Fixed Points for Multivalued Mappings in b-metric Spaces

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1 Applied Mathematical Sciences, Vol. 10, 2016, no. 59, HIKARI Ltd, Fixed Points for Multivalued Mappings in b-metric Spaces Seong-Hoon Cho Department of Mathematics, Hanseo University Chungnam, , South Korea Copyright c 2016 Seong-Hoon Cho. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The notion of generalized α-ψ-contraction multivalued mappings of Ćirić-Berinde type is introduced and some fixed point theorems for such mappings are established. An example is given. Mathematics Subject Classification: 47H10, 54H25 Keywords: fixed point, multivalued mapping, b-metric space 1 Introduction and preliminaries In 1989, Bakhtin [5] introduced the notion of b-metric spaces and extended the Banach s contraction principle to b-metric spaces (see also [11]). Since then, some authors ( for example, [2, 8, 9, 11, 12, 13, 14, 17] and references therein ) obtaind fixed point theorems in b-metric spaces. In 2012, Samet et al. [16] introduced the notions of α-ψ-contractive mapping and α-admissible mappings in metric spaces and obtained corresponding fixed point results, which are generalizations of ordered fixed point results (see [16]). And then, by using their idea, some authors investigated fixed point results in the field. Recently, Jlieli et al. [14] gave a generalization and extention of the result of [16] to the case of multivalued mappings defined on b-metric spaces by introducing α-ψ-contraction of Ćirić type.

2 2928 Seong-Hoon Cho Very recently, Joseph et al. [13] introduced the concept of Hardy-Roger type contraction mappings and obtained a fixed point theorem for such mappings in b-metric spaces. The purpose of this paper is to introduce the concept of α-ψ-contraction multivalued mappings of Ćirić-Berinde type and to establish fixed point theorems for such mappings and to generalize the results of [13, 14]. Recall that some definitions, notations and basic properties in b-metric spaces, which are found in [5, 11, 12, 14]. Let X be a set and let s 1. A function d : X X [0, ) is called a b-metric space if, for all x, y, z X, the following conditions are satisfied: (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x); (3) d(x, z) s[d(x, y) + d(y, z)]. The triplet (X, d, s) is called a b-metric space. It is well known that a metric space implies b-metric space with s = 1 but, in general, the converse is not true (see [14]). Let (X, d, s) be a b-metric space, Y X and let {x n } be a sequence of points in X. Then we say that (1) {x n } is convergent to x X if and only if lim n d(x n, x) = 0; (2) {x n } is Cauchy if and only if lim n,m d(x n, x m ) = 0; (3) (X, d, s) is complete if and only if every Cauchy sequence in X is convergent in X. (4) Y is closed if and only if for each sequence {x n } of points in Y with lim n d(x n, x) = 0, we have x Y. Lemma 1.1. [9] Let (X, d, s) be a b-metric space, and let {x n } be a sequence of points in X. Then the following are satisfied: (1) if {x n } is convergent, then the limit is unique; (2) each convergent sequence is Cauchy. Remark 1.1. [9] Let (X, d, s) be a b-metric space. Then the following are satisfied:

3 Fixed points for multivalued mappings in b-metric spaces 2929 (1) d is generally not continuous; (2) d does not generally induce a topology on X. Let (X, d, s) be a b-metric space. We denote by CL(X) the class of nonempty closed subsets of X. Let H(, ) be the generalized Pompeiu-Hausdorff b-metric on CL(X), i.e., for all A, B CL(X), { max{sup H(A, B) = a A d(a, B), sup b B d(b, A)}, if the maximum exists,, otherwise, where d(a, B) = inf{d(a, b) : b B} is the distance from the point a to the subset B. For A, B CL(X), let δ(a, B) = sup x A d(x, B). Then we have δ(a, B) H(A, B) for all A, B CL(X). From now on, let M(x, y) = max{d(x, y), d(x, T x), d(y, T y), 1 2s and m(x, y) = max{d(x, y), d(x, T x), d(y, T y), 1 {d(x, T y) + d(y, T x)}} 2s for a multivalued map T : X CL(X) and x, y X. d(x, T x)(1 + d(x, T x)) {d(x, T y) + d(y, T x)}, } 1 + d(x, y) Let s 1, and we denote by Ψ the family of all strictly increasing functions ψ : [0, ) [0, ) such that n=1 sn ψ n (t) < for each t > 0, where ψ n is the n-th iteration of ψ. It is known that if ψ Ψ, then ψ(t) < t for all t > 0. Let ψ(t) = ct for all t 0, where c (0, 1). s Then ψ Ψ. Lemma 1.2. Let s 1 and let ψ Ψ be such that Then sψ(t) < t for all t > 0. ψ(st) sψ(t) for all t > 0. Proof. Suppose that there exists t 0 > 0 such that t 0 sψ(t 0 ). Since ψ is strictly increasing function, we have t 0 sψ(t 0 ) sψ(sψ(t 0 )) s 2 ψ 2 (t 0 ) s n ψ n (t 0 ). Since lim n s n ψ n (t 0 ) = 0, we have t 0 = 0 which is a contradiction.

4 2930 Seong-Hoon Cho Remark 1.2. Let (X, d, s) be a b-metric space and let α : X X [0, ) be a function. Let {x n } be a sequence in X such that α(x n, x n+1 ) 1 for all n N and lim n d(x n, x) = 0. Consider the following conditions: (1) α(x n, x) 1 for all n N; (2) lim n inf α(x n, x) 1; (3) there exists a subsequence {x n(k) } of {x n } such that Then we have (1) = (2) = (3). α(x n(k), x) 1 for all k N. Let X be a set, and let T : X N(X) be a multivalued mapping, where N(X) is the class of all nonempty subsets of X. Then, we say that (1) T is called α -admissible [4] if α(x, y) 1 implies α (T x, T y) 1, where α (T x, T y) = inf{α(a, b) : a T x, b T y}; (2) T is called α-admissible [15] if, for each x X and y T x with α(x, y) 1, we have α(y, z) 1 for all z T y. Lemma 1.3. Let X be a set, and let T : X N(X) be a multivalued mapping. If T is α -admissible, then it is α-admissible. Lemma 1.4. [10] Let (X, d, s) be a b-metric space, and let B CL(X). If a X and d(a, B) < c, then there exists b B such that d(a, b) < c. Lemma 1.5. [12] Let (X, d, s) be a b-metric space, and let A CL(X) and x X. Then we have where A denotes the closure of A. d(x, A) = 0 if and only if x A = A, Let (X, d, s) be a b-metric space. Then we say that (1) a mapping f : X X is continuous if and only if, for each x X and {x n } X with lim n d(x n, x) = 0, lim n d(fx n, fx) = 0;

5 Fixed points for multivalued mappings in b-metric spaces 2931 (2) a multivalued mapping T : X CL(X) is continuous if and only if, for each x X and {x n } X with lim n d(x n, x) = 0, lim n H(T x n, T x) = 0; (3) a multivalued mapping T : X CL(X) is h-upper semicontinuous [14] if and only if, for each x X and {x n } X with lim n d(x n, x) = 0, lim n δ(t x n, T x) = 0; (4) a function f : X [0, ) is called upper semicontinuous if and only if, for each x X and {x n } X with lim n d(x n, x) = 0, we have lim sup fx n fx; n (5) a function f : X [0, ) is called lower semicontinuous if and only if, for each x X and {x n } X with lim n d(x n, x) = 0, we have fx lim n inf fx n. Remark 1.3. Let (X, d, s) be a complete b-metric space, and let T : X CL(X) be a multivalued map. Then T is continuous implies that it is h-upper semicontinuous. For a multivalued map T : X N(X), let f T : X [0, ) be a function defined by f T (x) = d(x, T x). 2 Fixed point theorems We investigate the existence of fixed points for multivalued mappings in b- metric spaces. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. A multivalued mapping T : X CL(X) is called generalized α-ψ-contraction of Ćirić-Berinde type if, for all x, y X with α(x, y) 1, where ψ Ψ and L 0. H(T x, T y) ψ(m(x, y)) + Ld(y, T x) (2.1) Theorem 2.1. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) is generalized α-ψ-contraction of Ćirić-Berinde type. Also, suppose that the following are satisfied: (1) T is α-admissible;

