Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2011, Article ID 745451,19 pages doi:10.1155/2011/745451

Research Article

Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

Tanakit Thianwan

School of Science, University ofPhayao, Phayao 56000, Thailand Correspondence should be addressed to Tanakit Thianwan, tanakit.th@up.ac.th Received 31 May 2011; Revised 26 August 2011; Accepted 30 August 2011 Academic Editor: Victor M. Perez Garcia

Copyright © 2011 Tanakit Thianwan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach space E which has a Frechet differentiable norm or its dual E* has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Frechet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.

1. Introduction and Preliminaries

Let C be a nonempty closed convex subset of real normed linear space X. Let T : C ^ C be a mapping. A point x € C is called a fixed point of T if and only if Tx = x. The set of all fixed points of a mapping T is denoted by F (T). A self-mapping T : C ^ C is said to be nonexpansive if ||T(x) - T(y)|| < ||x - y|| for all x,y € C. A self-mapping T : C ^ C is called asymptotically nonexpansive if there exists a sequence {kn} c [1, <x>),kn ^ 1asn ^ to such that

||Tn(x) - Tn(y)\\ < kn||x-y|| (1.1)

for all x,y € C and n > 1. A mapping T : C ^ C is said to be uniformly L-Lipschitzian if there exists a constant L> 0 such that

\\T"(x) - Tn{y) || < L\\x - y||

for all x,y e C and n > 1. T is uniformly Holder continuous if there exist positive constants L and a such that

||Tn(x) - Tn(y) || <L||x - y||a (1.3)

for all x,y e C and n > 1. T is termed as uniformly equicontinuous if, for any e > 0, there exists 6 > 0 such that

||Tn(x) - Tn(y) || < e (1.4)

whenever ||x-y\\ < 6 for all x,y e C and n > 1 or, equivalently, T is uniformly equicontinuous if and only if

||Tn(xn) - Tn(yn)|| 0 (1.5)

whenever \\xn - yn\\ ^ 0 as n ^ to.

It is easy to see that if T is an asymptotically nonexpansive, then it is uniformly L-Lipschitzian with the uniform Lipschitz constant L = sup{kn : n > 1}.

Remark 1.1. It is clear that asymptotically nonexpansiveness ^ uniformly L-Lipschitz ^ uniformly Holder continuous ^ uniformly equicontinuous.

However, their converse fail in the presence of the following example. Example 1.2 (see [1]). Define T : [0,1] ^ [0,1] by Tx = (1 - x3/2)2/3 for all x e [0,1].

Fixed-point iteration process for nonexpansive self-mappings including Mann and Ishikawa iteration processes has been studied extensively by various authors [2-8]. For nonexpansive nonself-mappings, some authors (see [9-13]) have studied the strong and weak convergence theorems in Hilbert spaces or uniformly convex Banach spaces. In [14], Tan and Xu introduced a modified Ishikawa iteration process:

xn+1 = (1 - bn)xn + bnT( (1 - Jn)xn + JnTxn), n > 1, (1.6)

to approximate fixed points of nonexpansive self-mappings defined on nonempty closed convex bounded subsets of a uniformly convex Banach space X. The mapping T remains self-mapping of a nonempty closed convex subset C of a uniformly convex Banach space. If, however, the domain C of T is a proper subset of X (and this is the case in several applications) and T maps C into X then, the sequence {xn} generated by (1.6) may not be well defined. More precisely, Tan and Xu [14] proved weak convergence of the sequences generated by (1.6) to some fixed point of T in a uniformly convex Banach space which satisfies Opial's condition or has a Frechet differentiable norm.

Note that each lp (1 < p < to) satisfies Opial's condition, while all Lp do not have the property unless p = 2 and the dual of reflexive Banach spaces with a Frechet differentiable norm has the Kadec-Klee property. It is worth mentioning that there are uniformly convex Banach spaces, which have neither a Frechet differentiable norm nor Opial property; however, their dual does have the Kadec-Klee property (see [15,16]).

In 2005, Shahzad [11] extended Tan and Xu's result [14] to the case of nonexpansive nonself-mapping in a uniformly convex Banach space. He studied weak convergence of the modified Ishikawa type iteration process:

X„+1 = P((1 - bn)Xn + bnTP( (1 - Jn)xn + JnTXn)), n > 1, (1.7)

in a uniformly convex Banach space whose dual has the Kadec-Klee property. The result applies not only to Lp spaces with (1 < p < to) but also to other spaces which do not satisfy Opial's condition or have a Frechet differentiable norm. Meanwhile, the results of [11] generalized the results of [14].

The class of asymptotically nonexpansive self-mappings is a natural generalization of the important class of nonexpansive mappings. Goebel and Kirk [17] proved that if C is a nonempty closed convex and bounded subset of a real uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.

