Research Article Fixed Points for Multivalued Mappings in b-metric Spaces

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1 Abstract and Applied Analysis Volume 5, Article D 7874, 7 pages Research Article Fixed Points for Multivalued Mappings in b-metric Spaces Mohamed Jleli, Bessem Samet, Calogero Vetro, and Francesca Vetro 3 Department of Mathematics, College of Science, King Saud University, P.O. Box 455, Riyadh 45, Saudi Arabia Dipartimento di Matematica e nformatica, UniversitàdegliStudidiPalermo,ViaArchirafi34,93Palermo,taly 3 Dipartimento Energia, ngegneria dell nformazione e Modelli Matematici (DEM), UniversitàdegliStudidiPalermo, Viale delle Scienze, 98 Palermo, taly Correspondence should be addressed to Bessem Samet; bessem.samet@gmail.com Received 6 June 4; Revised 6 July 4; Accepted 3 July 4 Academic Editor: Poom Kumam Copyright 5 Mohamed Jleli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. n, Samet et al. introduced the notion of α-ψ-contractive mapping and gave sufficient conditions for the existence of fixed points for this class of mappings. The purpose of our paper is to study the existence of fixed points for multivalued mappings, under an α-ψ-contractive condition of Ćirić type, in the setting of complete b-metric spaces. An application to integral equation is given.. ntroduction The theory of multivalued mappings has an important role in various branches of pure and applied mathematics because of its many applications, for instance, in real and complex analysis as well as in optimal control problems. Over the years, this theory has increased its significance and hence in the literature there are many papers focusing on the discussion of abstract and practical problems involving multivalued mappings. As a matter of fact, amongst the various approaches utilized to develop this theory, one of the most interesting is based on methods of fixed point theory, often in view of the constructive character of fixed point theorems, especially in its metric branch (see, for instance, []). Thus, Nadler [] was the first author who combined the notion of contraction (see condition () below) with multivalued mapping by establishing the following fixed point theorem. Theorem (see []). Let (X, d) be a complete metric space and let T : X CL(X) be a multivalued mapping satisfying H(Tx,Ty) kd(x,y), () for all x, y X, wherek is a constant such that k (, ) and CL(X) denotes the family of nonempty closed subsets of X. Then T has a fixed point; that is, there exists a point z Xsuch that z Tz. Later on, many authors discussed this result and gave their generalizations, extensions, and applications; see, for instance, [3 7]. On the other hand, the concept of metric space has been generalized in different directions to better cover much more general situations, arising in computer science and others (see, for instance, [3, 8]). Here, we deal with the notion of b-metric space, which is a metric space satisfying a relaxed form of triangle inequality; see Bakhtin [9] andczerwick []. Many researchers followed this idea and proved various results in the b-metric setting [ 4]. n, Samet et al. [5] introduced the notion of α-ψcontractive mapping and gave sufficient conditions for the existence of fixed points for this class of mappings. n this paper, we study the existence of fixed points for multivalued mappings, under an α-ψ-contractiveconditionofćirićtype, in the setting of complete b-metric spaces. Also, we consider a complete b-metric space endowed with a partial ordering. An inspiration for the paper is the recent work of Mohammadi et al. [6] in which some good ideas are suggested to the reader. Finally, an application to integral equation is given.. Preliminaries Let R + denote the set of all nonnegative real numbers and let N denote the set of positive integers. From [9,, 7, 8]weget

