Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2010, Article ID 307245,19 pages doi:10.1155/2010/307245

Research Article

Convergence Theorems on a New Iteration Process for Two Asymptotically Nonexpansive Nonself-Mappings with Errors in Banach Spaces

Murat Ozdemir, Sezgin Akbulut, and Hukmi Kiziltunc

Department of Mathematics, Faculty of Science, Ataturk University, Erzurum 25240, Turkey Correspondence should be addressed to Murat Ozdemir, mozdemir@atauni.edu.tr Received 21 October 2009; Accepted 20 January 2010 Academic Editor: Guang Zhang

Copyright © 2010 Murat Ozdemir et al. 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 introduce a new two-step iterative scheme for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for this iterative scheme in a uniformly convex Banach space. The results presented extend and improve the corresponding results of Chidume et al. (2003), Wang (2006), Shahzad (2005), and Thianwan (2008).

1. Introduction

Let E be a real normed space and K be a nonempty subset of E. A mapping T : K ^ K is called nonexpansive if \\Tx - Ty\\ < \\x - y\\ for all x,y e K. A mapping T : K ^ K is called asymptotically nonexpansive if there exists a sequence {kn} c [1, to) with kn ^ 1 such that \\Tnx - Tny\\ < kn\\x - y\\ for all x,y e K and n > 1. T is called uniformly L-Lipschitzian if there exists a real number L> 0 such that \\Tnx - Tny\\ < L\\x - y\\ for all x,y e K and n > 1. 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}.

Iterative techniques for nonexpansive and asymptotically nonexpansive mappings in Banach spaces including Mann type and Ishikawa type iteration processes have been studied extensively by various authors; see [1-8]. However, if the domain of T, D(T), is a proper subset of E (and this is the case in several applications), and T maps D(T) into E, then the iteration processes of Mann type and Ishikawa type studied by the authors mentioned above, and their modifications introduced may fail to be well defined.

A subset K of E is said to be a retract of E if there exists a continuous map P : E ^ K such that Px = x, for all x e K. Every closed convex subset of a uniformly convex Banach

space is a retract. A map P : E ^ K is said to be a retraction if P2 = P. It follows that if a map P is a retraction, then Py = y for all y e R(P), the range of P.

The concept of asymptotically nonexpansive nonself-mappings was firstly introduced by Chidume et al. [4] as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.

Definition 1.1 (see [4]). Let K be a nonempty subset of real normed linear space E. Let P : E ^ K be the nonexpansive retraction of E onto K. A nonself mapping T : K ^ E is called asymptotically nonexpansive if there exists sequence {kn} c [1, to), kn ^ 1 (n ^ to) such that

|| T(PT)n-1x - T(PT)n-1y || < kn ||x - y || (1.1)

for all x,y e K 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|| (1.2)

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

In [4] , they study the following iterative sequence:

xn+1 = p((1 - an)xn + anT(PT)n-1xn), x1 e K, n > 1 (1.3)

to approximate some fixed point of T under suitable conditions. In [9], Wang generalized the iteration process (1.3) as follows:

xn+1 = P( (1 - an)xn + anTi(PT1)n-1y„j,

x d.4)

yn = p((1 - a'n)xn + dnT1(PT1)n-xx^, x1 e K, n > 1,

where Ti,T2 : K ^ E are asymptotically nonexpansive nonself-mappings and {an}, {a!n} are sequences in [0,1]. He studied the strong and weak convergence of the iterative scheme (1.4) under proper conditions. Meanwhile, the results of [9] generalized the results of [4]. In [10], Shahzad studied the following iterative sequence:

xn+1 = P((1 -an)xn + anTP[(1 -¡¡n)xn + ¡¡nTxn]),x1 e K, n > 1, (1.5)

where T : K ^ E is a nonexpansive nonself-mapping and K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P, nonexpansive retraction.

Recently, Thianwan [11] generalized the iteration process (1.5) as follows:

Xn+1 = P( (1 - an - Jn)xn + ttnTP( (1 - pn)yn + pnTyn) + JnUn),

yn = P( (1 - a'n - jn)xn + a'nTP( (1 - fa) xn + ¡¡Jxn) + Vn), X1 e K, n > 1,

where {an}, {¡n}, {yn},{an}, {¡„}, {jD are appropriate sequences in [0,1] and {un}, {vn} are bounded sequences in K. He proved weak and strong convergence theorems for nonexpansive nonself-mappings in uniformly convex Banach spaces.

The purpose of this paper, motivated by the Wang [9], Thianwan [11] and some others, is to construct an iterative scheme for approximating a fixed point of asymptotically nonexpansive nonself-mappings (provided that such a fixed point exists) and to prove some strong and weak convergence theorems for such maps.