In 1991, the modified Mann iteration which was introduced by Schu [18] generates a sequence {xn} in the following manner:

xn+i = (1 - an)xn + anTnxn, n > 1, (1.8)

where {an} is a sequence in the interval (0,1) and T : C ^ C is an asymptotically nonexpansive mapping. To be more precise, Schu [18] obtained the following weak convergence result for an asymptotically nonexpansive mapping in a uniformly convex Banach space which satisfies Opial's condition.

Theorem 1.3 (see [18]). Let X be a uniformly convex Banach space satisfying Opial's condition, 0 /C c X closed bounded and convex, and T : C ^ C asymptotically nonexpansive with sequence {kn} c [1, to) for which £ TO=1(kn -1) < to and {an} e [0,1] is bounded away. Let {xn} beasequence generated in (1.8). Then, the sequence {xn} converges weakly to some fixed point of T.

Since then, Schu's iteration process has been widely used to approximate fixed points of asymptotically nonexpansive self-mappings in Hilbert space or Banach spaces (see [6,14, 19, 20]).

In 1994, Tan and Xu [21] obtained the following results.

Theorem 1.4 (see [21]). Let X be a uniformly convex Banach space whose norm is Frechet differentiable, C a nonempty closed and convex subset of X, and T : C ^ C an asymptotically nonexpansive mapping with a sequence {kn} c [1, to) such that £TO=1(kn -1) < to such that F(T) is nonempty. Let {xn} be sequence generated in (1.8), where {an} is a real sequence bounded away from 0 and 1. Then, the sequence {xn} converges weakly to some point in F(T).

In 2001, Khan and Takahashi [22] constructed and studied the following Ishikawa iteration process:

xn+1 = (1 - an)xn + anTn yn,

yn = (1 - Pn)xn + PnT^xn, n > 1,

where T1, T2 are asymptotically nonexpansive self-mappings on C with ^°°=1(kn - 1) < ° (rate of convergence) and 0 < an, ¡n < 1.

Note that the rate of convergence condition, namely, ^°°=1(kn - 1) < ° has remained in extensive use to prove both weak and strong convergence theorems to approximate fixed points of asymptotically nonexpansive maps. The conditions like Opial's condition, Kadec-Klee property, or Frechet differentiable norm have remained key to prove weak convergence theorems.

In 2010, Khan and Fukhar-Ud-Din [23] established weak convergence of Ishikawa iterates of two asymptotically nonexpansive self-mappings without any condition on the rate of convergence associated with the two mappings. They got that the following new weak convergence theorem does not require any of Opial's condition, Kadec-Klee property or Frechet differentiable norm.

Theorem 1.5 (see [23]). Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let T1,T2 : C — C be asymptotically nonexpansive maps with sequences {kn}, {ln} c [1, °) such that limnkn = 1, limnln = 1, respectively. Let the sequence {xn} be as in (1.9) with 6 < an,fin < 1 - 6. for some 6 e (0,1/2). If F(T1) n F(T2) / 0, then {xn} converges weakly to a common fixed point of T1 and T2.

The concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [24] in 2003 as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.

Definition 1.6 (see [24]). Let C be a nonempty subset of a real normed linear space X. Let P : X — C be a nonexpansive retraction of X onto C. A nonself-mapping T : C — X is called asymptotically nonexpansive if there exists a sequence {kn} c [1, °), kn — 1 as n —> °o such that

||T(PT)n-1x - T(PT)n-1y|| < kn||x - y|| (1.10)

for all x,y e C and n > 1. T is said to be uniformly L-Lipschitzian if there exists a constant L> 0 such that

||t(PT)n-1x - T(PT)n-1y|| < L||x - y|| (1.11)

for all x,y e C and n > 1.

By studying the following iteration process:

x1 e C, xn+1 = P((1 - an)xn + anT(PT)n-1xn), (1.12)

Chidume et al. [24] got the following weak convergence theorem for asymptotically nonexpansive nonself-mapping.

Theorem 1.7 (see [24]). Let X be a real uniformly convex Banach space which has a Frechet differentiable norm and C a nonempty closed convex subset of X. LetT : C — X bean asymptotically nonexpansive map with sequence {kn} c [1, °) such that ^°°=1(k2 - 1) < ° and F(T) / 0. Let

{an} c (0,1) be such that e < 1 - an < 1 - e,for all n > 1 and some e> 0. From an arbitrary x1 e C, define the sequence {xn} by (1.12). Then, {xn} converges weakly to some fixed point of T.

If T is a self-mapping, then P becomes the identity mapping so that (1.10) and (1.11) reduce to (1.1) and (1.2), respectively. Equation (1.12) reduces to (1.8).