2 Abstract and Applied Analysis some basic definitions, lemmas, and notations concerning the b-metric space. Definition. Let X be a nonempty set and let s be a given real number. A function d : X X R + is said to be a b-metric if and only if for all x, y, z Xthe following conditions are satisfied: () d(x, y) = if and only if x=y; () d(x, y) = d(y, x); (3) d(x, z) s[d(x, y) + d(y, z)]. Then, the triplet (X, d, s) is called a b-metric space. t is an obvious fact that a metric space is also a b-metric space with s=,buttheconverseisnotgenerallytrue.to support this fact, we have the following example. Example 3. Consider the set X = [,] endowed with the function d:x X R + defined by d(x, y) = x y for all x, y X.Clearly,(X,d,)is a b-metricspacebutitisnot ametricspace. Let (X,d,s)be a b-metric space. The following notions are natural deductions from their metric counterparts. (i) A sequence {x n } X converges to x X if lim n + d(x n,x)=. (ii) A sequence {x n } Xis said to be a Cauchy sequence if, for every given ε>, there exists n(ε) N such that d(x m,x n )<εfor all m, n n(ε). (iii) A b-metric space (X,d,s)is said to be complete if and only if each Cauchy sequence converges to some x X. From the literature on b-metric spaces, we choose the following significant example. Example 4 (see []). Let p (, ). Consider the space L p ([, ]) of all real functions f : [,] R such that f(t) p dt < +, endowed with the functional d : L p ([, ]) L p ([, ]) R defined by d(f,g)=( f (t) g(t) p /p dt) () Then, (X, d, /p ) is a b-metric space. f, g L p ([, ]). Next, we collect some lemmas and notions concerning the theory of multivalued mappings on b-metric spaces. We recall that CB(X) denotes the class of nonempty closed and bounded subsets of X.ForA, B CB(X), define the function H : CB(X) CB(X) R + by H (A, B) = max {δ (A, B),δ(B, A)}, (3) where δ (A, B) = sup {d (a, B),a A}, (4) δ (B, A) = sup {d (b, A),b B} with d (a, C) = inf {d (a, x),x C}. (5) Note that H is called the Hausdorff b-metric induced by the b-metric d. We recall the following properties from [, 3, 8]; see also [] and the references therein. Lemma 5. Let (X,d,s)be a b-metric space. For any A, B, C CB(X) and any x, y X, one has the following: (i) d(x, B) d(x, b),foranyb B; (ii) δ(a, B) H(A, B); (iii) d(x, B) H(A, B),foranyx A; (iv) H(A, A) = ; (v) H(A, B) = H(B, A); (vi) H(A, C) s(h(a, B) + H(B, C)); (vii) d(x, A) s(d(x, y) + d(y, A)). Remark 6. The function H : CL(X) CL(X) R + is a generalized Hausdorff b-metric; that is, H(A, B) = + if max{δ(a, B), δ(b, A)} does not exist. Lemma 7. Let (X, d, s) be a b-metric space. For A CL(X) and x X,onehas d (x, A) = x A=A, (6) where A denotes the closure of the set A. Lemma 8. Let (X, d, s) be a b-metric space and A, B CL(X). Then, for each h>and for each a Athere exists b(a) B such that d(a, b(a)) < h H(A, B) if H(A, B) >. Finally, to prove our results we need the following class of functions. Let s be a real number; we denote by Ψ s the family of strictly increasing functions ψ:[,+ ) [,+ )such that + n= s n ψ n (t) <+ for each t>, (7) where ψ n denotes nth iterate of the function ψ. tiswell known that ψ(t) < t for all t>. An example of function ψ Ψ s is given by ψ(t) = ct/s for all t,wherec (, ). Definition 9. A multivaluedmappingt : X CL(X) is said to be α-admissible, with respect to a function α:x X [, + ), ifforeachx Xand y Txwith α(x, y), we have α(y, z) for all z Ty. Definition. Let (X, d, s) be a b-metric space and let δ(, ) be as in (4).Then,amultivaluedmappingF : X CL(X) is said to be h-upper semicontinuous at x X,ifthefunction δ (Fx, Fx ) := sup {d (y, Fx ) :y Fx} (8) is continuous at x. Clearly, F is said to be h-upper semicontinuous, whenever F is h-upper semicontinuous at every x X.