Let E be a normed space, K a nonempty convex subset of E, P : E ^ K the nonexpansive retraction of E onto K, and T1,T2 : K ^ E be two asymptotically nonexpansive nonself-mappings. Then, for given xi e K and n > 1, we define the sequence {xn} by the iterative scheme:

xn+1 = P((1 - an - Jn)xn + anT1(PT1)n-1P('(1 - pn)yn + ¡nT1(PTi)n-ly^j + jnUn),

yn = p((1 - a'n - yn)xn + anT2(PT2)n-1P((1 - ¡¡n)xn + ¡¡nT2(PT2)n-1x„) + ^Vn),

where {an}, {¡n}, {jn}, {a'n}, {¡„}, {jh} are appropriate sequences in [0,1] satisfying an + ¡n + jn = 1 = a'n + ¡'n + Yn and {un}, {vn} are bounded sequences in K. Clearly, the iterative scheme (1.7) is generalized by the iterative schemes (1.4) and (1.6). Now, we recall the well-known concepts and results.

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

2 (x + y)

6e(s) = inf 1 -

||x|| = ||y|| = 1, £ = ||x - y|| . (1.8)

A Banach space E is uniformly convex if and only if 6E(e) > 0 for all e e (0,2].

A Banach space E is said to satisfy Opial's condition [12] if for any sequence {xn} in E, xn —^ x implies that

limsup||xn - x|| < limsup||xn - y|| (1.9)

n -^œ n -^œ

for all y e E with y /x, where xn — x denotes that {xn}converges weakly to x.

The mapping T : K ^ E with F(T) = 0 is said to satisfy condition (A) [13] if there is a nondecreasing function f : [0, œ) ^ [0, œ) with f (0) = 0, f (t) > 0 for all t e (0, œ) such that

||x - Tx||> f (d(x,F(T))) (1.10)

for all x e K, where d(x,F(T)) = inf{||x - p|| : p e F(T)}; (see [13, page 337]) for an example of nonexpansive mappings satisfying condition (A).

Two mappings T1,T2 : K ^ E are said to satisfy condition (A') [14] if there is a

nondecreasing function f : [0, to) ^ [0, to) with f (0) = 0, f (t) > 0 for all t e (0, to) such that

2(||x - T1X|| + ||x - T2xy) > f (d(x,F(T))) (1.11)

for all x e K, where d(x,F(T)) = inf{||x - p|| : p e F(T) = F(T1) n F(T2)}.

Note that condition (A') reduces to condition (A) when T1 = T2 and hence is more general than the demicompactness of T1 and T2 [13]. A mapping T : K ^ K is called: (1) demicompact if any bounded sequence {xn} in K such that {xn - Txn} converges has a convergent subsequence, (2) semicompact (or hemicompact) if any bounded sequence {xn} in K such that {xn - Txn} ^ 0asn ^ to has a convergent subsequence. Every demicompact mapping is semicompact but the converse is not true in general.

Senter and Dotson [13] have approximated fixed points of a nonexpansive mapping T by Mann iterates, whereas Maiti and Ghosh [14] and Tan and Xu [5] have approximated the fixed points using Ishikawa iterates under the condition (A) of Senter and Dotson [13]. Tan and Xu [5] pointed out that condition (A) is weaker than the compactness of K. Khan and Takahashi [6] have studied the two mappings case for asymptotically nonexpansive mappings under the assumption that the domain of the mappings is compact. We shall use condition (A') instead of compactness of K to study the strong convergence of {xn} defined in (1.7).

In the sequel, we need the following usefull known lemmas to prove our main results.

Lemma 1.2 (see [5]). Let {an}, {bn}, and {Sn} be sequences of nonnegative real numbers satisfying the inequality

an+i < (1 + 6n)an + bn, n > 1. (1.12)

n=\ bn< to and%n=\ 6n < to, then

(i) limnexists;

(ii) In particular, if {an} has a subsequence which converges strongly to zero, then limn^nan = 0.

Lemma 1.3 (see [2]). Suppose that E is a uniformly convex Banach space and 0 < p < tn < q < 1 for all n > 1. Suppose further that {xn} and {yn} are sequences of E such that

lim sup || xn || < r, limsup||yn|| < r, lim \\tnxn + (1 - tn)yn|| = r (113)

hold for some r > 0. Then limn ^TO||xn - ynH = 0.