In 2006, Wang [25] generalizes the iteration process (1.12) as follows: x1 e C,

where T1,T2 : C ^ X are asymptotically nonexpansive nonself-mappings and {an}, {pn} are real sequences in [0,1). He studied the strong and weak convergence of the iterative scheme (1.13) under proper conditions. Meanwhile, the results of [25] generalized the results of [24].

Recently, an iterative scheme which is called the projection type Ishikawa iteration for two asymptotically nonexpansive nonself-mappings was defined and constructed by Thianwan [26]. It is given as follows:

where {an} and {pn} are appropriate real sequences in [0,1).

In [26] , Thianwan gave the following weak convergence theorem.

Theorem 1.8. Let X be a uniformly convex Banach space which satisfies Opial's condition and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1, T2 : C ^ X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn}, {ln} c [1, go) such that X^=1(k (ln -1) < g, respectively, and F(T1) n F(T2) — 0. Suppose

that {an} and {pn} are real sequences in [e, 1 - e] for some e e (0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, {xn} and {yn} converge weakly to a common fixed point of T1 and T2.

The iterative schemes (1.14) and (1.13) are independent: neither reduces to the other. If T1 = T2 and p n — 0 for all n > 1, then (1.14) reduces to (1.12). It also can be reduces to Schu process (1.8).

Inspired and motivated by the recent works, we prove some new weak convergence theorems of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.

Now, we recall some well-known concepts and results.

Let X be a Banach space with dimension X > 2. The modulus of X is the function 6x : (0,2] ^ [0,1] defined by

x„+i = P((1 - a„)x„ + a„Ti(PTi)" !y„), yn = P((1 - ßn)xn + ßnTi(PT2)n-1x^, n > 1,

(1.13)

xn+1 = p( (1 - an)yn + anT1(PT1)n 1y^, yn = P((1 - ßn)xn + ßnTi(PT2)n-1xn), n > 1,

(1.14)

ÖX(e)= f - 2(x + y) : llxll = 1,||y|| = 1,e = ||x - y||}

(1.15)

Banach space X is uniformly convex if and only if SX(e) > 0 for all e € (0,2]. It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that a Banach space X is said to satisfy Opial's condition [27] if weakly

as n —> to and x = y implying that

limsup||xn -x|| < limsup|xn -y\\. (1.16)

n — <x> n — m

The norm of X is said to be Frechet differentiable if for each x € X with ||x|| = 1 the limit

liml|X + - "X" (1.17)

t—o t v '

exists and is attained uniformly for y, with ||y || = 1. In the case of Frechet differentiable norm, it has been obtained in [21] that

1 2 1 2 (h,J(x)) + 2||x||2 < 2||x + h||2

(1.18)

<(h,J (x)) + - ||x||2 + b(||h||)

for all x, h in E, where J is the normalized duality map from E to E* defined by

J(x) = {x* e E* : <x,s*) = ||x||2 = ||x*||2}, (1.19)

(•, ■) is the duality pairing between E and E* and b is an increasing function defined on [0, to) such that lim^0 b(t)/t = 0.

A subset C of X is said to be retract if there exists continuous mapping P : X ^ C such that Px = x for all x e C. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping P : X ^ X is said to be a retraction if P2 = P. If a mapping P is a retraction, then Pz = z for every z e R(P), range of P. A set C is optimal if each point outside C can be moved to be closer to all points of C. It is well known (see [28]) that

(1) if X is a separable, strictly convex, smooth, reflexive Banach space, and if C c X is an optimal set with interior, then C is a nonexpansive retract of X;

(2) a subset of lp, with 1 <p < to, is a nonexpansive retract if and only if it is optimal.

Note that every nonexpansive retract is optimal. In strictly convex Banach spaces, optimal sets are closed and convex. Moreover, every closed convex subset of a Hilbert space is optimal and also a nonexpansive retract.

Recall that weak convergence is defined in terms of bounded linear functionals on X as follows.

A sequence {x„} in a normed space X is said to be weakly convergent if there is an x e X such that lim„f (xn) = f (x) for every bounded linear functional f on X. The element x is called the weak limit of {xn}, and we say that {xn} converges weakly to x. In this paper, we use ^ and ^ to denote the strong convergence and weak convergence, respectively.

A Banach space X is said to have the Kadec-Klee property if, for every sequence {xn} in X, xn —^ x and \\xn\\ ^ ||x|| together imply \\xn - x|| ^ 0; for more details on Kadec-Klee property, the reader is referred to [29, 30] and the references therein.

In the sequel, the following lemmas are needed to prove our main results.