3 Abstract and Applied Analysis 3 3. Fixed Point Theory in b-metric Spaces We study the existence of fixed points for multivalued mappings, by adapting the ideas in [6] to the b-metric setting. Definition. Let (X, d, s) be a b-metric space. A multivalued mapping T : X CL(X) is said to be an α-ψ-contraction of Ćirić type if there exist a function α:x X [,+ )and afunctionψ Ψ s such that, for all x, y Xwith α(x, y), the following condition holds where M(x,y) H (Tx, Ty) ψ (M (x, y)), (9) = max {d (x, y), d (x, Tx),d(y,Ty), [d (x, Ty) + d (y, Tx)]}. s () Now, we are ready to state and prove our first main theorem. Theorem. Let (X,d,s)be a complete b-metric space and let T : X CL(X). Assume that there exist two functions α:x X [,+ ) and ψ Ψ s such that T is an α-ψ-contraction of Ćirić type. Also, suppose that the following conditions are satisfied: (i) T is an α-admissible multivalued mapping; (ii) there exist x Xand x Tx such that α(x,x ) ; (iii) Tish-upper semicontinuous. Then T has a fixed point. Proof. By condition (ii) there exist x X and x Tx such that α(x,x ).Clearly,ifx =x or x Tx,we deduce that x is a fixed point of T andsowecanconclude theproof.now,weassumethatx =x and x Tx and hence d(x,tx )>.First,from(9),wededuce <d(x,tx ) H(Tx,Tx ) f max{d(x,x ), d(x,tx )} = d(x,tx ),thenwehave <d(x,tx ) ψ(d(x,tx )) < d (x,tx ), () which is a contradiction. Thus, max {d (x,x ),d(x,tx )} = d (x,x ), (3) and since ψ is strictly increasing, we have <d(x,tx ) ψ(d (x,x )) <ψ(τd (x,x )), (4) where τ>is a real number. This ensures that there exists x Tx (obviously, x =x )suchthat <d(x,x ) <ψ(τd (x,x )). (5) Since T is α-admissible, from condition (ii) and x Tx, we have α(x,x ).fx Tx,thenx is a fixed point. Assume that x Tx ;thatis,d(x,tx )>. Next, from (9),we deduce <d(x,tx ) H(Tx,Tx ) ψ(max {d (x,x ),d(x,tx ),d(x,tx ), s [d (x,tx )+d(x,tx )]}) =ψ(max {d (x,x ),d(x,tx )}). (6) f max{d(x,x ), d(x,tx )} = d(x,tx ),thenwehave <d(x,tx ) ψ(d(x,tx )) < d (x,tx ), (7) which is a contradiction. Thus, max {d (x,x ),d(x,tx )} = d (x,x ), (8) and since ψ is strictly increasing, we have <d(x,tx ) ψ(d(x,x )) < ψ (τd (x,x )). (9) This ensures that there exists x 3 Tx (obviously, x 3 such that =x ) <d(x,x 3 )<ψ (τd (x,x )). () terating this procedure, we construct a sequence {x n } X such that x n Tx n,x n+ Tx n,α(x n,x n+ ), ψ(max {d (x,x ),d(x,tx ),d(x,tx ), <d(x n,tx n ) d(x n,x n+ )<ψ n (τd (x,x )), () s [d (x,tx )+d(x,tx )]}) () Let m>n.then n N. ψ(max {d (x,x ),d(x,x ),d(x,tx ), [d (x,x )+d(x,tx )]}) =ψ(max {d (x,x ),d(x,tx )}). d(x n,x m ) m k=n m k=n s k n+ d(x k,x k+ ) s k ψ k (τd (x,x )), ()

4 4 Abstract and Applied Analysis and so {x n } is a Cauchy sequence in X. Hence, there exists z Xsuch that x n z. From d (z, Tz) s [d (z, x n+ )+d(x n+, Tz)] (3) sd(z,x n+ )+sδ(tx n,tz), since T is h-upper semicontinuous, passing to limit as n +,weget d (z, Tz), (4) which implies d(z, Tz) =. Finally, since Tz is closed we obtain that z Tz;thatis,zisafixed point of T. n view of Theorem,wehavethefollowingcorollary. Corollary 3. Let (X, d, s) be a complete b-metric space and let T : X CL(X). Assume that there exist two functions α:x X [,+ )and ψ Ψ s such that α (x, y) H (Tx, Ty) ψ(m (x, y)) x, y