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

Li-Li Wan*

Correspondence: 15882872311@163.com School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China

Abstract

In this paper, we prove the demiclosed principle for total asymptotically nonexpansive nonself mappings in hyperbolic spaces. Then we obtain convergence theorems of the mixed Agarwal-O'Regan-Sahu type iteration for total asymptotically nonexpansive nonself mappings. Our results extend some results in the literature. MSC: 47H09; 49M05

Keywords: total asymptotically nonexpansive nonself mappings; hyperbolic space; A-convergence

1 Introduction

One of the fundamental and celebrated results in the theory of nonexpansive mappings is 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 T: C ^ X is a nonexpansive non-self mapping, then I - T is demiclosed at 0, that is, for any sequence {xn} in C if xn ^ x weakly and || (I - T)xn || ^ 0, then (I - T)x = 0 (where I is the identity mapping in X). Later, Chidume etal. [2] proved the demiclosed principle for asymptotically nonexpansive non-self mappings in uniformly convex Banach spaces. Recently, Chang et al. [3] proved the demiclosed principle for total asymptotically nonexpansive nonself mappings in CAT(0) spaces. It is well known that the demiclosed principle plays an important role in studying the asymptotic behavior for nonexpansive mappings. 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 our result to approximate common fixed points of total asymptotically nonexpansive nonself mappings in hyperbolic spaces, using the mixed Agarwal-O'Regan-Sahu type iterative scheme [4]. Our results extend and improve the corresponding results of Chang etal. [3], Nanjaras and Panyanak [5], Chang etal. [6], Zhao etal. [7], Khan etal. [8] and many other recent results.

In this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [9]. Concretely, (X, d, W) is called a hyperbolic space if (X, d) is a metric space and W: X x X x [0,1] ^ X a function satisfying

(I) Vx,y,z e X, VX e [0,1], d(z, W(x,y, X)) < (1- X)d(z,x) + Xd(z,y);

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©2015 Wan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(II) Vx,y e X, VA.i, X2 e [0,1], d(W(x,y, X1), W(x,y, X2)) = \Xi- X2\ -d(x,y);

(III) Vx, y e X, VX e [0,1], W(x, y, X)= W(y, x,(1-X));

(IV) Vx, y, z, w e X, VX e [0,1], d(W(x, z, X), W(y, w, X)) < (1 - X)d(x, y) + Xd(z, w).

If a space satisfies only (I), it coincides with the convex metric space introduced by Taka-hashi [10]. The concept of hyperbolic spaces in [9] is more restrictive than the hyperbolic type introduced by Goebel and Kirk [11] since (I)-(III) together are equivalent to (X, d, W) being a space of hyperbolic type in [11]. But it is slightly more general than the hyperbolic space defined in Reich and Shafrir [12] (see [9]). This class of metric spaces in [9] covers all normed linear spaces, R-trees in the sense of Tits, the Hilbert 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 of examples can be found in [9] (see also [11-13]).

A hyperbolic space is uniformly convex [16] if for u,x,y e X, r >0, and e e (0,2] there exists a S e (0,1] such that

d(V(x,y,0,u^ < (1- S)r,

provided that d(x, u) < r, d(y, u) < r, and d(x, y) > er.

A map n : (0, to) x (0,2] — (0,1] is called modulus of uniform convexity if S = n(r, e) for given r >0. The function n is monotone if it decreases with r (for a fixed e), that is,

n(r2, e) < n(r1, e), Vr2 > r1 > 0.