Lemma 1.9 (see [31]). Let p > 1, r > 0 be two fixed numbers. Then, a Banach space X is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g : [0, to) ^ [0, to), g(0) = 0 such that

||Ax + (1 - A)y||p <A\\x\\p + (1 - A) ||y||p - wp(X)g(||x - y||) (1.20)

for all x, y in Br = {x e X : \\x\\ <r}, A e [0,1], where

Wp(A) = A(1 - A)p + Ap(1 - A). (1.21)

Lemma 1.10 (see [24]). Let X be a uniformly convex Banach space and C a nonempty closed convex subset of X, and let T : C ^ X be an asymptotically nonexpansive mapping with a sequence {kn}c [1, to) and kn ^ 1 as n ^ to. Then, I - T is demiclosed at zero; that is, if xn ^ x weakly and xn - Txn ^ 0 strongly, then x e F (T).

Lemma 1.11 (see [26]). Let X be a uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2 : C ^ X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn}, {ln} c [1, to) such that XTO=!(k n 1) < to^ ¿j n=1 (ln - 1) < to, respectively, and F(T1) n F(T2) = 0. Suppose that {an} and {fin} are real sequences in [0,1). From an arbitrary xi e C, define the sequence {xn} by (1.14). If q e F(T1) n F(T2), then limn^to \\xn - q\\ exists.

Lemma 1.12 (see [26]). Let X be a uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1 ,T2 : C ^ X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn}, {ln} c [1,to) such that STO=1(k n 1) < to^ ¿J n=1 (ln - 1) < to, respectively, and F(T1) n FT) = 0. Suppose that {an} and {fin} are real sequences in [e, 1 - e] for some e e (0,1). From an arbitrary x1 e C, define the sequence {xn} by (1.14). Then, limn^to \\xn - TxW = limn^to \\xn - T2xn\\ = 0.

Lemma 1.13 (see [16]). Let X be a real reflexive Banach space such that its dual X* has the Kadec-Klee property. Let {xn} be a bounded sequence in X and x*,y* e ww(xn), where ww(xn) denotes the set of all weak subsequential limits of {xn}. Suppose that limn ^to \\txn + (1 - t)x* - y*\\ exists for all t e [0,1]. Then, x* = y*.

We denote by r the set of strictly increasing, continuous convex functions j : R+ ^ R+ with j(0) = 0. Let C be a convex subset of the Banach space X. A mapping T : C ^ C is said to be type (j) [32] if y e r and 0 < a < 1,

Y(||aTx + (1 - a)Ty - T(ax + (1 - a)y)||) < ||x - y|| - ||Tx - Ty|| (1.22)

for all x, y in C. Obviously, every type (y) mapping is nonexpansive. For more information about mappings of type (y), see [33-35].

Lemma 1.14 (see [36,37]). Let X be a uniformly convex Banach space and C a convex subset of X. Then, there exists y e r such that for each mapping S : C ^ C with Lipschitz constant L,

||aSx + (1 - a)Sy - S(ax + (1 - a)y)|| < Lr-^||x - y\\ - L||Sx - Sy|^ (1.23)

for all x,y e C and 0 < a < 1.

2. Main Results

In this section, we prove weak convergence theorems of the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.

Firstly, we deal with the weak convergence of the sequence {xn} defined by (1.14) in a real uniformly convex Banach space X whose dual X* has the Kadec-Klee property. In order to prove our main results, the following lemma is needed.

Lemma 2.1. Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2 : C ^ X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn}, {ln} c [1, to) such that (k n 1) < то, ¿J n=1 (ln - 1) < to, respectively, and F(T1) n FT) = 0. Suppose that {an} and {fin) are real sequences in [e, 1 - e] for some e e (0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, for all u,v e F (Ti) n F(T2), the limit limn \\txn - (1 - t)u - v|| exists for all t e [0,1].

Proof. It follows from Lemma 1.11 that the sequence {xn} is bounded. Then, there exists R> 0 such that {xn} c BR(0)nC.Let an(t) := \\txn + (1-t)u-v\\ where t e [0,1].Then,limnan(0) = \\u - v\\ and, by Lemma 1.11, limnan(1) = limn\\xn - v\ exists. Without loss of the generality, we may assume that limn\\xn - v\ = r for some positive number r. Let x e C and t e (0,1). For each n > 1, define An : C ^ C by

Anx = P((1 - an)yn(x) + anT1(PT1)n-1yn(x)), (2.1)

yn(x) = p((1 - fin)x + finT2(PT2)n-1 x). (2.2)

Setting kn = 1 + sn and ln = 1 + tn. For x,z e C, we have \\Anx - Anz\\ = ||p((1 - an)yn(x) + anT1(PT1)n-1yn(x))

