Abstract

This paper is for the purpose of introducing and studying a class of new two-step viscosity iteration approximation methods for finding fixed points of set-valued nonexpansive mappings in spaces. By means of some properties and characteristic to space and using Cauchy-Schwarz inequality and Xu’s inequality, strong convergence theorems of the new two-step viscosity iterative process for set-valued nonexpansive and contraction operators in complete spaces are provided. The results of this paper improve and extend the corresponding main theorems in the literature.

1. Introduction

In [1], the fixed point theory in spaces was first introduced and studied by Kirk. Further, Kirk [1] presented that each nonexpansive (single-valued) mapping on a bounded closed convex subset of a complete space always has a fixed point. On the other hand, fixed point theory for set-valued mappings has been applied to applied sciences, game theory, and optimization theory. This promotes the rapid development of fixed point theory for single-valued (set-valued) operators in spaces, and it is natural and particularly meaningful to extensively study fixed point theory of set-valued operators. Particularly, some old relative works on Ishikawa iterations for multivalued mappings can be found in [2–4]. For more detail, we refer to [5–14] and the references therein.

Definition 1. Let be a nonlinear operator on a metric space and be a set-valued operator, where is a nonempty subset and is the family of nonempty bounded closed subsets of . Then
(i) is said to be a contraction, if there exists a constant such that Here, is called nonexpansive when in (1).
(ii) is said to be a nonexpansive, if where is Hausdorff distance on , i.e.,

Recently, Shi and Chen [5] first considered the following Moudafi’s viscosity iteration for a nonexpansive mapping with and a contraction mapping in space : and where and is an any given element in a nonempty closed convex subset . is called unique fixed point of contraction . Shi and Chen [5] showed that defined by (4) converges strongly to as , where in space satisfies the following property : for all , that is, an extra condition on the geometry of spaces is requested, where for . Further, the authors also found that the sequence generated by (5) converges strongly to under some suitable conditions about . Afterwards, based on the concept of quasilinearization introduced by Berg and Nikolaev [15], Wangkeeree and Preechasilp [6] explored strong convergence results of (4) and (5) in spaces without the property and presented that the iterative processes (4) and (5) converges strongly to such that is the unique solution of the following variational inequality:

In [16], Panyanak and Suantai extended (4) and (5) to being a set-valued nonexpansive mapping from to . That is, for each , let a set-valued contraction on be defined by By Nadler’s [17] theorem, it is easy to know that there exists such that is a fixed point of , which does not have to be unique, and i.e., for each , there exists such that Further, when the contraction constant coefficient of is and satisfying some suitable conditions, Panyanak and Suantai [16] proved strong convergence of one-step viscosity approximation iteration defined by (10) or the following iterative process in spaces: and for all , where is a set-valued nonexpansive operator from to , the family of nonempty compact subsets of , is a contraction, and . Moreover, Chang et al. [7] affirmatively answered the open question proposed by Panyanak and Suantai [16, Question 3.6]: “If and satisfying the same conditions, does converge to ”, where denotes the set of all fixed points of .

On the other hand, Piatek [18] introduced and studied the following two-step viscosity iteration in complete spaces with the nice projection property : where is an given element and satisfying some suitable conditions and the contraction coefficient of is .

Based on the ideas of Wangkeeree and Preechasilp [6] and Piatek [18] intensively, Kaewkhao et al. [19] omit the nice projection property . We note that the two-step viscosity iteration (12) is also considered and studied by Chang et al. [8] when the property is not satisfied and , which is due to the open questions in [19], where the property depends on whether its metric projection onto a geodesic segment preserves points on each geodesic segment, that is, for every geodesic segment and , implies , where denotes the metric projection from onto . For more works on the convergence analysis of (viscosity) iteration approximation method for (split) fixed point problems, one can refer to [20–27].

Motivated and inspired mainly by Panyanak and Suantai [16] and Piatek [18] and so on, we consider the following two-step viscosity iteration for set-valued nonexpansive operator : where is a nonempty closed convex subset of complete space , is an given element and , is a contraction mapping, and satisfying for any .

By using the method due to Chang et al. [7, 8], the purpose of this paper is to prove some strong convergence theorems of the viscosity iteration procedure (13) in complete spaces. Hence, the results of Chang et al. [7, 8] and many others in the literature can be special cases of main results in this paper.

2. Preliminaries

Throughout of this paper, let be a metric space. A geodesic path joining to (or, more briefly, a geodesic from to ) is a map such that , and for each . In particular, is a isometry and . The image of is called a geodesic segment (or metric) joining and if unique is bespoke by . The space is called a geodesic space when every two points in are joined by a geodesic, and is called uniquely geodesic if there is exactly one geodesic joining and for any . A subset of is said to be convex if includes every geodesic segment joining any two of its points. A geodesic triangle in a geodesic space consists of three points in (the vertices of ) and a choice of three geodesic segments (the edge of ) joining them. A comparison triangle for geodesic triangle in is a triangle in the Euclidean plane such that A point is said to be a comparison point for if . Similarly, we can give the definitions of comparison points on and .

