Demiclosed principle and convergence theorems for total asymptoticallynonexpansive nonself mappings in hyperbolic spaces

Abstract

In this paper, we prove the demiclosed principle for total asymptoticallynonexpansive nonself mappings in hyperbolic spaces. Then we obtain convergencetheorems of the mixed Agarwal-O’Regan-Sahu type iteration for totalasymptotically nonexpansive nonself mappings. Our results extend some results inthe literature.

MSC: 47H09, 49M05.

1 Introduction

One of the fundamental and celebrated results in the theory of nonexpansive mappingsis Browder’s demiclosed principle[1] which states that if X is a uniformly convex Banach space,C is a nonempty closed convex subset of X, and if is a nonexpansive nonself mapping, then is demiclosed at 0, that is, for any sequence in C if weakly and , then (where I is the identity mapping inX). Later, Chidume et al.[2] proved the demiclosed principle for asymptotically nonexpansive nonselfmappings in uniformly convex Banach spaces. Recently, Chang et al.[3] proved the demiclosed principle for total asymptotically nonexpansivenonself mappings in CAT(0) spaces. It is well known that the demiclosed principleplays an important role in studying the asymptotic behavior for nonexpansivemappings. The purpose of this paper is to extend Chang’s result from CAT(0)spaces to the general setup of uniformly convex hyperbolic spaces. We also apply ourresult to approximate common fixed points of total asymptotically nonexpansivenonself mappings in hyperbolic spaces, using the mixed Agarwal-O’Regan-Sahutype iterative scheme [4]. Our results extend and improve the corresponding results of Chang etal.[3], Nanjaras and Panyanak [5], Chang et al.[6], Zhao et al.[7], Khan et al.[8] and many other recent results.

In this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [9]. Concretely, is called a hyperbolic space if is a metric space and a function satisfying

• (I) , , ;

• (II) , , ;

• (III) , , ;

• (IV) , , .

If a space satisfies only (I), it coincides with the convex metric space introducedby Takahashi [10]. The concept of hyperbolic spaces in [9] is more restrictive than the hyperbolic type introduced by Goebel andKirk [11] since (I)-(III) together are equivalent to being a space of hyperbolic type in [11]. But it is slightly more general than the hyperbolic space defined inReich and Shafrir [12] (see [9]). This class of metric spaces in [9] covers all normed linear spaces, ℝ-trees in the sense of Tits, theHilbert ball with the hyperbolic metric (see [13]), Cartesian products of Hilbert balls, Hadamard manifolds (see [12, 14]), and CAT(0) spaces in the sense of Gromov (see [15]). A thorough discussion of hyperbolic spaces and a detailed treatment ofexamples can be found in [9] (see also [11–13]).

A hyperbolic space is uniformly convex[16] if for , , and there exists a such that

provided that , , and .

A map is called modulus of uniform convexity if for given . The function η is monotone ifit decreases with r (for a fixed ϵ), that is,

A subset C of a hyperbolic space X is convex if for all and .

Let be a metric space and let C be a nonemptysubset of X. C is said to be a retract of X, ifthere exists a continuous map such that , . A map is said to be a retraction, if. If P is a retraction, then for all y in the range of P. Recallthat a nonself mapping is said to be a -total asymptotically nonexpansive nonselfmapping if there exist nonnegative sequences , with , , and a strictly increasing continuous function with such that

(1)

where P is a nonexpansive retraction of X onto C. It iswell known that each nonexpansive mapping is an asymptotically nonexpansive mappingand each asymptotically nonexpansive mapping is a -total asymptotically nonexpansive mapping.

is said to be uniformlyL-Lipschitzian if there exists a constant such that

2 Preliminaries

We now give the concept of △-convergence and collect some of its properties.Let be a bounded sequence in a hyperbolic spaceX. For , we define

The asymptotic radiusof is given by

The asymptotic radiusofwith respect to is given by

The asymptotic centerof is the set

The asymptotic centerofwith respect to is the set

Recall that a sequence in X is said to △-converge to if x is the unique asymptotic center of for every subsequence of . In this case we call x the△-limit of.

Lemma 1[17, 18]

Letbe a complete uniformly convex hyperbolic space with monotone modulus of uniformconvexity andCa nonempty closed convex subset ofX. Then every bounded sequenceinXhas a unique asymptotic center with respect toC.

Lemma 2[17]

Letbe a uniformly convex hyperbolic space with monotone modulus of uniformconvexityη. Letandbe a sequence infor some. Ifandare sequences inXsuch that, , andfor some. Then

Lemma 3[3]

Let, , andbe sequences of nonnegative numbers such that

Ifand, thenexists.

3 Main results

We shall prove that a total asymptotically nonexpansive nonself mapping in a completeuniformly convex hyperbolic space X with monotone modulus of uniformconvexity is demiclosed. We need the following notation:

where C is a closed convex subset which contains the bounded sequence and .

