Abstract

The purpose of this paper is using the viscosity approximation method to study the strong convergence problem for a family of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of the family of nonexpansive mappings are proved which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.

1. Introduction

Throughout this paper, we assume that is a CAT(0) space, is the set of positive integers, is the set of real numbers, is the set of nonnegative real numbers, and is a nonempty closed and convex subset of a complete CAT(0) space .

A mapping is called a nonexpansive mapping, if

It is well-known that one classical way to study nonexpansive mappings is to use the contractions to approximate nonexpansive mappings. More precisely, take and define a contraction by where is an arbitrary fixed element. In the case of having a fixed point, Browder [1] proved that converged strongly to a fixed point of that is nearest to in the framework of Hilbert spaces. Reich [2] extended Browder’s result to the setting of a uniformly smooth Banach space and proved that converged strongly to a fixed point of .

Halpern [3] introduced the following explicit iterative scheme (3) for a nonexpansive mapping on a subset of a Hilbert space: He proved that the sequence converged to a fixed point of .

Fixed-point theory in CAT(0) spaces was first studied by Kirk (see [4, 5]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the fixed-point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed. In 2012, using Moudafi’s viscosity approximation methods, Shi and Chen [6] studied the convergence theorems of the following Moudafi’s viscosity iterations for a nonexpansive mapping : They proved that defined by (4) and defined by (5) converged strongly to a fixed point of in the framework of CAT(0) space which satisfies the property .

Motivated and inspired by the researches going on in this direction, especially inspired by Shi and Chen [6], the purpose of this paper is to study the strong convergence theorems of Moudafi’s viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaces. We prove that the implicit and explicit iteration algorithms both converge strongly to the same point such that , which is the unique solution to the variational inequality (35), where is the set of common fixed points of the family of nonexpansive mappings.

2. Preliminaries and Lemmas

In this paper, we write for the unique point in the geodesic segment joining from to such that

Lemma 1 (see [7]). A geodesic space is a CAT(0) space if and only if the following inequality is satisfied for all and . In particular, if are points in a CAT(0) space and , then

Lemma 2 (see [8]). Let be a CAT(0) space, , and . Then By induction, one writes

Lemma 3. Let be a CAT(0) space; then, for any sequence in satisfying and for any , the following conclusions hold:

Proof. It is obvious that (11) holds for . Suppose that (11) holds for some . Next we prove that (11) is also true for . From (8) and (10) we have This implies that (11) holds.
Next, we prove that (12) holds.
Indeed, it is obvious that (12) holds for . Suppose that (12) holds for some . Next we prove that (12) is also true for .
In fact, we have From (7) and (10) and the assumption of induction, we have This completes the proof of (12).

Clearly, every CAT(0) space is strictly convex: if, in , and , then whenever . Dhompongsa et al. [9] showed the following conclusion which is called Condition (A):(A)if and belong to and for all , where , then for all .

The concept of -convergence introduced by Lim [10] in 1976 was shown by Kirk and Panyanak [11] in CAT(0) spaces to be very similar to the weak convergence in Banach space setting. Now, we give the concept of -convergence.

Let be a bounded sequence in a CAT(0) space . For , we set The asymptotic radius of is given by and the asymptotic center of is the set It is known from Proposition 7 of [12] that, in a complete CAT(0) space, consists of exactly one point. A sequence is said to -converge to if for every subsequence of .

The uniqueness of an asymptotic center implies that a CAT(0) space satisfies Opial’s property; that is, for given such that   -converges to and given with ,

Lemma 4 (see [11]). Every bounded sequence in a complete CAT(0) space always has a -convergent subsequence.
Berg and Nikolaev [13] introduced the concept of quasilinearization as follows. Let one denote a pair by and call it a vector. Then quasilinearization is defined as a map defined by It is easily seen that and for all . One says that satisfies the Cauchy-Schwarz inequality if for all .
Let be a nonempty closed convex subset of CAT(0) space . The metric projection is defined by
Recently, Dehghan and Rooin [14] presented a characterization of metric projection in CAT(0) spaces as follows.

Lemma 5. Let be a nonempty convex subset of a complete CAT(0) space ,   and . Then if and only if

Lemma 6 (see [15]). Let be a complete CAT(0) space, let be a sequence in , and . Then   -converges to if and only if for all .

Lemma 7 (see [16]). Let be a complete CAT(0) space. Then, for all , the following inequality holds:

Lemma 8 (see [16]). Let be a complete CAT(0) space. For any and , let . Then, for any , the following inequality holds:

Lemma 9 (see [17]). Let be a sequence of nonnegative real numbers satisfying the property , where and such that(i),(ii) or .
Then converges to zero as .

3. Viscosity Approximation Iteration Algorithms

In this section, we present the strong convergence theorems of Moudafi’s viscosity approximation implicit and explicit iteration algorithms for a family of nonexpansive mappings in CAT(0) spaces.

Lemma 10. Let be a nonempty closed convex subset of a complete CAT(0) space and let be a given sequence in such that and ; one defines a sequence as follows: Then the following holds:(i);(ii) is nonexpansive;(iii)for any , the sequence converges uniformly to an element , writing , where is a bounded subset of .

Proof. (i) For each we introduce thus
(ii) We will show by induction that is nonexpansive for all . Since , is nonexpansive. Suppose is nonexpansive. We consider Thus is nonexpansive.
(iii) In view of that , for any , we have This implies that the sequence converges uniformly to an element . Since is closed, .

Lemma 11. Let be a nonempty closed convex subset of a complete CAT(0) space , and let be a family of nonexpansive mappings satisfying . Define by for all , where with . Then is nonexpansive and .

Proof. For any , we have This implies that is nonexpansive.
It is easy to see that . We only show that . Let . For given , from Lemma 10(iii) we have In view of that we obtain that for all . By condition (A), for all . Thus we complete the proof of Lemma 10.

Now we are in a position to state and prove our main results.

Theorem 12. Let be a closed convex subset of a complete CAT(0) space , and let be a family of nonexpansive mappings satisfying . Let be a contraction on with coefficient , and let be as in Lemma 10. Suppose the sequence is given by for all , where satisfies . Then converges strongly to such that , which is equivalent to the following variational inequality:

Proof. We will divide the proof of Theorem 12 into five steps.
Step1. The sequence defined by (34) is well defined for all .
In fact, let us define the mapping by For any , from Lemma 2, we have This implies that is a contraction mapping. Hence, the sequence is well defined for all .
Step 2. The sequence is bounded.
For any , from Lemma 3, we have that Then This implies that Hence is bounded.
Step 3. , where .
From (34) and , we have From Lemma 10, we get
Step4. The sequence contains a subsequence converging strongly to such that , which is equivalent to (35).
Since is bounded, by Lemma 4, there exists a subsequence of (without loss of generality we denote it by ) which -converges to a point .
First we claim that . Since every CAT(0) space has Opial’s property, if , we have This is a contraction, and hence .
Next we prove that converges strongly to . Indeed, it follows from Lemma 8 that It follows that and thus Since   -converges to , by Lemma 6 we have It follows from (46) that converges strongly to .
Next we show that solves the variational inequality (35). Applying Lemma 1, for any , we have This together with Lemma 10(ii) implies that Taking the limit through , we can obtain On the other hand, from (20) we have From (50) and (51) we have That is, solves the inequality (35).
Step 5. The sequence converges strongly to .
Assume that as . By the same argument, we get that which solves the variational inequality (35); that is, Adding up (53) and (54), we get that Since , we have that , and so . Hence the sequence converges strongly to , which is the unique solution to the variational inequality (35).
This completes the proof.