Abstract

Let K be a nonempty closed and convex subset of a complete CAT(0) space. Let , be a family of multivalued demicontractive mappings such that . A Krasnoselskii-type iterative sequence is shown to -converge to a common fixed point of the family . Strong convergence theorems are also proved under some additional conditions. Our theorems complement and extend several recent important results on approximation of fixed points of certain nonlinear mappings in CAT spaces. Furthermore, our method of the proof is of special interest.

1. Introduction

A metric space is said to be a CAT space if it is geodesically connected and if every geodesic triangle in is at least as “thin” as its comparison triangle in the Euclidean space. It is well known that pre-Hilbert spaces, -trees (see [1]), and Euclidean buildings (see, e.g., [2]) are among examples of CAT spaces. For a thorough discussion of these spaces and the fundamental role they play in various branches of mathematics see Bridson and Haefliger [1] or Burago et al. [3]. Fixed point theory in CAT spaces was first studied by Kirk (see [4, 5]). He showed that every nonexpansive mapping defined on a nonempty closed convex and bounded subset of a CAT space always has a fixed point. Since then, the fixed point theory for single-valued and multivalued mappings has received much attention (see, e.g., [6–13]). In 1976, Lim [14] introduced a notion of convergence in a general metric space which he called -convergence (see Definition 8). In 2008, Kirk and Panyanak [15] specialized Lim’s concept to CAT spaces and showed that many results which involve weak convergence (e.g., Opial property and Kadec-Klee property) have precise analogs in this setting. Later on, Dhompongsa and Panyanak [16] obtained -convergence theorems for the Picard, Mann, and Ishikawa iterations involving one mapping in the CAT space setting.

In [17], Chidume et al. introduced the class of multivalued -strictly pseudocontractive mappings which is a generalization of the class of multivalued nonexpansive mappings in Hilbert spaces. They constructed a Krasnoselskii-type algorithm sequence and showed that it is an approximate fixed point sequence of the map. In particular, they proved the following theorem.

Theorem 1 (Theorem 3.1 of [17]). Let be a nonempty, closed, and convex subset of a real Hilbert space . Suppose that is a multivalued -strictly pseudocontractive mapping such that . Assume that for all . Let be a sequence defined by where and . Then, .

Very recently, Chidume and Ezeora extended the result of Chidume et al. [17] to a finite family of multivalued -strictly pseudocontractive mappings in real Hilbert spaces. The following theorem is their main result.

Theorem 2 (Theorem 2.2 of [18]). Let be a nonempty, closed, and convex subset of a real Hilbert space , and let be a finite family of multivalued -strictly pseudocontractive mappings, , such that . Assume that, for all , . Let be sequence defined by where , and , such that , where . Then, .

Remark 3. In Theorem 2.2 of [18], the condition that , such that , where , restricts the class of operators for which the theorem is applicable. For instance, if , then the theorem is not applicable to the family of the mappings for which , since there is no , , such that .
In [19], Isiogugu and Osilike proved weak and strong convergence theorems for the class of multivalued demicontractive mappings which contains the class of -strictly pseudocontractive mappings for which the fixed point set is nonempty. They proved the following theorem in the setting of real Hilbert spaces.

Theorem 4 (Theorem 3.1 of [19]). Let be a nonempty closed convex subset of a real Hilbert space H. Suppose that is a demicontractive mapping from into the family of all proximinal subsets of with and for all . Suppose (I-T) is weakly demiclosed at zero. Then, the Mann type sequence defined by converges weakly to , where and is a real sequence in satisfying (i) and (ii) .

It is our purpose in this paper to prove strong and -convergence theorems for a Krasnoselskii-type algorithm sequence to a common fixed point of a finite family of demicontractive mappings in the setting of CAT spaces. In our results, the condition imposed on , in Theorem 2.2 of [18] is reduced to the condition , where the rest of the , , can be chosen arbitrarily in (0; 1). Thus, our result is applicable to all classes of demicontractive mappings. Furthermore, our theorems extend and improve the results of Chidume and Ezeora [18], Chidume et al. [17], and Isiogugu and Osilike [19] and complement the results of Dhompongsa and Panyanak [16], Dhompongsa et al. [9], Leustean [11], Shahzad and Markin [13], and Sokhuma [20] and results of a host of other authors on iterative approximation of fixed points in CAT spaces.