AsubsetC of a hyperbolic space X is convex if W(x, y, X) e C forall x, y e C and X e [0,1]. Let (X, d) be a metric space and let C be a nonempty subset of X. C is said to be a retract of X, if there exists a continuous map P: X — C such that Px = x, Vx e C.A map P: X — C is said to be a retraction, if P2 = P.If P is a retraction, then Py = y for all y in the range of P. Recall that a nonself mapping T: C — X is said to be a ({vn}, |^n}, Z)-total asymptotically nonexpansive nonselfmapping if there exist nonnegative sequences {vn}, {¡in} with vn — 0, ¡xn — 0, and a strictly increasing continuous function Z : [0, (x>) ^ [0, to) with Z (0) = 0 such that

d(T(PT)n-1x, T(PT)n-1y) < d(x,y) + VnZ(d(x,y)) + ¡in, Vn > 1,x,y e C, (1)

where P is a nonexpansive retraction of X onto C. It is well known that each nonexpansive mapping is an asymptotically nonexpansive mapping and each asymptotically nonexpansive mapping is a ({vn}, {^n}, Z)-total asymptotically nonexpansive mapping. T: C ^ X is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that

d(T(PT)n-1x, T(PT)n-1y) < Ld(x,y), Vn > 1,x,y e C.

2 Preliminaries

We now give the concept of A-convergence and collect some of its properties. Let {xn} be a bounded sequence in a hyperbolic space X. For x e X,we define

r(x, {xn}) = limsup d(x, xn).

The asymptotic radius r({xn}) of {xn} is given by

r({xn}) = inf{r(x, {xn}): x e X}.

The asymptotic radius rC({xn}) of {xn} with respect to C c X is given by

rC({xn}) = inf{r(x, {xn}): x e C}.

The asymptotic center A({xn}) of {xn} is the set

A( {xn}) = {x e X: r(x, {xn}) = r ({xn})}.

The asymptotic center AC({xn}) of {xn} with respect to C c X is the set

Ac ({xn}) = {x e C: r(x, {xn}) = C{xn})}.

Recall that a sequence {xn} in X is said to A-converge tox e X if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case we callx the A-limitof {xn}.

Lemma 1 [17, 18] Let (X, d, W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity and C a nonempty closed convex subset ofX. Then every bounded sequence {xn} in X has a unique asymptotic center with respect to C.

Lemma 2 [17] Let (X, d, W) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity n. Letx e Xand{an} be a sequence in [a, b] for some a, b e (0,1). If {xn} and {yn} are sequences in X such that limsupn—TO d(xn,x) < c, limsupn—TO d(yn,x) < c, and limn—TO d(W(xn,yn,an),x) = cforsome c > 0. Then

lim d(xn,yn) = 0.

Lemma 3 [3] Let {an}, {bn}, and {cn} be sequences of nonnegative numbers such that

an+1 < (1 + bn)an + cn, "in > 1. If bn < ro and X^i cn < ro, then limn—TO an exists. 3 Main results

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

{xn} ^ o if and only if $(«) = inf $(x),

where C is a closed convex subset which contains the bounded sequence {xn} and $(x) := lim supn—TO d(xn, x).

Theorem 1 (Demiclosed principle for total asymptotically nonexpansive nonself mappings in hyperbolic spaces) Let (X, d, W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity q. Let C be a nonempty closed and convex subset ofX. Let T: C — X be a uniformly L-Lipschitzian and (|^nj, {vn}, Z)-total asymptotically nonexpansive nonself mapping. P is a nonexpansive retraction ofX onto C. Let {xn} cC be a bounded approximate fixed point sequence, i.e., limn—TO d(xn, Txn) = 0 and {xn} ^p. Then we have T(p) = p.

Proof By the definition, {xn} ^ p if and only if AC({xn}) = {p}. By Lemma 1, we have A({xn}) = {p}. Since limn—TO d(xn, Txn) = 0, by induction we can prove that

lim d{xn, T(PT)m-1xn) = 0 for each m > 1. (2)

n—>TO v 7

In fact, it is obvious that the conclusion is true for m = 1. Suppose the conclusion holds for m > 1, now we prove that it is also true for m + 1. Indeed, since T is uniformly L-Lipschitzian, we have

d(xn, T(PT)mxn) < d(xn, T(PT)m-1xn) + d(T(PT)m-1xn, T(PT)mxn)

< d(xn, T(PT)m-1xn) + Ld(xn, PTxn)

< d(xn, T(PT)m-1xn) + Ld(xn, Txn) — 0 as n — to.