-P((1 - an)yn(z) + anT1(PT1)n-1 y,(z)) || < ||(1 - an){yn(x) - yn(z))

+ an(T1 (PT1)n-1yn(x) - T1(PT1)n-1yn(z))H

< (1 - an)\\yn{x) - yn(z)|| + ankn\\yn(x) - yn(z)\\

< (1 - an) \\ (1 - fin) (x - z) + finT2(PT2)n-1 (x - z) \\ + ankn\\ (1 - fin) (x - z)+ finT2(PT2)n-1(x - z) \\

< (1 - an)(1 - fin)\\x - z\\ + (1 - an)finln\\x - z|| + ankn( 1 - fin) \\x - z\\ + anfinknln\\x - z\\

= (1 - an - fin + anfin)\\x - z\\ + (1 - an)fin(1 + tn)\\x - z\\ + an(1 + Sn)( 1 - fin)\\x - z\\ + anfin(1 + sn)(1 + tn)\\x - z\\

= \\x - z\\ + fintn\\x - z\\ + anSn\\x - z\\ + anfintnSn\\x - z\\

< (1 + tn + Sn + tnSn)\\x - z\\.

Set Sn,m •— -An+m- \An+m-2 • -'—n, n,m > 1 and bn,m = \\Sn,m(txn + (1 - t)u) - (tSn,mxn + (1 - t)u) \\, where 0 < t < 1. Also,

\\Sn,mx - Sn,my\\ < \\An+m-1(An+m-2 ' ' ' Anx) - An+m-\( An+m-2 ' ' ' Any)\\

< (1 + tn+m-1 + Sn+m-1 + tn+m-1Sn+m-1)

\ \ An+m-2 (An+m-3 ' ' ' Anx) - An+m-2( An+m-3 ' ' ' Any) \ \

< n i1 + tj + Sj + ¥i)\\x - y\\ j=n

for all x,y e C and Snm xn — xn+m, Snm x* — x* for all x* e F(T\) n F(T2).

Applying Lemma 1.14 with x — xn, y — u, S — Snm and using the facts that limn^^ tn — limn ^^ (ln - 1) — 0, lim n ^ra Sn — limn^^ (kn - 1) — 0, and limn^ \\xn - x*\ exist for all x* e F(T\) n F(T2), we obtain limn^^ bnm — 0. Observe that

an+m(t) — \\txn+m + (1 - t)u - v\\

— \\tSn,mxn + (1 - t)u - Sn,mV\\

— \\Sn,mv - (tSn,mxn + (1 - t)u)\\

— \\Sn,mv - Sn,m(txn + (1 - t)u) + Sn,m(txn + (1 - t)u)

- (tSn,mxn + (1 - t)u)\\

< \Sn,mv - Sn,m(txn + (1 - t)u)\ + bn,m

= \\Sn,m(tXn + (1 - t)u) - Sn,mV\\ + bn,m

< n (1 + tj + Sj + tjSj)\\txn + (1 - t)u - v\\ + bn j=n

< n (1 + tj + Sj + tjSj)Un(t) + bnfm-j=n

Consequently,

lim sup am(t) = lim sup an+m(t) < limsup( bnm + n(1 + tj + Sj + tjSj )an(t)j, (2.6)

m ^^ m ^^ m j=n J

lim sup an(t) < liminf an(t). (2.7)

This implies that limn ^^ an(t) exists for all t e [0,1]. This completes the proof. □

Theorem 2.2. Let X be a real uniformly convex Banach space which has a Frechet differentiable norm and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2 : C ^ X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn}, {ln} c [1, to) such that ^™=1(kn - 1) < TO,^n=1(Zn - 1) < <x>, respectively, and F(T1) n F(T2) / 0. Suppose that {an} and {fin} are real sequences in [e, 1 - e] for some e e (0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, {xn} converges weakly to a fixed point of T\ and T2.

Proof. Set x = p\ - p2 and h = t(xn - pi) in (1.18). By using Lemmas 1.11, and 2.1 and the same proof of Lemma 4 of Osilike and Udomene [7], we can show that, for every p1,p2 e F(T1) n F(T2),

{p - q,Jp - pa)) = 0, (2.8)

for all p,q e ww(xn). Since E is reflexive and {xn} is bounded, we from Lemma 1.13 conclude that ww(xn) c F(Ti) for each i = 1,2. Let p,q e ww(xn). It follows that p,q e F(T1) n F(T2); that is,

lip - qII2 = {p - q,J(p - q)) = 0. (2.9)

Therefore, p = q. This completes the proof. □

Theorem 2.3. Let X be a real uniformly convex Banach space such that its dual X* has the Kadec-Klee property and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2 : C ^ X be two asymptotically nonexpansive nonself-mappings of C with sequences {k n }, {ln} c [1, to) such that y, TO=1(kn 1) < TO^ TO=1 (ln - 1) < to, respectively, and F(T1) n F(T2)/ 0. Suppose that {an} and {pn} are real sequences in [e, 1 - e] for some e e (0,1). Let {xn} and {yn} be the sequences defined by (1.14). Then, {xn} converges weakly to a fixed point of T1 and T2.