Lemma 1.4 (see [4]). Let E be a uniformly convex Banach space, K a nonempty closed convex subset of E, and T : K ^ E be a nonexpansive mapping. Then, (I-T) is demiclosed at zero, that is, if xn ^ x weakly and xn - Txn ^ 0 strongly, then x e F (T), where F(T) is the set fixed point of T.

2. Main Results

We shall make use of the following lemmas.

Lemma 2.1. Let E be a normed space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K ^ E be two asymptotically nonexpansive nonself-mappings of E with sequences {k„}, {l„} c [1, to) such that £TO=1(k„ - 1) < to, £(l„ - 1) < to, respectively and F(T1)nF(T2) := {x e K : T1x = T2x = x} = 0. Suppose that {u„}, {v„} are bounded sequences in K such that £ Y„ < to, % TO=1 Y„ < to. Starting from an arbitrary x1 e K, define the sequence {x„} by the recursion (1.7). Then, lim„ ^TO\\x„ - p\\ exists for all p e F (T1) n F (T2).

Proof. Let p e F(T1) n F(T2). Since {u„} and {v„} are bounded sequences in K, we have

r = max] sup||u„ - p|,sup|v„ - p|| [. (2.1)

„>1 „>1

Set a„ = (1 -¡5„)y„ + p„Tx(VTx)„-xy„ and 6„= (1 -p'„)x„ + p'„T2(PT2)n-1x„. Firstly, we note that

JnjynTynT1\ 1) y„

IK - p|| = ||(1 - P„)y„ + P„T1(PT1 )„-1y„ - p||

< j3„||T1(PT1)n-1y„ -p|| + (1 - #„) ||y„ -p||

< P„kn\yn- py + C1 - P„) ¡y„- py

< k„\y„ - p^

|K - p|| = ||(1 - p'n)x„ + pnT1(PTi)n-Xx„ - p||

< #„||T2(PT2)n-1xn - p|| + (1 - pn)||x„ - p||

< pnl„\x„ - p|| + (1 - p'„)||x„ - p||

< ln ||x„ - p||.

From (1.7) and (2.3), we have

||y„ - p|| = ||p((1 - a„ - y'„)x„ + a„T2(PT2)n-1PS„ + y„v„) - p||

< ||(1 - a„ - y„)x„ + a'„T2(PT2)n-1PSn + y„v„ - p||

< a„||T2(PT2)n-1P6„ - p|| + (1 - a„ - Y„)||x„ - p|| + r„||v„ - p|| (2.4)

< a„ln\&n - p! + (1 - a'„ - Y„) ||x„ - p! + Y„||v„ - p!

< ann^n - p! + (1 - a'„ - Y„) ||x„ - p! + Y„r

< l„||x„ - p! + Y„r.

Substituting (2.4) into (2.2), we obtain

IK - p|| < kn\yn - p|| < knl2n\\xn - pW + knj'nr. (2.5)

It follows from (1.7) and (2.5) that

\xn+i - p\\ = ||p((1 - an - Jn)xn + anTi(PTi)n-1P<Jn + YnUn) - p||

< || (1 - an - Jn)xn + anTi(PTi)n-1P<Jn + YnUn - p||

< an||T1(PT1)n-1Pan - p|| + (1 - an - ||xn - p|| + Yn||un - p||

< anknWon - p|| + (1 - an - jn) Wxn - p|| + Yn||un - p|| (2.6)

< an(k.n^||xn - p|| + k2y'nr) + (1 - an - Hxn - p|| + Jnr

< k2nllWxn - pW + k2nYnr + Ynr

= (1+(i2n -1) k -1)+(in -1)+k -1)) ||xn - p||+k rn+rn)r.

Note that ^TO=1 kn - 1 < to and ^TO=1 ln - 1 < to are equivalent to ^TO=1 - 1 < to and Y!n=1 in -1 < to, respectively. Since £TO=1 Yn < to and £TO=1 Yn < to, we have £TO=1(knYn + Jn)r < to. We obtained from (2.6) and Lemma 1.2 that limn^TO\\xn - p|| exists for all p e F(T). This completes the proof. □

Lemma 2.2. Let E be a normed space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K ^ E be nonself uniformly L1-Lipschitzian, L2-Lipschitzian, respectively. Suppose that {un}, {vn} are bounded sequences in K such that £TO=1 Yn < to, £TO=1 Yn < to. Starting from an arbitrary x1 e K, define the sequence {xn} by the recursion (1.7) and set Cn = \\xn - T1(PT1)n-1xn\\,C'n = \\xn - T2(PT2)n-1xn\\ for all n > 1. If limn ^o-Cn = limn = 0, then

lim \\xn - T1xn\\ = lim \\xn - T2xn\\ = 0. (2.7)

Proof. Since {un}, {vn} are bounded, it follows from Lemma 2.1 that {un - xn} and {vn - xn} are all bounded. We set

r1 = sup{\ \un - xn\\ : n > 1}, r2 = sup{\vn - xn\\ : n > 1},

r3 = sup{\\un-1 - xn-1\\ : n > 1}, r = max{r : i = 1,2,3}.