Equation (2) is proved. Hence for each x e X and m > 1, from (2) we have

$(x) := limsupd(xn,x) = limsupd(T(PT)m-1xn,x). (3)

n—TO n—TO

Taking x = T(PT)m-1p, m > 1 in (3), then by (1) we get

$(T(PT)m-1p) = limsupd(T(PT)m-1xn, T(PT)m-1p)

< limsupd(xn,p) + vmZ(d(xn,p)) + fim}.

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

limsup$(T(PT)m-1p) < $(p). (4)

m—TO

Assume that Tp = p. Then {T(PT)m-1p} does not converge to p, so we can find e0 > 0, for any k e N, that there exists m > k such that d(T(PT)m-1p,p) > e0. We can assume S0 e (0,2]. Then ^p+j e (0,2] and there exist 0 e (0,1] such that

1_n($(p) + 1,_fO < (5)

\ ^ ®(p)+1j ~ $(p) + 0 V '

By the definition of $ and (4), for the above 0, there exists N, M e N such that

d(p,xn) < $(p) + 0, Vn > N;

d(T(PT)m-1p,x„) < $(p) + e, Vk > W, m > M. For M, there exists m > M such that

d(T(pT)m-1p,p) > eo = ^ № + e) > *p7I • № + e).

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

d(V(p, t(PT)m-1p, 1),x.) < (1 - n($(p) + e, • (*(p) + e)

$(p)-e

$(p) + 0 = $(p)-0.

($(p) + 0)

Since z := W(p, T(PT)m 1p, 2) = p, we have got a contradiction with A({xn}) = {p}. It follows that Tp = p and the proof is completed. □

Theorem 2 Let C be a nonempty closed and convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity n. Let Ti : C ^ X, i = 1,2, be uniformly L-Lipschitzian and ({vn}, {^n}, Z)-total asymptotically nonexpansive nonself mappings. For arbitrarily chosen x1 e C, {xn} is defined as follows:

Xn+1 = PW(xn, T1(PT1)n-1yn, Un), n > 1,

n . ia„ i. n i.

! (6) yn = PW (Xn, T2(PT2)n-1Xn, ßn),

where Pisa nonexpansive retraction ofX onto C. Assume that F =: P| F(Ti) = 0 and the following conditions are satisfied:

(i) E^=1 Vn < ^ and E^=1 H-n < to;

(ii) there exist constants a, b e (0,1) such that |an}, {pn} c [a, b];

(iii) there exists a constant M >0 such that Z (r) < Mr, r > 0, then the sequence {xn} defined by (6) A-converges to a point in F.

Proof We divide our proof into three steps. Step 1. In the sequel, we shall show that

lim d(xn,p) exists for each p e F. (7)

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

d(yn,p) = d(PW(xn, T2(PT2)n-1Xn, Pn),p)

< d(W (xn, T2(PT2)n-1xn, Pn), p)

< (1 - Pn)d(xn,p) + Pnd(T2(PT2)n-1xn,p)

< (1 - Pn)d(xn,p) + Pn[d(xn,p) + Vnz (d(xmp)) + ^n]

< (1 + vnM)d(xn,p) + fin (8)

d(xn+1,p) = d(PW(xn, T1(PT1)n-1yn, an),p)

< d(W(xn, Ti(PTi)n-1yn, an),p)

< (1 - an)d(xn,p) + and(T1(PT1)n-1yn,p)

< (1 - an)d(xn,p) + an[d(yn,p) + Vnf (d(yn,p)) + ¡n]

< (1 - an)d(xn,p) + an(1 + vnM)[(1 + vnM)d(xn,p) + ^^ + an^n

< [1 +2vnM + vX)]d(xn,p) + (2 + VnM)iin. (9)