Theorem 1 (Demiclosed principle for total asymptotically nonexpansive nonselfmappings in hyperbolic spaces)

Letbe a complete uniformly convex hyperbolic space with monotone modulus of uniformconvexityη. LetCbe a nonempty closed and convex subset ofX. Letbe a uniformlyL-Lipschitzian and-total asymptotically nonexpansive nonselfmapping. Pis a nonexpansive retraction ofXontoC. Letbe a bounded approximate fixed point sequence, i.e.,and. Then we have.

Proof By the definition, if and only if . By Lemma 1, we have . Since , by induction we can prove that

(2)

In fact, it is obvious that the conclusion is true for . Suppose the conclusion holds for, now we prove that it is also true for. Indeed, since T is uniformlyL-Lipschitzian, we have

Equation (2) is proved. Hence for each and , from (2) we have

(3)

Taking , in (3), then by (1) we get

Letting and taking superior limit on the both sides, we have

(4)

Assume that . Then does not converge to p, so we can find, for any , that there exists such that . We can assume . Then and there exist such that

(5)

By the definition of Φ and (4), for the above θ, there exists such that

For M, there exists such that

Since X is uniformly convex and η is monotone, applying (5)we have

Since , we have got a contradiction with. It follows that and the proof is completed. □

Theorem 2LetCbe a nonempty closed and convex subset of a complete uniformly convex hyperbolicspaceXwith monotone modulus of uniform convexityη. Let, , be uniformlyL-Lipschitzian and-total asymptotically nonexpansive nonselfmappings. For arbitrarily chosen, is defined as follows:

(6)

wherePis a nonexpansive retraction ofXontoC. Assume thatand the following conditions are satisfied:

• (i) and;

• (ii) there exist constantssuch that;

• (iii) there exists a constantsuch that, ,

then the sequencedefined by (6) △-converges to a point in ℱ.

Proof We divide our proof into three steps.

Step 1. In the sequel, we shall show that

(7)

In fact, by conditions (1), (I), and (iii), we get

(8)

and

(9)

Combining (8) and (9), we have

(10)

where , . Furthermore, using the condition (i), we have

(11)

Consequently, a combination of (10), (11), and Lemma 3 shows that (7) isproved.

Step 2. We claim that

(12)

In fact, it follows from (7) that exists for each given . Without loss of generality, we assume that

(13)

By (8) and (13), we have

(14)

Noting

by (14) we have

(15)

Besides, by (10) we get

which yields

(16)

Now by (13), (15), (16), and Lemma 2, we have

(17)

Using the same method, we also have

(18)

By virtue of (18), we get

(19)

Combining (17) and (19), we obtain

(20)

Moreover, it follows from (17) that

(21)

Now by (18), (20), and (21), for each , we get

Therefore, (12) holds.

Step 3. Now we are in a position to prove the △-convergence of. Since is bounded, by Lemma 1, it has a uniqueasymptotic center . Let be any subsequence of with . Since , it follow from Theorem 1 that. By the uniqueness of asymptotic centers, we get. It implies that is the unique asymptotic center of for each subsequence of , that is, △-converges to . The proof is completed. □

Example 1 Let ℝ be the real line with the usual norm and let . Define two mappings by

and

It is proved in [[19], Example 3.1] that both and are asymptotically nonexpansive mappings with, . Therefore, they are total asymptoticallynonexpansive mappings with , , , . Additionally, they are uniformlyL-Lipschitzian mappings with . and . Let

(22)

Therefore, the conditions of Theorem 2 are fulfilled.

Example 2 Let ℝ be the real line with the usual norm and let . Define two mappings by

It is proved in [[20], Example 1] that both and are total asymptotically nonexpansive mappings with, , . Moreover, they are uniformly L-Lipschitzianmappings with . and . Let , be the same as in (22). Therefore, the conditions ofTheorem 2 are fulfilled.

Theorem 3Under the assumptions of Theorem 2, if one ofandis demi-compact, then the sequence defined by (6)converges strongly (i.e., in the metrictopology) to a common fixed point in ℱ.

Proof By (12) and the assumption that one of and is demi-compact, there exists a subsequence such that converges strongly to some point. Then by the continuity of and , we get

which implies that . It follows from (7) that exists and thus . The proof is completed. □

Theorem 4Under the assumptions of Theorem 2, if there exists anondecreasing functionwith, , such that

(23)

then the sequence defined by (6) converges strongly(i.e., in the metric topology) to a common fixedpoint in ℱ.

Proof By (12) and (23) we obtain . Since f is nondecreasing with, , , we have

(24)

Now we prove that is a Cauchy sequence in C. In fact, itfollows from (10) that, for any ,

where and . Then, for any and any positive integers n, m, weget

Since for each , , we obtain

where . It follows from (24) that

Thus is a Cauchy sequence in C. C iscomplete for it is a closed subset in a complete hyperbolic space. Without loss ofgenerality, we can assume that converges strongly to some point. It is easy to prove that ℱ is closed. Itfollows from (24) that . The proof is completed. □