Proof. It follows from Lemma 1.11 that the sequence {xn} is bounded. Then, there exists a subsequence {xn]} of {xn} converging weakly to a point x* e C. By Lemma 1.12, we have

lim llxn, - T\xnj II = 0 = lim ||xn, - T2xn ||. (2.10)

n^»11 j j II n^»11 j j II

Now, using Lemma 1.10, we have (I - T)x* = 0; that is, Tx* = x*. Thus, x* e F(Ti) n F№). It remains to show that {xn} converges weakly to x*. Suppose that [xni} is another subsequence of {xn} converging weakly to some y*. Then, y* e C and so x*,y* e ww(xn) n F(T1) n F(T2). By Lemma 2.1,

lim ||txn - (1 - t)x* - y*

(2.11)

exists for all t e [0,1]. It follows from Lemma 1.13 that x* = y*. As a result, ww(xn) is a singleton, and so {xn} converges weakly to a fixed point of T. □

In the remainder of this section, we deal with the weak convergence of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space without any of the Opial's condition, Kadec-Klee property, or Frechet differentiable norm.

Let T1 and T2 be two asymptotically nonexpansive nonself-mappings of C with [kn]c [1, to), limn^^ kn = 1, and {ln} c [1, to), limn^^ ln = 1, respectively. In the sequel, we take {tn} c [1, to), where tn = max{kn,ln}.

We start with proving the following lemma for later use.

Lemma 2.4. Let X be a uniformly convex Banach space and C a nonempty bounded closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T1,T2 : C ^ X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn}, {ln} c [1,to) such that kn ^ 1, ln ^ 1 as n ^ to, respectively, and F(T1) n F(T2)/ 0. Suppose that {an} and {fin} are real sequences in [e, 1 - e] for some e e (0,1). Then, for the sequence {xn} given in (1.14), we have that

lim \\xn - T\xn\\ = 0 = lim \\xn - T2xn

(2.12)

Proof. By setting tn = max{kn,ln}, then limn^to tn = 1 if limn^to kn = 1 = limn^to ln. Let p e F(T1) n F(T2). Since C is bounded, there exists Br(0) such that C c Br(0) for some r > 0. Applying Lemma 1.9 for scheme (1.14), we have

||yn - p||2 = ||p((1 - ßn)xn + ßnT2{PT2)n-lx^ - p"2 < ||(1 - ßn)(xn - p) + ßn(T2(PT2)n-1xn - p

12 Abstract and Applied Analysis

= (1 - fin)\\Xn - p\\2 + finQxn - p\\2 - Pn(1 - Pn)g(\\xn - T2(PT2)n-1Xn\\)

= (l - pn + Pnll)\\Xn - p\\2 - pn(1 - Pn)g(\\xn - T2(PT2)n-1Xn\\)

(2.13)

and so,

|xn+1 - p\\2 = \\p((1 - an)yn + anT1(PT1)n - p\\

< \\(1 - an)(yn - p) + an(T1(PTr)n-1yn - p)\\ = (1 - an)\\yn - p\2 + ank2n\\yn - p\2

- an(1 - an)g(\yn - T1(PT1)n-1y = (1 - an + ank\\yn - p\\2

- an(1 - an)g(\yn - T1(PT1)n-1y

< (1-an + ank2) ((1 - pn + Pnl2n) \\xn - p\\2- Pn(1 - Pn)g(\\xn - T2(PT2)n-1x,

- an(1 - an)g(||yn - T1(PT1)n-1yn = (^1 - an + ank^(l - pn + pnin) \\xn - p\\2

- (1 - an + ank2)pn (1 - Pn)g(\\xn - T2PT2)n-1 x

- an(1 - an)g{\yn - T1(PT1)n-1yn\\) = ((1 - an){ 1 - Pn) + (1 - an)pnl2n + (1 -pn)ank2n + ank2pjn) \\xn - p\\

- (1 - an + ank2)pn (1 - pn)g(\\xn - T2PT2)n-1 xn\\^

- an(1 - an)g{\yn - T1(PT1)n-1yn\\)