Combining (8) and (9), we have

d(xn+1,p) < (1 + <Jn)d(xmp) + fn, Vn > 1, (10)

where an = 2vnM + v^M2, fn = (2 + vnM)^n. Furthermore, using the condition (i), we have

y^ an < to and ^ fn < to. (1)

n=1 n=1

Consequently, a combination of (10), (11), and Lemma 3 shows that (7) is proved. Step 2. We claim that

lim d(xn, Tixn) = 0, i = 1,2. (12)

n—>to

In fact, it follows from (7) that limn—TO d(xn,p) exists for each given p e F. Without loss of generality, we assume that

lim d(xn,p) = c > 0. (13)

n—TO

By (8) and (13), we have

liminf d(yn, p) < lim sup d(yn, p) < lim {(1 + vnM)d(xn, p) + ¡A = c. (14)

n—TO n—TO n—TO'

Noting

d(T1(PT1)n 1yn,p) < d(yn,p) + vnt (d(yn,p)) + ¡n

< (1 + vnM)d(yn,p) + ¡n, Vn > 1,

by (14) we have

lim sup d(T1(PT1)n-1yn,p) < c. (15)

n—TO

Besides, by (10) we get

d(xn+1,p) = d(PW(xm T1(PT1)n-1yn, an),p) < (1 + 0n)d(xmp) + fn,

which yields

lim d(W(xn, T1(PT1)n-1yn,an),p) = c. (6)

n—TO v v ' '

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

lim d(xn, T1(PT1)n-1yn) = 0. (17)

n—TO v '

Using the same method, we also have

lim d(xn, T2(PT2)n-1xn) = 0. (18)

n—TO v '

By virtue of (18), we get

d(yn,xn) = d(PW(xn, T2(PT2)n -1x n, Pn), xn}

< 4W(xn, T2(PT2)n-1x n , P n , xn

< Pnd(T2(PT2)n-1xn,xn) — 0 as n — to. (19)

Combining (17) and (19), we obtain

d(xn, T1(PT1)n-1xn) < d(xn, T1(PT1)n-1yn) + d(T1(PT1)n-1yn, T1(PT1)n-1xn)

< d(xn, T1(PT1)n-1y^ + Ld(yn,xn) — 0 as n — to. (20)

Moreover, it follows from (17) that

d(xn+1,xn) = d(PW(xm T1(PT1)n-1yn, a^,x^

< 4 W(xn, T1(PT1)n-1y n , a n , xn

< and(T1(PT1)n-1yn,xn) — 0 as n — to. (21) Now by (18), (20), and (21), for each i = 1,2, we get

d(xn, Tixn) < d(xn,xn+1) + d(xn+1, Ti(PTi)nxn+1) + d(Ti(PTi)nxn+1, Ti(PTi)nxn)

+ d(Ti(PTi)n xn , Ti xn

= d(xn, xn+1) + ^Ti (PTi)n xn+1, Ti (PTi) n xn + d xn+1, Ti(PTi)nxn+1

+ d(Ti(PTi)n xn , Ti xn

< (1 + L)d(xn,xn+1) + d(xn+1, Ti(PTi)nxn+1)

xn, x^ ^ 0 as n — to.

Therefore, (12) holds.

Step 3. Now we are in a position to prove the A-convergence of {xn}. Since {xn} is bounded, by Lemma 1, it has a unique asymptotic center AC({xn}) = {x*}. Let {un} be any subsequence of {xn} with AC({un}) = {u}. Since limn—TO d(xn, T1xn) = limn—TO d(xn, T2xn) =

0, it follow from Theorem 1 that u e F. By the uniqueness of asymptotic centers, we get x* = u. It implies that x* is the unique asymptotic center of {un} for each subsequence {un} of {xn}, that is, {xn} A-converges to x* e F. The proof is completed. □

Example 1 Let R be the real line with the usual norm | • | and let C = [-1,1]. Define two mappings Ti, T2 : C — C by

Tx |-2 sin f, x e [0,1], lX "If sin f, x e [-1,0),

Tx = fx, x e [0,1], l-x, x e [-1,0).

It is proved in [19, Example 3.1] that both T1 and T2 are asymptotically nonexpansive mappings with kn = 1, Vn > 1. Therefore, they are total asymptotically nonexpansive mappings with vn = ¡¡n = 0, Vn > 1, Z (r) = r, Vr > 0. Additionally, they are uniformly L-Lipschitzian mappings with L = 1. F(T1) = {0} and F(T2) = {0 < x < 1}. Let

a. = ~-r, Pn = ~--, Vn > 1. (22)

2n + 1 3n + 1

Therefore, the conditions of Theorem 2 are fulfilled.