2. Preliminaries

Let be a metric space. A geodesic path joining and is a continuous map from a closed interval to such that , and for all . In particular, the mapping is an isometry and . The image of is called a geodesic segment joining and . When it is unique, this geodesic segment is denoted by . The space is called a geodesic space if any two points of are joined by a geodesic, and is said to be uniquely geodesic if there is exactly one geodesic joining and . A subset of is said to be convex if, for all , the segment remains in .

A geodesic triangle in a geodesic metric space consists of three points in (the vertices of ) and a geodesic segment between each pair of points (the edges ). A comparison triangle for in is a triangle in the Euclidean plane such that , for . A geodesic metric space is called a CAT space if all geodesic triangles satisfy the following comparison axiom.

Let be a geodesic triangle in , and let be its comparison triangle in . Then, is said to satisfy CAT inequality, if, for all and all comparison points , If , , are points in CAT space, and if is the midpoint of the segment , then, the CAT inequality implies This is the inequality of Bruhat and Tits [21]. In fact (cf. [1], p.163), a geodesic space is a CAT space if and only if it satisfies the inequality.

We now collect some elementary facts about CAT spaces.

Lemma 5 (see, e.g., [16]). Let be a CAT space. Then (i) is uniquely geodesic.(ii)For each and , there exists a unique point such that

For convenience, from now on, we will use the notation for the unique point satisfying (5).

Also, for such that and , we will use the notation to denote the unique point satisfying In particular, taking , we compute the point as follows.

From the illustration above, , where denotes the unique point such that , and .

Thus, we have , where denotes the unique point satisfying , and . Hence we have .

Extending this notation up to some , we use to denote the unique point satisfying where , , such that , , , .

Remark 6. The metric convex combinations defined above in (7) are similar to that defined on a Hilbert ball by Kopecká and Reich in [22], where the authors defined the metric convex combinations for self-maps , , on a Hilbert ball.

Lemma 7 (see, e.g., Lemmas 2.4 and 2.5 in [16]). Let be a CAT space. For and , the following inequalities hold: (i),(ii), where .

We now give the -convergence together with some of its basic properties.

Let be a bounded sequence in a CAT space . For , we set . The asymptotic radius of is given by and the asymptotic center of is the set It is well known that, in a CAT space, consists of exactly one point.

Definition 8. A sequence in a CAT space is said to -converge to if is the unique asymptotic center of every subsequence of . In this case we write and is called the -limit of .

Lemma 9. (i) (See, e.g., [15]). Every bounded sequence in a complete CAT space has a -convergent subsequence.
(ii) (See, e.g., [23]). If is a nonempty closed and convex subset of a complete CAT space and if is a bounded sequence in , then the asymptotic center of is in .
(iii) (See, e.g., [16]). If is a bounded sequence in a complete CAT   with and is a subsequence of with and the sequence converges, then .

Let be a geodesic metric space. We denote by the collection of all nonempty closed and bounded subsets of . Let be the Hausdorff metric with respect to the metric distance ; that is, for all , where is the distance from the point to the subset .

Let be a multivalued mapping on . A point is called a fixed point of if . The set is called the fixed point set of .

Definition 10. Let be a geodesic metric space. A multivalued mapping is said to be (i)nonexpansive if (ii)quasinonexpansive if and (iii)demicontractive if and there exists such that where and ,(iv)hemicontractive if in (iii) above; that is, It is clear that every multivalued nonexpansive mapping with nonempty fixed point set is quasinonexpansive, and every quasinonexpansive mapping is demicontractive mapping.
The following example shows that the class of demicontractive mappings strictly contains the class of quasinonexpansive mappings.

Example 11. Let (the set of real numbers with the usual metric). Define by Then, , and is demicontractive mapping which is not quasinonexpansive.
Indeed, for each , we have which implies that is not quasinonexpansive.
We also have that Thus, Hence, is a demicontractive mapping with constant .

3. Main Results

We start by proving the following lemmas.

Lemma 12. Let be a CAT(0) space. Let , and , , such that . Then, the following inequality holds:

Proof. The proof is by induction. For , the result follows from Lemma 7(ii). For simplicity, we will give the proof for . From Lemma 7(ii), we have that Now, suppose (19) holds up to some ; that is, Then, from Lemma 7 we have Using the induction hypothesis, we have Hence, by induction we have that (19) is true. The proof is complete.

Lemma 13. Let be a nonempty closed convex subset of a complete CAT space . Let , be a family of demicontractive mappings with constants , such that . Suppose that for all . For arbitrary , define a sequence by where , ,  ,   , , such that , and . Then, exists for all , and for all .