA(ζ) = . The family of all approximate quantities of the metric linear space (X,d) is denoted by W(X). Now, for A,B∈W(X)andα∈[, ], define Dα(A,B)=H(Aα,Bα), and d∞(A,B)= sup α∈[,] Dα(Aα,Bα). It is well known that d∞is a metric on W(X). In case that (X,d) is a (non-necessarily linear) metric space, we also define Dα(A,B)=H(Aα,Bα), whenever A,B∈IXand Aα,Bα∈CB(X), α∈[,]. In the sequel the letter Nwill denote the set of positive integer numbers. The following well-known properties on the Hausdorff metric (see e.g. []) will be useful in the next section. Lemma . Let (X,d)be a metric space and let A,B∈CB(X)with H(A,B)<r,r>.If a∈A,then there exists b ∈Bsuchthatd(a,b)<r. Lemma . Let (X,d)be a metric space and let {An}∞ n= be a sequence in CB(X)such that limn→∞ H(An,A)=,for some A ∈CB(X). If ξn∈An,for all n ∈N,and d(ξn,ξ)→, then ξ∈A. Now, let Xbe an arbitrary set and let Ybe a metric space. A mapping Tis called fuzzy mapping if Tis a mapping from Xinto IY. In fact, a fuzzy mapping Tis a fuzzy subset on X×Ywith membership function T(ζ). The value T(ζ)(ξ) is the grade of membership of ξin T(ζ). If (X,d) is a metric space and Tis a (fuzzy) mapping from Xinto IX,wesaythatξ∈X is a fixed point of Tif ξ∈T(ξ). We conclude this section with the notion of contractiveness that will be used in our main result. Definition . (compare []) Let ε∈(,∞]. A function ψ:[,ε)→[, ) is said to be a MT-function if it satisfies Mizoguchi-Takahashi’s condition (i.e.,lim supr→t+ψ(r)<,for all t∈[,ε)). Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 3 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 Clearly, if ψ:[,ε)→[,) is a nondecreasing function or a nonincreasing function, then it is a MT-function. So the set of MT-functions is a rich class. 3 Fixed points of fuzzy mappings Fixed-point theorems for locally contractive mappings were studied, among others, by Edelstein [], Beg and Azam [], Holmes [], Hu [], Hu and Rosen [], Ko and Tasi [], Kuhfitting []andNadler[]. Heilpern [] established a fixed-point theorem for fuzzy contraction mappings in metric linear spaces, which is a fuzzy extension of Banach’s contraction principle. Afterwards Azam et al. [, ], and Lee and Cho [] further extended Banach’s contraction principle to fuzzy contractive mappings in Heilpern’s sense. In our main result (Theorem . below) we establish a common fixed-point theorem for a sequence of generalized fuzzy uniformly locally contraction mappings on a complete metric space without the requirement of linearity. This is a generalization of many conventional results of the literature. Let ε∈(,∞], and λ∈(, ). A metric space (X,d)issaidtobeε-chainable if given ζ,ξ∈X,thereexistsanε-chain from ζto ξ(i.e., a finite set of points ζ=ζ,ζ,ζ,...,ζm=ξ such that d(ζj–,ζj)<ε, for all j=,,...,m). A mapping T:X→Xis called an (ε,λ)uniformly locally contractive mapping if ζ,ζ∈Xand  < d(ζ,ζ)<ε,impliesd(Tζ,Tξ)≤ λd(ζ,ξ). A mapping T:X→W(X) is called an (ε,λ) uniformly locally contractive fuzzy mapping if ζ,ξ∈Xand  < d(ζ,ξ)<ε,implyd∞(T(ζ),T(ξ)) ≤λd(ζ,ξ). We remark that a globally contractive mapping can