X. (5) Also, suppose that the following conditions are satisfied: (i) T is an α-admissible multivalued mapping; (ii) there exist x Xand x Tx such that α(x,x ) ; (iii) T is h-upper semicontinuous. Then T has a fixed point. Proof. Condition (5) ensures that condition (9) holds for all x, y X with α(x, y). Thus,T is an α-ψ-contraction of Ćirić type and by Theorem the multivalued mapping T has afixedpoint. Notice that one can relax the h-upper semicontinuity hypothesis on T, by introducing another regularity condition as shown in the next theorem. Theorem 4. Let (X, d, s) be a complete b-metric space and let T : X CL(X). Assume that there exist two functions α:x X [,+ ) and ψ Ψ s such that T is an α-ψ-contraction of Ćirić type. Also, suppose that the following conditions are satisfied: (i) T is an α-admissible multivalued mapping; (ii) there exist x Xand x Tx such that α(x,x ) ; (iii) for a sequence {x n } in X with α(x n,x n+ ) for all n N {}and x n x X,thenα(x n,x) for all n N {}. f ψ(t) < t/s for all t>,thent has a fixed point. Proof. By condition (ii) there exist x Xand x Tx such that α(x,x ). Proceeding as in the proof of Theorem, we obtain a sequence {x n } that converges to some z Xsuch that x n Tx n, x n+ Tx n and α(x n,x n+ ) for all n N {}. By condition (iii), we get α(x n,z) for all n N {}. f z Tz, then the proof is concluded. Assume d(z, Tz) >. From x n z,wededucethat (i) the sequences {d(x n,z)},{d(x n,tx n )},and{d(z, Tx n )} converge to ; (ii) lim sup n + d(x n, Tz) sd(z, Tz). These facts ensure that there exists N Nsuch that max {d (x n,z),d(x n,tx n ),d(z, Tz), (6) s [d (x n, Tz) + d (z, Tx n )]} = d (z, Tz), for all n Nwith n N.SinceT is an α-ψ-contraction of Ćirićtype,foralln N,wehave d (z, Tz) s[d(z,x n+ )+d(x n+,tz)] sd (z, x n+ )+sh(tx n,tz) sd (z, x n+ )+sψ(d (z, Tz)). (7) From ψ(t) < t/s,lettingn +,weget d (z, Tz) sψ(d (z, Tz)) <d(z, Tz), (8) which implies d(z, Tz) =. Finally, since Tz is closed we obtain that z Tz;thatis,zisafixed point of T. n view of Theorem 4,wehavethefollowingcorollary. Corollary 5. Let (X, d, s) be a complete b-metric space and let T : X CL(X). Assume that there exist two functions α:x X [,+ )and ψ Ψ s such that α (x, y) H (Tx, Ty) ψ (M (x, y)) x, y X. (9) Also, suppose that the following conditions are satisfied: (i) T is an α-admissible multivalued mapping; (ii) there exist x Xand x Tx such that α(x,x ) ; (iii) for a sequence {x n } in X with α(x n,x n+ ) for all n N {}and x n x X,thenα(x n,x) for all n N {}. f ψ(t) < t/s for all t>,thent has a fixed point. All results in the paper may be stated with respect to a self-mapping T:X X.Forinstance,andforourfurther use, we consider the following version of Theorem 4. Corollary 6. Let (X, d, s) be a complete b-metric space and let T:X X. Assume that there exist two functions α : X X [,+ )and ψ Ψ s such that, for all x, y X with α(x, y), the following condition holds d (Tx, Ty) ψ (M (x, y)). (3) Also, suppose that the following conditions are satisfied: (i) x, y X, α(x, y) implies α(tx, Ty) ; (ii) there exists x Xsuch that α(x,tx ) ; (iii) for a sequence {x n } in X with α(x n,x n+ ) for all n N {}and x n x X,thenα(x n,x) for all n N {}. f ψ(t) < t/s for all t>,thent has a fixed point.

