∆−Convergence Theorems for Multivalued Non-expansive Mappings in Hyperbolic Spaces

The study for fixed point problem involve that multivalued contractions and non-expansive mappings used the Hausdorff metric was initiated by Markin [1,2] later, different iterative processes have used to approximate the fixed points of multivalued non-expansive mappings in Banach space, many scholars have made extensive research in [1-17]. An interesting and rich fixed point theory for such mappings was developed which has applications in control theory, convex optimization, differential inclusion, and economics [3]. But the hyperbolic space has no set up the theory of multivalued non-expansive mappings fixed point. In order to define the concept of multivalued non-expansive mapping in the general setup of Banach spaces, we first collect some basic concepts.


Introduction
The study for fixed point problem involve that multivalued contractions and non-expansive mappings used the Hausdorff metric was initiated by Markin [1,2] later, different iterative processes have used to approximate the fixed points of multivalued non-expansive mappings in Banach space, many scholars have made extensive research in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. An interesting and rich fixed point theory for such mappings was developed which has applications in control theory, convex optimization, differential inclusion, and economics [3]. But the hyperbolic space has no set up the theory of multivalued non-expansive mappings fixed point. In order to define the concept of multivalued non-expansive mapping in the general setup of Banach spaces, we first collect some basic concepts.
Let E be a real Banach space. A subset K is called proximinal if for each x ∈ E, there exists an element k ∈ K such that ( ) It is known that weakly compact convex subsets of a Banach space and closed convex subset of a uniformly convex Banach space are proximinal sub-set of K by P(K). Consistent with [1], let CB(K) be the class of all nonempty bounded and closed subset of K. Let  (1) Moreover, F (T)=F (P T ).
Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [18], defined below, which is restrictive than the hyperbolic type introduced in [19] and more general than the concept of hyperbolic space in [20].
A hyperbolic space is a metric space (X, d) together with a mapping , for all x, y, z, w ∈ X and α, β ∈ [0, 1]. A nonempty subset K of a hyperbolic space X is convex if W (x, y, α) ∈ K for all x, y ∈ K and α ∈ [0, 1]. The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric [21], Hadamard manifolds as well as CAT(0) spaces in the sense of Gromov [22].
A hyperbolic space is uniformly convex [23] if for any r>0 and ε ∈ (0, 2] there exists a δ ∈ (0, 1] such that for all u, x, y ∈ X , we have for For given ,is known as a modulus of uniform convexity of X. We call η monotone if it decreases with r (for a fixed ε), i.e., )). , In the sequel, let (X, d) be a metric space and let K be a nonempty subset of X. We shall denote the fixed point set of a mapping T by The purpose of this paper is that iteration scheme of multivalued non-expansive mappings in Banach spaces is extended to hyperbolic spaces and to prove some ∆−convergence theorems of the mixed type iteration process for approximating a common fixed point of two multivalued non-expansive mappings and other two non-expansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [6][7][8][9][11][12][13]15,16,19,21,[24][25][26]. In order to define the concept of ∆−convergence in the general setup of hyperbolic spaces, we also collect some basic concepts and Lemmas. Lemma 1.3: [26] Let (X, d, W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then every bounded sequence {x n } in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
Recall that a sequence {x n } in X is said to ∆−converge to x ∈ X if x is the unique asymptotic center of {u n } for every subsequence {u n } of {x n }. In this case, we write :  ( Proof: The proof of Theorem 2.1 is divided into three steps: Step: First we prove that ) , For any given p∈F, since , is a multivalued nonexpansivemapping, S i , i=1,2 is a non-expansive mapping, by condition (1 ) ( , ) (1 ) ( , )

Competing Interests
The author declares that he has no competing interests.

Author's Contributions
Author contributed equally and significantly in this research work.