6 2932 Seong-Hoon Cho (2) there exists x 0 X and x 1 T x 0 such that α(x 0, x 1 ) 1; (3) either T is h-upper semicontinuous or f T is lower semi-continuous. Proof. Let x 0 X, and let x 1 T x 0 be such that α(x 0, x 1 ) 1. Let c be a real number with d(x 0, x 1 ) < c. If x 0 = x 1, then x 1 is a fixed point, and so the proof is finished. Let x 0 x 1. If x 1 T x 1, then x 1 is a fixed point. Let x 1 / T x 1. Then d(x 1, T x 1 ) > 0. From (2.1) we obtain 0 < d(x 1, T x 1 ) H(T x 0, T x 1 ) ψ(m(x 0, x 1 )) + Ld(x 1, T x 0 ) =ψ(max{d(x 0, x 1 ), d(x 0, T x 0 ), d(x 1, T x 1 ), 1 2s {d(x 0, T x 1 ) + d(x 1, T x 0 )}, d(x 0, T x 0 )(1 + d(x 0, T x 0 )) }) + Ld(x 1, T x 0 ) 1 + d(x 0, x 1 ) ψ(max{d(x 0, x 1 ), d(x 0, x 1 ), d(x 1, T x 1 ), 1 2 {d(x 0, x 1 ) + d(x 1, T x 1 )}, d(x 0, x 1 )}) =ψ(max{d(x 0, x 1 ), d(x 1, T x 1 )}). (2.2) If max{d(x 0, x 1 ), d(x 1, T x 1 )} = d(x 1, T x 1 ), then from (2.2) we have 0 < d(x 1, T x 1 ) ψ(d(x 1, T x 1 )) < d(x 1, T x 1 ) which is a contradiction. Thus, max{d(x 0, x 1 ), d(x 1, T x 1 )} = d(x 0, x 1 ), and from (2.2) we have d(x 1, T x 1 ) ψ(d(x 0, x 1 )) < ψ(c). By Lemma 1.2, there exists x 2 T x 1 such that d(x 1, x 2 ) < ψ(c). Since T is α-admissible and α(x 0, x 1 ) 1 and x 2 T x 1, we have α(x 1, x 2 ) 1. If x 2 T x 2, then x 2 is a fixed point. Let x 2 T x 2. Then d(x 2, T x 2 ) > 0.

7 Fixed points for multivalued mappings in b-metric spaces 2933 From (2.1) we obtain 0 < d(x 2, T x 2 ) H(T x 1, T x 2 ) ψ(max{d(x 1, x 2 ), d(x 1, T x 1 ), d(x 2, T x 2 ), 1 2s {d(x 1, T x 2 ) + d(x 2, T x 1 )}, d(x 1, T x 1 )(1 + d(x 1, T x 1 )) }) + Ld(x 2, T x 1 ) 1 + d(x 1, x 2 ) ψ(max{d(x 1, x 2 ), d(x 1, x 2 ), d(x 2, T x 2 ), 1 2s {d(x 1, T x 2 ) + d(x 2, x 2 )}, d(x 1, x 2 )}) ψ(max{d(x 1, x 2 ), d(x 2, T x 2 ), 1 2 {d(x 1, x 2 ) + d(x 2, T x 2 )}}) =ψ(max{d(x 1, x 2 ), d(x 2, T x 2 )}). If max{d(x 1, x 2 ), d(x 2, T x 2 )} = d(x 2, T x 2 ), then we have d(x 2, T x 2 ) ψ(d(x 2, T x 2 )) < d(x 2, T x 2 ), which is a contradiction. Thus, max{d(x 1, x 2 ), d(x 2, T x 2 )} = d(x 1, x 2 ), and hence we have d(x 2, T x 2 ) ψ(d(x 1, x 2 )) < ψ 2 (c). Hence, there exists x 3 T x 2 such that d(x 2, x 3 ) < ψ 2 (c). Since T is α-admissible and x 2 T x 1 and α(x 1, x 2 ) 1, we have α(x 2, x 3 ) 1. By induction, we obtain a sequence {x n } X such that, for all n N {0}, α(x n, x n+1 ) 1, x n+1 T x n, x n x n+1 and d(x n, x n+1 ) < ψ n (c). Let ɛ > 0 be given. Since n=1 sn ψ n (c) <, there exists N N such that s n ψ n (c) < ɛ. For all m > n N, we have n N d(x n, x m ) m s k ψ k (c) k=n < s n ψ n (c) < ɛ. n N