5 Abstract and Applied Analysis 5 4. Fixed Point Theory in Ordered b-metric Spaces The study of fixed points in partially ordered sets has been developed in [5, 9 ] as a useful tool for applications on matrix equations and boundary value problems. n this section, we give some results of fixed point for multivalued mappings in the setting of ordered b-metric spaces. n fact, a b-metric space (X, d, s) may be naturally endowed with a partial ordering; that is, if (X, ) is a partially ordered set, then (X,d,s, ) is called an ordered b-metric space. We say that x, y X are comparable if x yor y xholds. Also, let A, B X;then A Bwhenever for each a Athere exists b Bsuch that a b. Theorem 7. Let (X,d,s, ) be a complete ordered b-metric space and let T : X CL(X). Assume that there exists a function ψ Ψ s such that H (Tx, Ty) ψ (M (x, y)), (3) for all x, y X with Tx Ty. Also, suppose that the following conditions are satisfied: (i) there exist x Xand x Tx such that Tx Tx ; (ii) for each x Xand y Txwith Tx Ty,wehave Ty Tz for all z Ty; (iii) T is h-upper semicontinuous. Then T has a fixed point. Proof. Define the function α:x X [,+ )by α(x,y)={ if Tx Ty otherwise. (3) Clearly, the multivalued mapping T is α-admissible. n fact, for each x X and y Tx with α(x, y), wehave Tx Tyand by condition (ii) we obtain that Ty Tz for all z Ty. This implies that α(y, z) for all z Ty.Also, by condition (3), T is an α-ψ-contraction of Ćirićtype.Thus all the hypotheses of Theorem aresatisfiedandt has a fixed point. Also inthiscase, onecan relaxtheh-upper semicontinuity hypothesis on T, by using condition (iii) in Theorem 4. Precisely we state the following result. Theorem 8. Let (X,d,s, ) be a complete ordered b-metric space and let T : X CL(X). Assume that there exists a function ψ Ψ s such that H (Tx, Ty) ψ(m (x, y)), (33) for all x, y X with Tx Ty. Also, suppose that the following conditions are satisfied: (i) there exist x Xand x Tx such that Tx Tx ; (ii) for each x Xand y Txwith Tx Ty,wehave Ty Tz for all z Ty; (iii) for a sequence {x n } in X with Tx n Tx n+ for all n N {}and x n x X,thenTx n Tx for all n N {}. f ψ(t) < t/s for all t>,thent has a fixed point. Following the same ideas in [6], we propose the following results, which provide an interesting alternative to partial ordering. Theorem 9. Let (X,d,s)be a complete b-metric space, x X, andlett : X CL(X). Assume that there exists a function ψ Ψ s such that H (Tx, Ty) ψ(m (x, y)), (34) for all x, y X with x Tx Ty. Also, suppose that the following conditions are satisfied: (i) there exist x Xand x Tx such that x Tx Tx ; (ii) for each x Xand y Txwith x Tx Ty,wehave x Ty Tzfor all z Ty; (iii) T is h-upper semicontinuous. Then T has a fixed point. Proof. Define the function α:x X [,+ )by α(x,y)={ if x Tx Ty otherwise. (35) Clearly, the multivalued mapping T is α-admissible. n fact, for each x Xand y Txwith α(x, y), wehavex Tx Ty and by condition (ii) we obtain that x Ty Tzfor all z Ty. This implies that α(y, z) for all z Ty.Also,by condition (34), T is an α-ψ-contraction of Ćirić type. Thus all the hypotheses of Theorem are satisfied and T has a fixed point. The following result is a consequence of Theorem 4; in order to avoid repetition we omit the proof that is similar to the one of Theorem 9. Theorem. Let (X, d, s) be a complete b-metric space, x X, andlett : X CL(X). Assume that there exists a function ψ Ψ s such that H (Tx, Ty) ψ(m (x, y)), (36) for all x, y X with x Tx Ty. Also, suppose that the following conditions are satisfied: (i) there exist x Xand x Tx such that x Tx Tx ; (ii) for each x Xand y Txwith x Tx Ty,wehave x Ty Tzfor all z Ty; (iii) for a sequence {x n } in X with x Tx n Tx n+ for all n N {}and x n x X,thenx Tx n Txfor all n N {}. f ψ(t) < t/s for all t>,thent has a fixed point.