Discrete Dynamics in Nature and Society

Let an = (1 - fin)yn + pnT\{PT\)n-lyn and 6n = (1 - f>'nX + f>'nT2(PT2)n-1Xn. Then, we have

— n-

We find the following from (1.7) and (2.10):

\\Vn - xn\\ = \\p((1 - a'n - y)xn + a'nT2(PT2)n-1PÖn + yVn) - x — 11(1 - a'n - Yn)Xn + a'nT2(PT2)n-1P6n + jnVn - Xn\\

\\On - Xn\\ = \\( 1 - ßn)Vn + ßnTl(PTX)n-Vyn - Xn\

— ßn\\T1(PT1)n-1yn - T1(PT1)n-1Xn\\ + ßn\\T1(PT1 )n-1Xn - Xn\\ + (1 - ßn) \\yn - Xn\\

— (L + 1)\\y

\\6n - Xn\\ = \\ (1 - ß'n)Xn + ß'nT2(PT2)n-1Xn - Xn\\

— ßn\\T2(PT2)n-1Xn - Xn\\ (2.10)

< a'n\\T2(PT2)n-1P6n - T2(PT2)n-1Xn\\

+ a'n\\T2(PT2)n-1 Xn - Xn\\ + rn\\vn - Xn\\ (Z11)

< L2\6n - Xn\\ + C'n + Ynr

< L2Cn + Cn + rnr

= (L2 + 1)cn + i r. Substituting (2.11) into (2.9), we get

\\an - Xn\\ < (L1 + 1)(L2 + 1)Cn + (L1 + 1)ynr + Cn. (2.12)

It follows from (1.7) and (2.12) that

||Xn+1 Xn || < ||p((1 - an - Tn)xn + anT1(PT1)n lPon + YnU

< ||T1(PT1)n-1Pan - xn|| + Ynllun - xn||

< ||T1(PT1)n-1Pan - T1(PT1)n-1Xn|| + ||T1(PT1)n-1Xn - Xn|| + Ynr

< L^On - Xn|| + Cn + Ynr

< L1((L1 + 1)(L2 + 1)Cn + (L1 + 1)Ynr + Cn) + Cn + Ynr

= (L1 + 1)Cn + L\(L\ + 1)(L2 + 1)Cn + L\(L\ + 1)y4 r + Ynr.

Using (2.11) and (2.13), we obtain

(2.13)

|K-1 - Xn|| = ||( 1 - fin-\)yn-1 + ^n-1T1(PT1)n yn-1 - Xn ||

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

< L1 ¡yn-1 - Xn-1 y + Cn-1 + ||Xn - Xn-11| + ||yn-1 - Xn-1|| + ||Xn - Xn-11|

< (L1 + 1)[(L2 + 1)Cn-1 + Yn-1r] "(L1 + 1)Cn-1 + L1(L1 + 1)(L2 + 1)C'n-1~

+L1(L1 + 1)Yn-1r + Yn-1r = (2L1 + 3)Cn-1 + (2L1 + 1)(L1 + 1)(L2 + 1)Cn_1 + (2L1 + 1)(L1 + 1)Yn-1r + 2Yn-1r.

(2.14)

