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

- Li-Li Wan
^{8}Email author

**2015**:4

https://doi.org/10.1186/1687-1812-2015-4

© Wan; licensee Springer. 2015

**Received:**18 June 2014**Accepted:**26 November 2014**Published:**16 January 2015

## 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.

## Keywords

- total asymptotically nonexpansive nonself mappings
- hyperbolic space
- △-convergence

## 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 in*X*). 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.

- (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]).

*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
.

*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

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.

## 2 Preliminaries

*X*. For , we define

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*
.

*Let*
*be a complete uniformly convex hyperbolic space with monotone modulus of uniformconvexity and**C**a nonempty closed convex subset of**X*. *Then every bounded sequence*
*in**X**has a unique asymptotic center with respect to**C*.

**Lemma 2**[17]

*Let*

*be a uniformly convex hyperbolic space with monotone modulus of uniformconvexity*

*η*.

*Let*

*and*

*be a sequence in*

*for some*.

*If*

*and*

*are sequences in*

*X*

*such that*, ,

*and*

*for some*.

*Then*

**Lemma 3**[3]

## 3 Main results

*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)

*Let*
*be a complete uniformly convex hyperbolic space with monotone modulus of uniformconvexity**η*. *Let**C**be a nonempty closed and convex subset of**X*. *Let*
*be a uniformly**L*-*Lipschitzian and*
-*total asymptotically nonexpansive nonselfmapping*. *P**is a nonexpansive retraction of**X**onto**C*. *Let*
*be 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

*T*is uniformly

*L*-Lipschitzian, we have

*p*, so we can find , for any , that there exists such that . We can assume . Then and there exist such that

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

**Theorem 2**

*Let*

*C*

*be a nonempty closed and convex subset of a complete uniformly convex hyperbolicspace*

*X*

*with monotone modulus of uniform convexity*

*η*.

*Let*, ,

*be uniformly*

*L*-

*Lipschitzian and*-

*total asymptotically nonexpansive nonselfmappings*.

*For arbitrarily chosen*,

*is defined as follows*:

*where*

*P*

*is a nonexpansive retraction of*

*X*

*onto*

*C*.

*Assume that*

*and the following conditions are satisfied*:

- (i)
- (ii)
- (iii)

*then the sequence*
*defined by* (6) △-*converges to a point in* ℱ.

*Proof* We divide our proof into three steps.

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

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. □

*L*-Lipschitzian mappings with . and . Let

Therefore, the conditions of Theorem 2 are fulfilled.

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 3***Under the assumptions of Theorem *2, *if one of*
*and*
*is 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 4**

*Under the assumptions of Theorem*2,

*if there exists anondecreasing function*

*with*, ,

*such that*

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

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. □

## Declarations

### Acknowledgements

Supported by General Project of Educational Department in Sichuan (No. 13ZB0182)and National Natural Science Foundation of China (No. 11426190).

## Authors’ Affiliations

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