be regarded as an (∞,λ) uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive. Theorem . Let ε∈(,∞], (X,d)a complete ε-chainable metric space and {Ti}∞ i= asequence of fuzzy mappings from X into IXsuch that,for each ζ∈Xandi∈N,Ti(ζ)∈ CB(X). If ζ,ξ∈X,<d(ζ,ξ)<εimplies DTi(ζ),Tj(ξ)≤ψd(ζ,ξ)d(ζ,ξ), () for all i,j∈N,where ψ:[,ε)→[,) is a MT-function,then the sequence {Ti}∞ i= has a common fixed point,i.e.,there is ξ∗∈Xsuchthatξ∗∈Ti(ξ∗),for all i ∈N. Proof Let ξbe an arbitrary, but fixed element of X.Findξ∈Xsuch that ξ∈T(ξ).Let ξ=ζ(,),ζ(,),ζ(,),...,ζ(,m)=ξ∈T(ξ) be an arbitrary ε-chain from ξto ξ. (We suppose, without loss of generality, that ζ(,i)= ζ(,j),foreachi,j∈{,,,...,m}with i=j.) Since  < d(ζ(,),ζ(,))<ε,wededucethat DT(ζ(,)),T(ζ(,))≤ψd(ζ(,),ζ(,))d(ζ(,),ζ(,)) <ψd(ζ(,),ζ(,))d(ζ(,),ζ(,)) <d(ζ(,),ζ(,))<ε. Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 4 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 Rename ξas ζ(,).Sinceζ(,) ∈T(ζ(,)), using Lemma . we find ζ(,) ∈T(ζ(,))such that d(ζ(,),ζ(,))<ψd(ζ(,),ζ(,))d(ζ(,),ζ(,)) <d(ζ(,),ζ(,))<ε. Similarly we may choose an element ζ(,) ∈T(ζ(,))such that d(ζ(,),ζ(,))<ψd(ζ(,),ζ(,))d(ζ(,),ζ(,)) <d(ζ(,),ζ(,))<ε. Thus we obtain a set {ζ(,),ζ(,),ζ(,),...,ζ(,m)}of m+pointsofXsuch that ζ(,) ∈ T(ζ(,))and ζ(,j)∈T(ζ(,j)),forj=,,...,m,with d(ζ(,j),ζ(,j+))<ψd(ζ(,j),ζ(,j+))d(ζ(,j),ζ(,j+)) <d(ζ(,j),ζ(,j+))<ε, for j=,,,...,m–. Let ζ(,m)=ξ.Thusthesetofpointsξ=ζ(,),ζ(,),ζ(,),...,ζ(,m)=ξ∈T(ξ)is an ε-chain from ξto ξ.Renameξas ζ(,). Then by the same procedure we obtain an ε-chain ξ=ζ(,),ζ(,),ζ(,),...,ζ(,m)=ξ∈T(ξ) from ξto ξ. Inductively, we obtain ξn=ζ(n+,),ζ(n+,),ζ(n+,),...,ζ(n+,m)=ξn+ ∈Tn+(ξn) with d(ζ(n+,j),ζ(n+,j+))<ψd(ζ(n,j),ζ(n,j+))d(ζ(n,j),ζ(n,j+)) <d(ζ(n,j),ζ(n,j+))<ε,() for j=,,,...,m–. Consequently, we construct a sequence {ξn}∞ n= of points of Xwith ξ=ζ(,m)=ζ(,) ∈T(ξ), ξ=ζ(,m)=ζ(,) ∈T(ξ), ξ=ζ(,m)=ζ(,) ∈T(ξ), . . . ξn+ =ζ(n+,m)=ζ(n+,) ∈Tn+(ξn), for all n∈N. Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 5 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 For each j∈{,,,...,m–},wededucefrom()that{d(ζ(n,j),ζ(n,j+))}∞ n= is a decreasing sequence of non-negative real numbers and therefore there exists lj≥suchthat lim n→∞ d(ζ(n,j),ζ(n,j+))=lj. By assumption, lim supt→l+ jψ(t)<,sothereexistsnj∈Nsuch that ψ(d(ζ(n,j),ζ(n,j+))) < s(lj), for all n≥njwhere lim supt→l+ jψ(t)<s(lj)<. Now put Mj=maxmax i=,...,njψd(ζ(i,j),ζ(i,j+)),s(lj). Then, for every n>nj,weobtain d(ζ(n,j),ζ(n,j+))<ψd(ζ(n–,j),ζ(n–,j+))d(ζ(n–,j),ζ(n–,j+)) <s(lj)d(ζ(n–,j),ζ(n–,j+)) ≤Mjd(ζ(n–,j),ζ(n–,j+)) ≤(Mj)d(ζ(n–,j),ζ(n–,j+)) ≤··· ≤(Mj)n–d(ζ(,j),ζ(,j+)). Putting N=max{nj:j=,,,...,m–},wehave d(ξn–,ξn)=d(ζ(n,),ζ(n,m))≤ m–  j= d(ζ(n,j),ζ(n,j+)) < m–  j= (Mj)n–d(ζ(,j),ζ(,j+)), for all n>N+.Hence d(ξn,ξp)≤d(ξn,ξn+)+d(ξn+,ξn+)+···+d(ξp–,ξp) < m–  j= (Mj)nd(ζ(,j),ζ(,j+))+···+ m–  j= (Mj)p–d(ζ(,j),ζ(,j+)), whenever p>n>N+. Since Mj< , for all j∈{,,,...,m–}, it follows that {ξn}∞ n= is a Cauchy sequence. Since (X,d) is complete, there is ξ∗∈Xsuch that ξn→ξ∗.Soforeachδ∈(, ε]there is Mδ∈Nsuch that n>Mδimplies d(ξn,ξ∗)<δ. This in view of inequality ()implies D(Tn+(ξn),Ti(ξ∗)) < δ, for all i∈N.Consequently,H(Tn+(ξn),Ti(ξ∗))→. Since ξn+ ∈ Tn+(ξn)with d(ξn+,ξ∗)→, we deduce from Lemma . that ξ∗∈Ti(ξ∗), for all i∈N. This completes the proof.  Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 6 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 Corollary . Let ε∈(,∞], (X,d)a complete ε-chainable metric space and {Ti}∞ i= a sequence of fuzzy mappings from X into IXsuch that,for each ζ∈Xandi∈N,Ti(ζ)∈ CB(X). If ζ,ξ∈X,<d(ζ,ξ)<εimplies DTi(ζ),Tj(ξ)≤λd(ζ,ξ), for all i,j∈N,where λ∈(,), then the sequence {Ti}∞ i= has a common fixed point. Proof Apply Theorem . when ψis the MT-function defined as ψ(t)=λ, for all t∈ [,ε).  Corollary . Let ε∈(,∞], (X,d)a complete ε-chainable metric linear space and {Ti}∞ i= a sequence of fuzzy mappings from X into W(X)satisfying the following condition: ζ,ξ∈X,<d(ζ,ξ)<εimplies d∞Ti(ζ),Tj(ξ)≤ψd(ζ,ξ)d(ζ,ξ), for all i,j∈N,where ψ:[,ε)→[,) is a MT-function.Then the sequence {Ti}∞ i= has a common fixed point. Proof Since W(X)⊆CB(X)andD(Ti(ζ),Tj(ξ)) ≤d∞(Ti(ζ), Tj(ξ)), for all i,j∈N,theresult follows immediately from Theorem .. Corollary . Let ε∈(,∞], (X,d)a complete ε-chainable metric linear space and {Ti}∞ i= a sequence of fuzzy mappings from X into W(X)satisfying the following condition: ζ,ξ∈X,<d(ζ,ξ)<εimplies d∞Ti(ζ),Tj(ξ)≤λd(ζ,ξ), for all i,j∈N,where λ∈(,). Then the sequence {Ti}∞ i= has a common fixed point. Corollary . [] Let ε∈(, ∞], (X,d)a complete ε-chainable metric linear space and T, T,two fuzzy mappings from X into W(X)satisfying the following condition: ζ,ξ∈X,<d(ζ,ξ)<εimplies d∞Ti(ζ),Tj(ξ)≤ψd(ζ,ξ)d(ζ,ξ), for i,j=,,where ψ:[,ε)→[, ) is a MT-function.Then Tand Thave a common fixed point. Corollary . [, ]Let ε∈(, ∞], (X,d)a complete ε-chainable metric linear space and T:X→W(X)an (ε,λ)uniformly locally contractive fuzzy mapping.Then T has a fixed point. Corollary . Let ε∈(,∞], (X,d)a complete ε-chainable metric space and S be a multivalued mapping from X into CB(X)satisfying the following condition: ζ,ξ∈X,<d(ζ,ξ)<εimplies HS(ζ),S(ξ)≤ψd(ζ,ξ)d(ζ,ξ), where ψ:[ ,ε)→[, ) is a MT-function.Then S has a fixed point. Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 7 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 Proof Define a fuzzy mapping Tfrom Xinto IXas T(ξ)(t)=ift∈S(ξ)andT(ξ)(t)=, otherwise. Then T(ξ)=S(ξ), for all ξ∈X,soT(ξ)∈CB(X), for all ξ∈X.Since DT(ζ),T(ξ)=HT(ζ),T(ξ)=HS(ζ),S(ξ), for all ζ,ξ∈X, we deduce that condition ()ofTheorem. is satisfied for T.HenceThas afixedpointξ∗,i.e.,ξ∗∈T(ξ∗).Weconcludethatξ∗∈S(ξ∗). The proof is complete.  Corollary . [] Let ε∈(, ∞], (X,d)a complete ε-chainable metric space and S be a multivalued mapping from X into CB(X)satisfying the following condition: ζ,ξ∈X,<d(ζ,ξ)<εimplies HS(ζ),S(ξ)≤λd(ζ,ξ), where λ∈(,). Then S has a fixed point. Corollary . ([,], see also [, ]) Let (X,d)be a complete metric space,Samultivalued mapping from X into CB(X)and ψ:[,∞)→[, ) a MT-function such that H(Sζ,Sξ)≤ψd(ζ,ξ)d(ζ,ξ), for all ζ,ξ∈X.Then S has a fixed point in X. Proof Apply Corollary . with ε=∞. We conclude