Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces

Weprove strong convergence of the sequence generatedby implicit viscosity approximationmethod involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete R-trees without the endpoint condition.


Introduction
Let C be a nonempty closed convex subset of a Hilbert space E and T : C → C, the fixed point set of T is denoted by F(T), that is, F(T) = {x ∈ C : x = Tx}. A mapping T : C → C said to be a nonexpansive mapping if ||Tx − Ty|| ≤ ||x − y|| for all x, y ∈ C.
In the past few decades, several iterative schemes have been introduced to approximate fixed point of nonexpansive mappings.
Moudafi has shown that under certain conditions, the sequence {xn} not only converges strongly to a fixed point x * of T but also solves the following variational inequality: The viscosity approximation has attracted the attention of several mathematicians due to its applications in monotone inclusions, convex optimization, and linear programming, etc. In 2015, Xu et al. [2] modified the viscosity method and introduced an implicit midpoint rule for nonexpansive mappings as follows: x 0 ∈ C arbitrarily chosen The authors showed that viscosity implicit midpoint rule converges strongly to a fixed point of T and also solves the variational inequality (1). Ke and Ma [3] studied the generalized viscosity implicit rules of nonexpansive mappings. They introduced the following schemes and proved convergence theorems under certain assumptions imposed on the sequences of parameters αn , and βn: x n+1 = αn f (xn) + (1 − αn)T(βn xn + (1 − βn)x n+1 ), and x n+1 = αn xn + βn f (xn) + n T(sn xn + (1 − sn)x n+1 ).
Recently, many authors have generalized results on nonlinear mappings to more general spaces than Hilbert spaces such as CAT(0) spaces such as [4][5][6]. The study of viscosity approximation methods has been extended to various directions. Wu and Zhao [7] proved the viscosity approximation results for multivalued nonexpansive mappings in the setup of Hilbert and Banach spaces and obtained a relationship between fixed point of the such mappings and the solution of the variational inequality problem. Preechasilp [8] extended the results in [2] to CAT(0) spaces. Panyanak and Suantai [9] extended the results in [10] for a multivalued nonexpansive mapping in the setting of CAT(0) spaces. Xiong and Lan [11] and [12] studied a two-step viscosity iteration approximation methods for approximating the fixed points of multivalued nonexpansive mappings in CAT(0) spaces.
In this paper, motivated by the work in [8] and [11,12], the generalized implicit viscosity approximation scheme presented by Ke and Ma [3] is extended and generalized in the following ways: (a) underlying space possess a nonlinear structure; (b) a two step implicit viscosity scheme is introduced for a multivalued nonexpansive mapping. Consider the following two step implicit viscosity approximation method for multivalued nonexpansive mapping in the framework of CAT(0) spaces. Let αn , βn ∈ (0, 1), and x 0 ∈ C chosen arbitrarily. Define ) and x n+1 = αn f (xn) ⊕ (1 − αn)un; un ∈ Tyn; such that d(un , u n+1 ) ≤ d(yn , y n+1 ) . We prove the convergence of the above implicit viscosity iterative processes to the fixed point of the multivalued nonexpansive map under some appropriate conditions on the parameters. We also show that this fixed point solves a variational inequality. Now, we recall some definitions and lemmas to be used in the sequel. A subset C of a metric space (E, d) is called proximal if for any x ∈ E, there exists an element y ∈ C such that We denote by K(C) and CB(X), the collection of all nonempty compact subsets of C and the collection of all nonempty closed bounded subsets of C, respectively. The Hausdorff distance induced by metric d on E is given by Let T : C → 2 C , where 2 C is the collection of all nonempty subsets of C. An element x ∈ C is called a fixed point of T, if x ∈ Tx. The set of all fixed points of T will be denoted by F(T). Let x, y ∈ E. A geodesic path joining x to y (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, a] to E such that c(0) = x, c(a) = y, and d(c(s), c(t)) = |t − s| for all t, s ∈ [0, a]. In particular, c is an isometry and d(x, y) = a. The image of c is called a geodesic (or metric) segment joining x and y. When it is unique, the geodesic segment is denoted by [x, y]. The space E is called a geodesic if every two points of E are joined by a geodesic, and E is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ E. A subset C ⊂ X is said to be convex if C includes every geodesic segment joining any two of its points.
A geodesic triangle ∆(p, q, r) in a geodesic space (E, d) consists of three points p, q, r in E (called vertices of ∆) and a choice of three geodesic segments [p, q], [q, r], [r, p] ( edges of ∆) joining them. A comparison triangle for a geodesic triangle ∆(p, q, r) in E is a triangle ∆(p, q, r) in the Euclidean plane R 2 such that d R 2 (p, q) = d(p, q), d R 2 (q, r) = d(q, r) and d R 2 (r, p, ) = d(r, p). A triangle △(p, q, r) having vertices p, q, r ∈ R 2 is called a comparison triangle of ∆(p, q, r) in E.
A Suppose that ∆ is a geodesic triangle in E and ∆ is a comparison triangle for ∆. A geodesic space is said to be a CAT(0) space, if all geodesic triangles of appropriate size satisfy the following comparison axiom called CAT(0) inequality: Every CAT(0) space is a uniquely geodesic and any complete and simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples of CAT(0) spaces include pre-Hilbert spaces , R-trees, Euclidean buildings and complex Hilbert ball with a hyperbolic metric as special case (see, for example, [13][14][15] ).
Let C be a nonempty closed convex subset of a complete CAT(0) space (E, d). Then, for any x ∈ E there exists a unique point x 0 ∈ C such that [14] d(x, x 0 ) = inf{d(x, y) : y ∈ C}.
A point x 0 in C is said to be a unique nearest point of x ∈ E. The metric projection of E onto C is the mapping P C : E → C defined as: corresponding to each x in E, By Lemma 2.1 [13], for each x, y ∈ E and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that Throughout this paper, we shall use the notation tx ⊕ (1 − t)y for the unique point z satisfying (4). Finally, we give some known results in CAT(0) spaces needed in our results.
Berg and Nikolaev [19] introduced an important concept of quasilinearization as follows: ab and call it a vector. The quasilinearization is a map ⟨., .⟩ : It is known that a geodesically connected metric space is CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality [19]. Following are some important lemmas.
Lemma 1.0.6. [10] Let E be a complete CAT(0) space. Then for all u, x, y ∈ E, the following inequality Then, for each x, y ∈ E, we have The asymptotic radius r({xn}) of {xn} is given by A multivalued nonexpansive operator T :