8 2934 Seong-Hoon Cho Hence, {x n } is a Cauchy sequence in X. It follows from the completeness of X that there exists x X with lim d(x n, x ) = 0. n Suppose that T is h-upper semicontinuous. We have d(x, T x ) s[d(x, x n+1 ) + d(x n+1, T x )] sd(x, x n+1 ) + sδ(t x n, T x ). Since lim n d(x, x n+1 ) = lim n δ(t x n, T x ) = 0, by letting n in the above inequality, we obtain d(x, T x ) = 0, and so x T x. Assume that f T is lower semicontinuous. Then f T (x ) lim n inf f T (x n ). Hence, Thus, x T x. d(x, T x ) lim inf d(x n, T x n ) n lim d(x n, x n+1 ) lim s[d(x n, x ) + d(x, x n+1 )] = 0. n n Corollary 2.2. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) satisfies α(x, y)h(t x, T y) ψ(m(x, y)) + Ld(y, T x) for all x, y X, where L 0 and ψ Ψ. Also, suppose that conditions (1), (2) and (3) of Theorem 2.1 are satisfied. Remark 2.1. (1) Corollary 2.2 is a generalization of Theorem 3.4 of [15]. In fact, if s = 1 and L = 0, then Corollary 2.5 becomes to Theorem 3.4 of [15], whenever M(x, y) is replaced by m(x, y). (2) Corollary 2.2 is a generalization of Theorem 3 of [7]. In fact, if s = 1, α(x, y) = 1 for all x, y X and ψ(t) = kt for all t 0, where k [0, 1), then Corollary 2.2 becomes to Theorem 3 of [7], whenever M(x, y) is replaced by d(x, y). Let (X, ) be an ordered set and A, B X. We say that A B whenever for each a A, there exists b B such that a b. Corollary 2.3. Let (X,, d, s) be a complete ordered b-metric space. Suppose that a multivalued mapping T : X CL(X) satisfies H(T x, T y) ψ(m(x, y)) + Ld(y, T x)

9 Fixed points for multivalued mappings in b-metric spaces 2935 for all x, y X with T x T y (resp. T y T x), where L 0 and ψ Ψ. Assume that for each x X and y T x with T x T y (resp. T y T x), we have T y T z (resp. T z T y) for all z T y. Also, suppose that the following are satisfied: (1) there exists x 0 X and x 1 T x 0 such that T x 0 T x 1 (resp. T x 1 T x 0 ); (2) either T is h-upper semicontinuous or f T is lower semi-continuous. Remark 2.2. Corollary 2.3 is a generalization of Corollary 3.6 of [15]. If we have s = 1, then Corollary 2.3 becomes to Corollary 3.6 of [15], whenever M(x, y) is replaced by m(x, y). From Theorem 2.1 we obtain the following result. Corollary 2.4. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) is a generalized α-ψ-contraction of Ćirić-Berinde type. Also, suppose that T is an α -admissible and that conditions (2) and (3) of Theorem 2.1 are satisfied. Corollary 2.5. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) satisfies α (x, y)h(t x, T y) ψ(m(x, y)) + Ld(y, T x) for all x, y X, where L 0 and ψ Ψ. Also, suppose that T is an α -admissible and that conditions (2) and (3) of Theorem 2.1 are satisfied. Remark 2.3. Corollary 2.5 is a generalization of Theorem 2.2 of [6]. In fact, if s = 1 and α(x, y) = 1 for all x, y X and ψ(t) = kt for all t 0, where k [0, 1). If T is single valued map, then Corollary 2.5 becomes to Theorem 2.2 of [6], whenever M(x, y) is replaced by m(x, y). Theorem 2.6. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) is a generalized α-ψ-contraction of Ćirić-Berinde type, where ψ is upper semicontinuous such that the following condition holds:

10 2936 Seong-Hoon Cho ψ(st) sψ(t) for all t > 0. Also, suppose that the following are satisfied: (1) T is α-admissible; (2) there exists x 0 X and x 1 T x 0 such that α(x 0, x 1 ) 1; (3) for a sequence {x n } of points in X with α(x n, x n+1 ) 1 for all n N {0} and lim n d(x n, x) = 0, there exists a subsequence {x n(k) } of {x n } such that, for all k N {0}, α(x n(k), x) 1. Proof. Following the proof of Theorem 2.1, we obtain a sequence {x n } X with lim n d(x n, x ) = 0 for some x X such that, for all n N {0}, x n+1 T x n, x n x n+1 and α(x n, x n+1 ) 1. From (3) there exists a subsequence {x n(k) } of {x n } such that We have M(x n(k), x ) α(x n(k), x ) 1. (2.3) = max{d(x n(k), x ), d(x n(k), T x n(k) ), d(x, T x ), 1 2s {d(x n(k), T x ) + d(x, T x n(k) )}, d(x n(k), T x n(k) )[1 + d(x n(k), T x n(k) )] } 1 + d(x n(k), x ) max{d(x n(k), x ), d(x n(k), x n(k)+1 ), d(x, T x ), 1 2s {d(x n(k), T x ) + d(x, x n(k)+1 )}, d(x n(k), x n(k)+1 )[1 + d(x n(k), x n(k)+1 )] } 1 + d(x n(k), x ) M (x n(k), x ), where, M (x n(k), x ) = max{d(x n(k), x ), d(x n(k), x n(k)+1 ), d(x, T x ), 1 2s {s[d(x n(k), x )+ d(x, T x )] + d(x, x n(k)+1 )}, d(x n(k),x n(k)+1 )[1+d(x n(k),x n(k)+1 )] 1+d(x n(k),x ) }. We deduce that lim M (x n(k), x ) = d(x, T x ). n

11 Fixed points for multivalued mappings in b-metric spaces 2937 Since ψ is upper semicontinuous, From (2.1) and (2.3) we have lim ψ(m (x n(k), x )) ψ(d(x, T x )). n d(x, T x ) s[d(x, x n(k)+1 ) + d(x n(k)+1, T x )] =sd(x, x n(k)+1 ) + sh(t x n(k), T x ) sd(x, x n(k)+1 ) + sψ(m(x n(k), x )) + sld(x, x n(k)+1 )) sd(x, x n(k)+1 ) + sψ(m (x n(k), x )) + sld(x, x n(k)+1 )). (2.4) Letting k in the inequality (2.4), we obtain d(x, T x ) sψ(d(x, T x )). If d(x, T x ) > 0, then by Lemma 1.1, we obtain d(x, T x ) sψ(d(x, T x )) < d(x, T x ) which is a contradiction. Hence, d(x, T x ) = 0, and hence x is a fixed point of T. The following example shows that upper semicontinuity of ψ can not be dropped in Theorem 2.6. Example 2.1. Let X = [0, ), and let d(x, y) = x y 2 for all x, y 0. Then (X, d) be a b-metric space with s = 2. Let L 0 be fixed. Define a mapping T : X CL(X) by { 4, 1} (x = 0), 5 T x = { 3 x} (0 < x 1), 4 {2x} (x > 1). Let ψ(t) = { t (t 1), t (0 t < 1). Then ψ Ψ and ψ is a strictly increasing function, and ψ(st) sψ(t) for all t > 0. Let α : X X [0, ) be defined by α(x, y) = { 4 (0 x, y 1), 0 otherwise. Obviously, condition (3) of Theorem 2.6 is satisfied. Condition (2) of Theorem 2.6 is satisfied with x 0 = 1 4.