the paper with two examples to support Theorem . and Corollary .. Example . Let (X,d) be the compact, and thus complete, metric space such that X= [,], and d(x,y)=|x–y|, for all x,y∈X.Letλbe a constant such that λ∈[/,) and let {Tk}∞ k= be the sequence of fuzzy mappings defined from Xinto IXas follows: if x=, Tk(x)(y)=⎧ ⎪ ⎨ ⎪ ⎩ ify=, /kif  < y≤/,  if / < y≤, k∈N, if x=, Tk(x)(y)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ if≤y≤x/, λ/kif x/ < y≤x/, λ/kif x/ < y<x, ifx≤y≤, k∈N. For each x,y∈Xwith x=y,andi,j∈Nwe have DTi(x),Tj(y)=HTi(x),Tj(y)=H[,x/], [, y/]=  |x–y|. Hence, for ψ(t)=λ, the conditions of Corollary ., and hence of Theorem ., are satisfied for any ε∈(, ∞], whereas Xis not linear. Therefore all previous relevant fixed point results Corollaries .-. on metric linear spaces are not applicable. Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 8 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 Example . Let (X,d)bethecompletemetricspacesuchthatX=[,∞), d(x,x)=,for all x∈X,andd(x,y)=max{x,y}whenever x=y(in the sequel we shall write x∨yinstead of max{x,y}). Note that a sequence {xn}∞ n= is a Cauchy sequence in (X,d)ifandonlyifd(xn,)→. Moreover, x=  is the only non-isolated point of Xfor the topology induced by d. Let ψ:[,∞)→[, ) be the MT-function defined as ψ(t)=/ if  ≤t≤, t/(t+) ift>, and let {Tk}∞ k= be the sequence of fuzzy mappings defined from Xinto IXas follows: if  ≤x≤, Tk(x)(y)=ifx/k≤y≤x/k, otherwise, k∈N, if x>, Tk(x)(y)=ifx/k≤y<x/k( + x), otherwise, k∈N. Observe that, for  ≤x≤, Tk(x)=x k,x k, and, for x>, Tk(x)=x k,x k( + x). Therefore Tk(x)∈CB(X), for all x∈Xand k∈N(recall that each x=  is an isolated point for the induced topology, so every bounded interval belongs to CB(X)). We show that condition ()ofTheorem. is satisfied for ε=∞and ψas defined above. Indeed, let x,y∈Xwith x=yand j,k∈N. Assume without loss of generality that x>y. If x,y>,foreachb∈Tj(y),weobtain dTk(x),b=inf a∈Tk(x) (a∨b)≤x k( + x)∨b≤x k( + x)∨y j( + y). Similarly, for each a∈Tk(x),weobtain da,Tj(y)≤x k( + x)∨y j( + y). Consequently DTk(x),Tj(y)=HTk(x),Tj(y)≤x k( + x)∨y j( + y) ≤(x∨y) +(x∨y)=d(x,y) +d(x,y)d(x,y) =ψd(x,y)d(x,y). Ahmad et al. Journal of Inequalities and Applications 2014, 2014:74 Page 9 of 10 http://www.journalofinequalitiesandapplications.com/content/2014/1/74 If x>andy≤, we deduce, in a similar way, that DTk(x),Tj(y)=HTk(x),Tj(y)≤x k( + x)∨y j ≤x +x∨y ≤x +x∨x =x +x =(x∨y) +(x∨y)=d(x,y) +d(x,y)d(x,y) =ψd(x,y)d(x,y). Finally, if x,y≤, we deduce DTk(x),Tj(y)=HTk(x),Tj(y)≤x k∨y j ≤x∨y =ψd(x,y)d(x,y). We have shown that all conditions of Theorem . are satisfied (in fact x=istheonly fixed point of T). Competing interests The authors declare that they have no competing interests. Authors’ contributions The three authors contributed equally in writing this article. They read and approved the final manuscript. Author details 1Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, 44000, Pakistan. 2Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, 46022, Spain. 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