< ((1 - an) (1 - pn) + (1 - an)pnt2n + (1 - pn )ant2n + anpnti)\\xn - p\\2

- (1 - an + ank2)pn (1 - pn)g(\\xn - T2PT2)n-1 x (1 - an)g(\yn - T1(PT1)n-1y

Abstract and Applied Analysis 13

< ((1 - a„)(1 - pn)ti + (1 - an)pnti + (1 - Pn)anti + anpnti)\\xn - p\\2

- (1 + an(k2n - 1))0n(1 -pn)g(\\xn - T2(PT2)n-1Xn\\)

- an(1 - an)g(\yn - T1(PT1)n-1yn\\)

< ((1 - an) (1 - pn)ti + (1 - an)pnti + (1 - Pn)aji + anpnttyX - p\\2

- &(1 -Pn)g(\\xn - Tn(PTn)n-1Xn\\)

- an(1 - an)g{\yn - T1(PT1)n-1yn\\)

<\\xn - p\\2 + r(f4n - 1) - e2g(\\xn - Tn(PTn)n-1Xn\\)

- ¿2g(\\yn - T1(PTj)n-1yn\^.

(2.14)

From (2.14), we obtain the following two important inequalities:

\\xn+1 - p\\2 < \\xn - p\\2 + rft - 1) - e2g(\\xn - T2(PT2)n-1Xn|\), (2.15)

\\xn+1 - p\\2 < \\xn -p\\2 + riti - 1) - e2g(\yn - T1(PT1)n-1yn\\). (2.16)

Now, we prove that

lim \\xn - T2(PT2)n-1xn\\ = 0 = lim !!yn - T1(PT1)n-1 y„\i. (2.17)

Assume that limsup„^^ \\x„ - T2(PT2)„ lxn\\ > 0. Then, there exists a subsequence (use the same notation for subsequence as for the sequence) of {x„} and fi> 0 such that

xn - T2(PT2)n-1xn\\ > ¡d> 0.

(2.18)

By definition of g, we have

g(\\Xn - T2(PT2)n-1Xn ID > g(p)> 0. (2.19)

From (2.15), we have

\\xn+1 - p\\2 < \\xn - p\\2 + r(tAn - 1) - £2g(p)

2 t i \ ¿2 ¿2 = \\xn-p\\ + r( - ^ - -¿rg^ )- T

(2.20)

In addition, ^ — 1 and (s2/2r)g(p) > 0; there exists n0 > 1 such that (t^ - 1) < (¿2/2r)g(p) for all n > n0. From (2.20), we obtain

£ 2 2

2g(p) < \\Xn - p\\ - \\xn+1 - p\\ (2.21)

for all n > n0.

Let m > n0. It follows from (2.21) that

¿2 m m / 2 2\

yX gp <X (\xn-p\\ - \\xn+1-p\\)

n=no n=no (2.22)

= \xno - p\\2

By letting m —> to in (2.22), we obtain

to = \\xn0 -p\\2 < to (2.23)

which contradicts the reality. This proves that p = 0. Thus, lim sup^^ \\xn-T2(PT2)n xn\\ < 0. Consequently, we have

lim \xn - T2PT2)n-1xn\ = 0. (2.24)

n — toII II

Similarly, using (2.16), we may show that

lim \yn - T1PT1 )n-1yn\\ = 0. (2.25)

n — to II II

Using (2.24), we have

\\xn - yn\\ < fin\\xn - T2 (PT2 )n-1xn\\ -— 0 (as n -—to ). (2.26)

From (2.25), (2.26), and the uniform equicontinuous of T1 (see Remark 1.1), we have

||x„ - Ti(PTi)n-1x„|| < ||xn - yn|| + ||yn - Ti(PTi)n-1x„||

< ||xn - yn|| + ||yn - Ti(PTi)n-1y„|| (2.27)

+ ||Ti(PTi)n-1yn - Ti(PTi)n-1x„|| -— 0 (as n -— to).

\\Xn - Xn+i|| < (1 - an)\yn - Xn! + aJh^PTi^yn - Xn||

= (1 - an)|yn - Xn|| + an||T1(PT1)n-1yn - T1(PT1)n-1Xn + T1 (PT1)n-1Xn - Xn || < ||yn - Xn || + ||T1 (PT1)n-1yn - T1(PT1)n-1Xn|| + ||T1(PT1)n-1Xn - Xn||,

(2.28)

it follows from (2.26), (2.27), and the uniform equi-continuity of T1 (see Remark 1.1) that

lim \\Xn - Xn+1\\ = 0. (2.29)

Since limn —\Xn - T1(PT1)n-1Xn\ = 0 and again from the fact that T1 is uniformly equicontinuous mapping, by Using (2.29), we have

¡Xn+1 - Tl(PTl)n-1Xn+1|| = !Xn+1 - Xn + Xn - T1(PT1)n-1Xn + T1(PT1)n-1Xn - T1(PT1)n-1Xn+J

< \Xn+1 - Xn\ + ||T1(PT1)n-1Xn+1 - T1(PT1)n-1Xn|

+ ||T1(PT1)n 1Xn - Xny —> 0 (as n —>to).