12 2938 Seong-Hoon Cho We show that (2.1) is satisfied. Let x, y X be such that α(x, y) 1. Then, 0 x, y 1. If x = y, then obviously (2.1) is satisfied. Let x y. If x = 0 and 0 < y 1, then we obtain H(T x, T y) = H({ 4 5, 1}, 3 4 y) ( 1 20 )2 ψ(d(x, T x)) ψ(m(x, y)) + Ld(y, T x). Let 0 < x 1 and 0 < y 1, Then, we have Thus, (2.1) is satisfied. H(T x, T y) = d(t x, T y) =d( 3 4 x, 3 4 y) = 9 16 x y 2 =ψ(d(x, y)) ψ(m(x, y)) + Ld(y, T x). We now show that T is α-admissible. Let x X be given, and let y T x be such that α(x, y) 1. Then, 0 x, y 1. Obviously, α(y, z) 1 for all z T y whenever 0 < y 1. If y = 0, then T y = { 1, 1}. Hence, for all z T y, α(y, z) 1. 2 Hence T is α-admissible. Thus, all hypothesis of Theorem 2.6 are satisfied. However, T has no fixed points. Note that ψ is not upper semicontinuous. Corollary 2.7. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) satisfies α(x, y)h(t x, T y) ψ(m(x, y)) + Ld(y, T x) for all x, y X, where L 0 and ψ Ψ is upper semicontinuous such that the following condition holds: ψ(st) sψ(t) for all t > 0. Also, suppose that conditions (1), (2) and (3) of Theorem 2.6 are satisfied.

13 Fixed points for multivalued mappings in b-metric spaces 2939 Corollary 2.8. Let (X,, d, s) be a complete ordered b-metric space. Suppose that a multivalued mapping T : X CL(X) satisfies H(T x, T y) ψ(m(x, y)) + Ld(y, T x) for all x, y X with T x T y (resp. T y T x), where L 0 and ψ Ψ is upper semi-continuous and ψ(st) sψ(t) for all t > 0. Assume that for each x X and y T x with T x T y (resp. T y T x), we have T y T z (resp. T z T y) for all z T y. Also, suppose that the following are satisfied: (1) there exists x 0 X and x 1 T x 0 such that T x 0 T x 1 (resp. T x 1 T x 0 ); (2) for a sequence {x n } in X with x n x n+1 (resp. x n+1 x n ) for all n N {0} and lim n d(x n, x) = 0, there exists a subsequence {x n(k) } of {x n } such that, for all k N {0}, x n(k) x (resp. x x n(k) ). Remark 2.4. Corollary 2.8 is a generalization and extension of the result of [1] to multivalued mappings. Corollary 2.9. Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) is a generalized α-ψ-contraction of Ćirić-Berinde type, where ψ is upper semicontinuous such that the following condition holds: ψ(st) sψ(t) for all t > 0. Also, suppose that T is α -admissible and that conditions (2) and (3) of Theorem 2.6 are satisfied. Remark 2.5. By taking L = 0 in Corollary 2.9, replacing M(x, y) by m(x, y) and by applying Remark 1.2, Corollary 2.9 reduces to Theorem 2.6 of [3]. Corollary Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) satisfies α (x, y)h(t x, T y) ψ(m(x, y)) + Ld(y, T x)