Example 2 Let R be the real line with the usual norm | • | and let C = [0, to). Define two mappings T1, T2 : C ^ C by

T1x = sin x and T2x = x.

It is proved in [20, Example 1] that both T and T2 are total asymptotically nonexpansive mappings with vn = l^n = nr, — 1- Moreover, they are uniformly L-Lipschitzian mappings with L = 1- F(T1) = {0} and F(T2) = {0 < x < to}- Let |an}, |pnj be the same as in (22). Therefore, the conditions of Theorem 2 are fulfilled.

Theorem 3 Under the assumptions of Theorem 2, if one ofT1 and T2 is demi-compact, then the sequence defined by (6) converges strongly (i.e., in the metric topology) to a common fixed point in F.

Proof By (12) and the assumption that one of T1 and T2 is demi-compact, there exists a subsequence {xni} c {xn} such that {xni} converges strongly to some point p e C. Then by the continuity of T1 and T2, we get

d(p, Tip) = lim d(xn:, TiXn, ) = 0, i = 1,2,

n—>to ' '

which implies that p e F -It follows from (7) that limn—TO d(xn, p) exists and thus limn—TO d(xn,p) = 0. The proof is completed- □

Theorem 4 Under the assumptions of Theorem 2, if there exists a nondecreasingfunction f: [0, to) — [0, to) withf (0) = 0, f (r) > 0, Vr >0 such that

f (d(x, F)) < d(x, Tix) + d(x, T2x), Vx e C, (23)

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

Proof By (12) and (23) we obtain limn—TOf (d(xn, F)) = 0. Since f is nondecreasing with f (0) = 0, f (r) > 0, Vr > 0, we have

lim d(xn, F) = 0. (24)

n—>TO

Now we prove that {xn} is a Cauchy sequence in C. In fact, it follows from (10) that, for any p e F,

d(xn+i,p) < (1 + On)d(xn,p) + fn, Vn > 1,

where ^^ an < to and ^^ fn < to. Then, for anyp e F and any positive integers n, m, we get

d(xn+m, x^) << d(xn+m,p) + d(xn,p)

< (1 + &n+m-l)d(x n+m-1,p) + fn+m- 1 + d(xn, p).

Since for each x > 0,1 + x < ex,we obtain

d(%n+m, xn) < e d(xn+m-1,p) + fn+m— 1 + d(xn, p)

< eCT«+m-1+CT«+m-2d(xn+m-2,p) + eCT"+m-1 fn+m-2 + fn+m-1 + d(xn,p)

y^n+m-1 ^ y^n+m-1 ^ y^n+m-2 n

< ld(xmp) + ^i=«+1 fn + ^f="+2 fn+1 + • ••

+ e «+m 1 fn+m-2 + fn+m- 1 + d(xn, p) n+m-1

< (1+ K)d(xn,p)+Kj2 fi,

where K = e^s< to. It follows from (24) that

d(xn+m,xn) < (1 + K)d(xn, F) + K fi — 0 as n, m — to.

Thus {xn} is a Cauchy sequence in C. C is complete for it is a closed subset in a complete hyperbolic space. Without loss of generality, we can assume that {xn} converges strongly to some point p* e C. It is easy to prove that F is closed. It follows from (24) thatp* e F. The proof is completed. □

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Supported by GeneralProject of EducationalDepartment in Sichuan (No. 13ZB0182) and NationalNaturalScience

Foundation of China (No. 11426190).

Received: 18 June 2014 Accepted: 26 November 2014 Published: 16 Jan 2015

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10.1186/1687-1812-2015-4

Cite this article as: Wan: Demiclosed principle and convergence theorems for total asymptotically nonexpansive nonself mappings in hyperbolic spaces. Fixed PointTheory and Applications 2015, 2015:4

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