Definition 2. Suppose that is a geodesic triangle in and is a comparison triangle for . A geodesic space is said to be a space, if all geodesic triangles of appropriate size satisfy the following comparison axiom (i.e., inequality): Complete spaces are often called Hadamard spaces (see [28]). For other equivalent definitions and basic properties of spaces, one can refer to [29]. It is well known that every space is uniquely geodesic and any complete, simply connected Riemannina manifold having nonpositive sectional curvature is a space. Other examples for spaces include pre-Hilbert spaces [29], trees [9], Euclidean buildings [30], and complex Hilbert ball with a hyperbolic metric [31] as special case.
Let be a nonempty closed convex subset of a complete space . By Proposition 2.4 of [29], it follows that, for all , there exists a unique point such that Here, is said to be unique nearest point of in .
Assume that is a space. For all and , by Lemma 2.1 of Phompongsa and Panyanak [10], there exists a unique point such that We shall denote by the unique point satisfying (17). Now, we give some results about spaces for the proof of our main results.

Lemma 3 ([1, 10]). Let be a space. Then for each and ,

(i)

(ii)

(iii)

Lemma 4 ([11]). Suppose that is a space. Then for all and ,

Lemma 5 ([12]). Assume that and are two bounded sequences in a space and is a sequence in with . If then .

Lemma 6 ([32]). Suppose that nonnegative real numbers sequence is defined by where and are two sequences satisfying
(i) ; (ii) or .
Then .

Lemma 7 ([13]). Assume that is a closed convex subset of a complete space . If a set-valued nonexpansive operator satisfies endpoint condition , i.e., and for any (see [33]), then is closed and convex.

In [15], Berg and Nikolaev introduced the concept of quasilinearization. Now we denote a pair by , which is a vector. Define the quasilinearization by a map : as follows: One can easily know that for every . We say that a geodesic metric space satisfies the Cauchy-Schwarz inequality if From [15, Corollary 3], it is known that a geodesic space is a space if and only if satisfies the Cauchy-Schwarz inequality. Further, we give the following other properties of quasilinearization.

Lemma 8 ([14]). Assume that is a nonempty closed convex subset of a complete space . Then for and ,

Lemma 9 ([6]). For two points and in a space and any , letting , then for all , the following results hold:

(i) ;

(ii) and .

Definition 10. A continuous linear functional is said to be Banach limit on if

Lemma 11 ([34]). Suppose that, for real number and all Banach limits , satisfies Then .

Lemma 12 ([16]). Assume that is a complete space, is a nonempty closed convex subset, is a set-valued nonexpansive operator satisfying endpoint condition , and is a contraction with . Then we have following results:

(i) generated by (10) converges strongly to as , where .

(ii) In addition, if is a bounded sequence such that , where is the distance from a point to the set , then for all Banach limits ,

3. Main Results

Employing the preliminaries in the previous section, now we will study the strong convergence of the new two-step viscosity iteration (13) for set-valued nonexpansive operators in complete spaces.

Theorem 13. Assume that is a complete space, is a nonempty closed convex subset, is a set-valued nonexpansive operator satisfying endpoint condition , and is contraction with . If sequences satisfy

() , () , and () , then the sequence generated by (13) converges strongly to , where

Proof. The proof shall be divided into the following four steps.
Step 1. We first prove that sequences , , , and are bounded. In fact, setting , then from Lemma 3, we know and Thus, we obtain Hence, is bounded, so is . By (29), it is easy to know that is bounded. Since , one can easily know that the sequence is also bounded.
Step 2. We present that , , , , , and . Indeed, by applying Lemmas 3 and 4, we have and so From and the boundedness of , , and , we know It follows from Lemma 5 that Thus, By (36), now we know that Moreover, and as . By (36) and (37), we get Step 3. Now, we show that with satisfying Above all, since is compact for any , then . It follows from Lemma 7 that is closed and convex, which implies that is well defined for any . By Lemma 12 (i), we know that generated by (10) converges strongly to as . Then by Lemma 8, we know that is the unique solution of the following variational inequality: Next, since is bounded and , it follows from Lemma 12 (ii) that for all Banach limits , and so Further, implies that By Lemma 11, we have Step 4. will be verified. In fact, by Lemma 3 and (13), now we know and It follows from (21), Cauchy-Schwarz inequality, and Lemma 9 that From (50) and (49), we know Combining (51) and (48), we get i.e., where , , and Thus, from the conditions - and the inequality (41), it follows that , and Hence, it follows from Lemma 6 that . This implies that the proof is completed.