14 2940 Seong-Hoon Cho for all x, y X, where L 0 and ψ Ψ is upper semicontinuous such that the following condition holds: ψ(st) sψ(t) for all t > 0. Also, suppose that T is an α -admissible and that conditions (2) and (3) of Theorem 2.6 are satisfied. Remark 2.6. If we have M(x, y) = m(x, y) and L = 0 in Corollary 2.10, then Corollary 2.10 reduces to Theorem 2.1 of [8]. Corollary Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) satisfies H(T x, T y) a 1 d(x, y) + a 2 d(x, T x) + a 3 d(y, T y) + a 4 d(x, T y) + a 5 d(y, T x) +a 6 d(x, T x)(1 + d(x, T x)) 1 + d(x, y) + Ld(y, T x) (2.5) for all x, y X with α(x, y) 1, where a i 0, i = 1, 2, 3, 4, 5, 6 with a 1 + a 2 + a 3 + 2sa 4 + 2sa 5 + a 6 < 1, a 3 s + a 4 s 2 < 1 and L 0. Also, suppose that conditions (1), (2) and (3) of Theorem 2.6 are satisfied. Proof. Let k [0, 1) be such that a 1 + a 2 + a 3 + 2sa 4 + 2sa 5 + a 6 < k < 1, and let ψ(t) = kt for all t 0. s Then ψ Ψ, and ψ is upper semi continuous and ψ(st) sψ(t) for all t > 0. Let x, y X be any points such that α(x, y) 1.

15 Fixed points for multivalued mappings in b-metric spaces 2941 From (2.5) we have H(T x, T y) a 1 d(x, y) + a 2 d(x, T x) + a 3 d(y, T y) + a 4 d(x, T y) + a 5 d(y, T x) + a 6 d(x, T x)(1 + d(x, T x)) 1 + d(x, y) + Ld(y, T x) d(x, T y) a 1 d(x, y) + a 2 d(x, T x) + a 3 d(y, T y) + 2sa 4 2s d(x, T x)(1 + d(x, T x)) + a 6 + Ld(y, T x) 1 + d(x, y) a 1 d(x, y) + a 2 d(x, T x) + a 3 d(y, T y) + (2sa 4 + 2sa 5 ) max{ + a 6 d(x, T x)(1 + d(x, T x)) 1 + d(x, y) + Ld(y, T x) a 1 d(x, y) + a 2 d(x, T x) + a 3 d(y, T y) + (2sa 4 + 2sa 5 ) + a 6 d(x, T x)(1 + d(x, T x)) 1 + d(x, y) + Ld(y, T x) + 2sa 5 d(y, T x) 2s d(x, T y), 2s d(x, T y) + d(y, T x) 2s (a 1 + a 2 + a 3 + 2sa 4 + 2sa 5 + a 6 ) max{d(x, y), d(x, T x), d(y, T y), d(x, T x)(1 + d(x, T x)) } + Ld(y, T x) 1 + d(x, y) =ψ(m(x, y)) + Ld(y, T x). d(y, T x) } 2s d(x, T y) + d(y, T x), 2s Hence (2.1) is satisfied. As in the proof of Theorem 2.1, we can fined a Cauchy sequence {x n } X such that, for all n N {0}, α(x n, x n+1 ) 1, x n+1 T x n and x n x n+1. Let lim d(x n, x ) = 0 n for some x X. By condition (2), there exists a subsequence {x n(k) } of {x n } such that, for all k N {0}, α(x n(k), x ) 1.