(2.30)

In addition,

Xn+1 - T1(PT1)n

= ||Xn+1 - Xn + Xn - T1(PT1)n-2Xn + T1(PT1)n-2Xn - T1(PT1)n-2Xn+11|

< \Xn+1 - Xn\ + ||T1(PT1)n-2Xn - Xn || + ||T1(PT1)n-2Xn+1 - T1(PT1)n-2Xn|

< \\Xn+1 - Xn\ + ||T1(PT1)n-2Xn - Xn || + L\Xn+1 - Xn\\,

(2.31)

where L = sup{kn : n > 1}. It follows from (2.29) and (2.30) that

limllXn+i - Ti(PTi)n 2Xn+i\\ = 0.

(2.32)

We denote (PT1)1 1 to be the identity maps from C onto itself. Thus, by the inequality (2.30) and (2.32), we have

||Xn+1 - TiXn+l|| =

Xn+1 - Ti(PTi)n-1Xn+1 + Ti(PTi)n-1Xn+1 - TiXn+1

Xn+1 - Ti(PTi)n Xn+1 Xn+1 - Ti(PTi) Xn+1 Xn+1 - Ti(PTi)n-1 Xn+1

Xn+i - Ti(PTi)n Xn+1

Xn+1 - Ti(PTi)n-i Xn+1

+ ||Ti(PTi)n-1Xn+i - TiXn+i|| + ||Ti(PTi)1-1(PTi)n-1Xn+i - Ti(PTi)1-1 Xn+i | + l! (PTi)n-1Xn+1 - Xn+il

+ L|(PTi)(PTi)n-2Xn+i - P(Xn+i)

+ L||Ti(PTi)n 2Xn+i - Xn+i|| —> 0 (as n —> to),

(2.33)

which implies that limn— to||Xn - TiXn || = 0. Similarly, we may show that lim 0. The proof is completed.

n | Xn

Xn T2Xn^ = □

Our weak convergence theorem is as follows. We do not use the rate of convergence conditions, namely, XTO=i(kn - i) < to and ^TO=i(ln - i) < to in its proof.

Theorem 2.5. Let X be a uniformly convex Banach space and C a nonempty bounded closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let Ti,T2 : C — X be two asymptotically nonexpansive nonself-mappings of C with sequences {kn}, {ln} c [i,to) such that kn — i, l n —> i as n —> to, respectively, and F(Ti) n F(T2) = 0. Suppose that {an} and {ftn} are real sequences in [e, i - e] for some e e (0, i). Then, the sequence {Xn} given in (i.i4) converges weakly to a common fixed point of Ti and T2.

Proof. Since C is a nonempty bounded closed convex subset of a uniformly convex Banach space X, there exists a subsequence {Xnj} of {Xn} such that Xnj converges weakly to q e ww(Xn), where ww(Xn) denotes the set of all weak subsequential limits of {Xn}. This show that ww(Xn) / 0 and, by Lemma 2.4, limn — to ||Xnj - TiXnj || = limn — to ||Xnj - T2Xnj || = 0. Since I - Ti and I - T2 are demiclosed at zero, using Lemma i.i0, we have Tiq = q = T2q. Therefore, (Xn) c F(Ti) n F(T2). For any q e ww(Xn), there exists a subsequence {Xni} of {Xn} such that

q (as i —> to).

(2.34)

It follows from (2.24) and (2.34) that

T2(PT2)n'-1Xn, = (T2(PT2)n'-1Xn, - Xn) + XH] ^ q. (2.35)

Now, from (1.14), (2.34), and (2.35),

yn, = P((l - pn)xn, + pn,T2(PT2p-1Xn,) ^ q. (2.36)

Also, from (2.25) and (2.36), we have

Ti(PTi)n-1yn, = (Ti(PTi)nj-1yn, - Vn) + V, ^ q. (2.37)

It follows from (2.36) and (2.37) that

Xni+i = p((l - an^Vni + j(PTi)nj-1ynj)^q. (2.38)

Continuing in this way, by induction, we can prove that, for any m > 0,

Xn,+m ^ q. (2.39)

By induction, one can prove that U^=0{Xn,+m} converges weakly to q as j ^ to; in fact,

{Xn =0 {Xn, +m}7=1 gives that Xn ^ q as n ^ TO. This completes the prove. □

Acknowledgments

The author would like to thank the Thailand Research Fund, The Commission on Higher Education(MRG5380226), and University of Phayao, Phayao, Thailand, for financial support during the preparation of this paper. Thanks are also extended to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper.

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