6 6 Abstract and Applied Analysis 5. Application to ntegral Equation n this section, inspired by Cosentino et al. [] wegivea typical application of fixed point methods to the study of existence of solutions for integral equations. Briefly, we give the background and notation. Let X=C([,],R) be the set of real continuous functions defined on [, ], where >, and let d:x X [,+ )be given by d(x,y)= (x y) = sup (x (t) y(t)), (37) t [,] for all x, y X.Then(X, d, ) is a complete b-metric space. Consider the integral equation x (t) =p(t) + S (t, u) f (u, x (u)) du, (38) where f:[,] R R and p : [,] R are two continuous functions and S : [, ] [, ] [, + ) is a function such that S(t, ) L ([, ]) for all t [,]. Consider the operator T:X Xdefined by T (x)(t) =p(t) + S (t, u) f (u, x (u)) du. (39) Then we prove the following existence result. Theorem. Let X = C([, ], R). Suppose that the following conditions are satisfied: (i) there exist η:x X [,+ )and α:x X [, + ) such that if α(x, y) for x, y X,then,for every u [,]and some λ>,onehas f (u, x (u)) f(u,y(u)) η(x,y) x (u) y(u), S (t, u) η(x,y)du +λ ; (4) (ii) x, y X, α(x, y) implies α(tx, Ty) ; (iii) there exists x Xsuch that α(x,t(x )) ; (iv) if {x n } is a sequence in X such that α(x n,x n+ ) for all n N {}and x n x as n +,then α(x n,x) for all n N {}. Then the integral equation (38) has a solution in X. Proof. Clearly, any fixed point of (39) is a solution of (38).By condition (i), we obtain T (x)(t) T(y)(t) =[ S (t, u) [f (u, x (u)) f(u,y(u))] du ] [ S (t, u) f (u, x (u)) f(u,y(u)) du] [ S (t, u) η(x,y) x (u) y(u) du] [ S (t, u) η(x,y) (x y) du] = (x y) [ S (t, u) η(x,y)du]. Thenwehave (T (x) T (y)) (x y) and hence, for all x, y X,weobtain S (t, u) η(x,y)du, (4) (4) d(t(x), T (y)) d(x,y) +λ, (43) which implies that (3) holds true with ψ Ψ given by ψ(t) = t/( + λ) for all t.theotherconditionsof Corollary6 are immediately satisfied and hence the operator T has a fixed point, that is, a solution of the integral equation (38) in X. Remark. Notice that α:x X [,+ )defined by α(x,y)={ if x y otherwise (44) is an easy example of function suitable for Theorem. Clearly, as X=C([,],R), thenwecansaythatx yif and only if x(t) y(t) for all t [,],where denotes the usual order of real numbers. n this case, condition (ii) is satisfied by assuming that f is nondecreasing with respect to its second variable. Conflict of nterests The authors declare that there is no conflict of interests regarding the publication of this paper. Authors Contribution All authors contributed equally and significantly in writing this paper. All authors read and approved the paper.

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