16 2942 Seong-Hoon Cho From (2.5) we obtain d(x, T x ) s[d(x, x n(k)+1 ) + H(T x n(k), T x )] s[d(x n(k)+1, x ) + a 1 d(x n(k), x ) + a 2 d(x n(k), T x n(k) ) + a 3 d(x, T x ) + a 4 d(x n(k), T x ) +a 5 d(x, T x n(k) ) + a 6 d(x n(k), T x n(k) )(1 + d(x n(k), T x n(k) )) 1 + d(x n(k), x ) + Ld(x, T x n(k) )] s[d(x n(k)+1, x ) + a 1 d(x n(k), x ) + a 2 d(x n(k), x n(k)+1 ) + a 3 d(x, T x ) + a 4 d(x n(k), T x ) +a 5 d(x, x n(k)+1 ) + a 6 d(x n(k), x n(k)+1 )(1 + d(x n(k), x n(k)+1 )) 1 + d(x n(k), x ) s[d(x n(k)+1, x ) + a 1 d(x n(k), x ) + a 2 d(x n(k), x n(k)+1 ) + a 3 d(x, T x ) + Ld(x, x n(k)+1 )] +a 4 s[d(x n(k), x ) + d(x, T x )] + a 5 d(x, x n(k)+1 ) + a 6 d(x n(k), x n(k)+1 )(1 + d(x n(k), x n(k)+1 )) 1 + d(x n(k), x ) +Ld(x, x n(k)+1 )]. Letting k in above inequality, we obtain d(x, T x ) (a 3 s + a 4 s 2 )d(x, T x ) which implies d(x, T x ) = 0, becuse a 3 s + a 4 s 2 < 1. Hence x T x. Corollary Let (X, d, s) be a complete b-metric space, and let α : X X [0, ) be a function. Suppose that a multivalued mapping T : X CL(X) satisfies H(T x, T y) a 1 d(x, y) + a 2 d(x, T x) + a 3 d(y, T y) + a 4 d(x, T y) + a 5 d(y, T x) +a 6 d(x, T x)(1 + d(x, T x)) 1 + d(x, y) + Ld(y, T x) for all x, y X with α(x, y) 1, where a i 0, i = 1, 2, 3, 4, 5, 6 with a 1 + a 2 + a 3 + 2sa 4 + a 5 + a 6 < 1, a 3 s + a 4 s 2 < 1 and L 0. Also, suppose that conditions (1), (2) and (3) of Theorem 2.6 are satisfied. Remark 2.7. Corollary 2.12 is a generalization of Theorem 15 of [13]. If we have α(x, y) = 1 for all x, y X and L = 0, then Corollary 2.12 reduces to Theorem 15 of [13]. References [1] R.P. Agarwal, MA. El-Gebeily, D. O Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., 87 (2008),

17 Fixed points for multivalued mappings in b-metric spaces 2943 [2] H. Aydi, M.F. Bota, E. Karapinar, S. Mitrovic, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory and Applications, 2012 (2012), [3] M. Ali, T. Kamran, E. Karapinar, (α,ψ,ξ)-contractive multivalued mappings, Fixed Point Theory and Applications, 2014 (2014), 7. [4] J. Asl, S. Rezapour, N. Shahzad, On fixed points of α-ψ-contractive multifunctions, Fixed Point Theory and Applications, 2012 (2012), [5] I.A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., 30 (1989), Unianowsk, Gos. Ped. Inst [6] V. Berinde, Some remarks on a fixed point theorem for Ćirić type almost contractions, Carpathian J. Math., 25 (2009), [7] M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl., 326 (2007), [8] M.F. Bota, C. Chifu, E. Karapinar, Fixed point theorems for generalized (α -ψ)- Ćirić-type contractive multivalued operators in b-metric spaces, Journal of Nonlinear Science and Applications, 9 (2016), [9] M. Bota, A. Molnár, C. Varga, On Ekeland s variational principle in b- metric spaces, Fixed Point Theory, 12 (2011), [10] S. H. Cho, A fixed point theorems for Ćirić-Berinde type mapping in orbitally complete metric spaces, Carpathian J. Math., 30 (2014), [11] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), [12] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Univ. Modena, 46 (1998), [13] J.R. Joseph, D.D. Roselin, M. Marudai, Fixed point theorems on multi valued mappings in b-metric spaces, Spinger Plus, 5 (2016), [14] M. Jleli, B. Samet, C. Vetro, F. Vetro, Fixed points for multivalued mappings in b-metric spaces, Abstract and Applied Analysis, 2015 (2015), 1-7, Article ID

18 2944 Seong-Hoon Cho [15] B. Mohammadi, S. Rezapour, N. Shahzad, Some results on fixed points of α-ψ-ćirić generalized multifunctions, Fixed Point Theory and Applications, 2013 (2013), [16] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), [17] W. Sintunavarat, S. Plubtieng, P. Katchang, Fixed point results and applications on a b-metric space endowed with an arbitrary binary relation, Fixed Point Theory and Applications, 2013 (2013), Received: August 18, 2016; Published: October 16, 2016