Results
Now, we present the following convergence result of our iterative scheme (2).
Then the sequence generated by (2) converges strongly to a fixed point x * of the nonexpansive mapping T, which solves the variational inequality That is, x * ∈ P F(T) f (x * ).
Proof. We shall be divide the proof into four steps.

STEP 4
Next, we show that x * = P F (T)f (x * ). Let q ∈ F(T). Since T(x) is compact for any x ∈ C, T(x) ∈ CB(X). From Lemma 1.0.12 it follows that F(T) is closed and convex, which further implies that P F(T) z is well defined for any z ∈ E. Moreover, applying Lemma 1.0.2 we obtain that Applying the Banach limit, we have For any x * ∈ C such that xn → x * , we obtain Thus, from (23) and (24), we have Using (25) and the fact that xn → x * as n → ∞ in (22) we get that That is, Hence, from Lemma 1.0.8 x * = P F(T) f (x * ) and this completes the proof.
If T is a nonexpansive single-valued operator, then from Theorem (2.0.1), we can obtain the following result in [3].  converges strongly to a fixed point x * which is equivalent to the following variational inequality If f is an identity function then we obtain the following result.
∑︀ ∞ n=1 |α n+1 − αn| = ∞, A4 0 < ϵ ≤ βn ≤ β n+1 < 1 for all n ≥ 0 then the for u ∈ C the sequence generated by x n+1 ), converges strongly to a unique nearest point x * of u in F(T) which is equivalent to the following variational inequality Now, we prove some convergence results of the implicit viscosity approximation method in complete R-trees without the endpoint condition.
Every R-tree is a CAT(0) space which does not contain the Euclidean plane. Thus to avoid the endpoint condition, R−trees are preferred. Although an R−tree is not strong enough to make all nonexpansive mappings having the endpoint condition, but it is strong enough to make our theorems hold without this condition. Let C be closed convex subset of a complete R−tree (E, d). Let T : C → K(C) be a multivalued mapping. Then, there exists a single-valued mapping t : C → C such that tx ∈ Tx and d(tx, ty) ≤ H (Tx, Ty) for all x, y ∈ C, see [22] for details. In this case, we call t a nonexpansive selection of T. A multivalued mapping T : C → K(C) is called a quasi-nonexpansive mapping if Clearly, every multivalued nonexpanive mapping is quasi-nonexpansive. The following result is needed in our result.
Then the sequence generated by (29) converges strongly to a fixed point x * of T, which solves the variational inequality That is, x * ∈ P F(T) f (x * ).
Proof. The proof is similar to Theorem 2.0.7.

Remarks
Results presented in this paper extend corresponding results in [3] to multivalued nonexpansive mappings in the framework of CAT(0) spaces. Using Cauchy Schwarz inequality [19] and other results from [10], [8] etc. we have shown the strong convergence of the new viscosity method for the implicit rule of a multivalued nonexpansive mapping in CAT(0) space. However, in iterative method the condition d(xn , z n+1 ) ≤ d(xn , x n+1 ) on is imposed. Can these results be obtained without this condition? Furthermore, can these results be obtained when f is a multivalued contraction mapping or f satisfies other generalized contraction conditions? These are the open questions worth studying in the future.