+ Cn 1

Discrete Dynamics in Nature and Society Combine (2.13) with (2.14) yields that

\xn - (PT1 )n-1xJ = llx„ - T1(PT1)n-2Xn\

< \\ ( 1 — an-1 — jn—1)Xn-1 + an- 1T1(PT1)n-2Pan-1 + Yn—1 Wn—1 — T1(PT1)n— Xn

< an—1\\T1(PT1)n—2Pon—1 — T1PT1 )n—2Xn \\

+ (1 — an—1) \xn—1 — T1(PT1)n 2Xn\\ + jn—1lWn—1 — Xn—1I

< \\T1(PT1)n—2Pan—1 — T1(PT1)n—2Xn,

+ \\xn—1 — t1(pt1)n—2x

< L1|on— 1 — Xn I + \\ Xn—1 — T1(PT1)n 2Xn—1 \\

+ \\T1(PT1)n—2Xn — T1(PT1)n—2Xn—1\\ + jn—1r

(2L1 + 3)Cn—1 + (2L1 + 1)(L1 + 1)(L + 1)Cn—1 +(2L1 + 1)(L1 + 1)Yn—1r + 2jn—1r

+ Cn—1 + (L1 + 1)Cn—1 + L1(L1 + 1)(L2 + 1)Cn_1

+ L1L + 1)jn—1r + 2jn—1r

2(L1 + 1)2Cn—1 + 2L1L + 1)2(L2 + 1)Cn_1

+ 2L1(L1 + 1)2jn—1r + 2(L1 + 1)jn—1r,

from which it follows that

|Xn — T1Xn| = \\ Xn — T1 (PT1)n—1Xn + T1(PT1)n—1Xn — T1Xn\

< Cn + 2L1(L1 + 1)2Cn—1 + 2L2(L1 + 1)2(L2 + 1)Cn_1

+ 2L2(L1 + 1)2jn—1r + 2L1L1 + 1)jn—1r.

(2.15)

< \\Xn — T1 (PT1)n—1Xn\\ + \\T1 (PT1)n—1Xn — T1Xn\\

< Cn + L1\\(PT1)n—1Xn — Xn \\ (2.16)

It follows from lim„— TOC„ = lim„— TOC„ = 0 that lim„— TO\\x„ - T1x„\ = 0. Similarly, we can show that lim„—TO\\x„ - T2x„\\ = 0. This completes the proof. □

Lemma 2.3. Let E be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K — E be two asymptotically nonexpansive nonself-mappings of E with sequences {k„}, {l„} c [1, to) such that £^=1(k„-1) < to, m=i(l„-1) < to, respectively, and F(T1)nF (T2) = 0. Suppose that {a„}, {fi„}, {j„], {a„}, {fi„}, {j„} are appropriate sequences in [0,1] satisfying a„+p„+j„ = 1 = d„+p'n +j'n, and {u„}, {v„} are bounded sequences in K such that £TO=1 Y„ < to, %TO=1 Yn < to. Moreover, 0 < a < a„,a'„,p„,pl„ < b < 1 for all „ > 1 and some a,b e (0,1). Starting from an arbitrary x1 e K, define the sequence {x„} by the recursion (1.7). Then,

lim \\x„ - Tx\\ = lim \\x„ - T2x„\\ = 0. (2.17)

„ —> to „ —> TO v '

Proof. Leto„ = (1-p„)y„ +pnT\(PTi)„-lyn and 6„ = (1-^„)x„ +£'„T2(PT2)„-1x„.ByLemma 2.1, we see that lim„—TO\\x„ - p\\ exists. Assume that lim„—TO\\x„ - p\\ = c. If c = 0, then by the continuity of T1 and T2 the conclusion follows. Now, suppose c > 0. Taking lim sup on both sides in the inequalities (2.2), (2.3), and (2.4), we have

lim sup 11 c„ - p|| < c, limsup||6„ - p|| < c, limsup||y„ - p|| < c, (2.18)

„ — TO „ — TO „ — TO

respectively. Next, we consider

||T1(PT1)„-1Po„ - p + Y„(u„ - x„)|| < ||T1(PT1 )„-1Po„ - p| + Y„\\u„ - x„\\ ii ii ii ii (2.19)

< k„||o„ - p|| + Y„r.

Taking lim sup on both sides in the above inequality and using (2.18), we get

lim sup 11T1 (PT1) „-1 Po„ - p + Y„(u„ - x„)|| < c. (2.20)

„ — TO " "

Observe that

||x„ - p + Y„(u„ - x„)|| < ||x„ - p|| + Y„\\u„ - x„\\ < ||x„ - p|| + Y„r, (2.21) which implies that

lim sup||x„ - p + Y„(u„ - x„) || < c. (2.22)

„ — TO

limsup„—TO\\x„+1 - p\\ = c means that

lim infH a„ (T1 (PT1)„-1 Pon - p + Yn(u „ xn ^ + (1 - a„)(x„ - p + Y„(u„ - x„)) I > c. (2.23)

On the other hand, by using (2.23) and (2.5), we have

||a„(X1(PX1)"-1 Pan - p + Yn(un - xn)j + (1 - an){Xn - p + Jn(un - Xn)) ||

< an|T1(PT1)n-1Pan - p| + (1 - «n)^Xn - p! + Jn\\un - Xn\\

< anknlan - p| + (1 - an)lxn - p| + Jn\\un - Xn\\ (2.24)

< ankn(knl2nlXn - pl + knj'nr) + (1 - «n)||Xn - p|| + JnT

< k2nnX - p1 + kninr + JnT-Therefore, we have

lim sup 11 ajT1(PT1)n 1Pon - p + jn(un - Xn)) + (1 - «n)(Xn - p + Jn(un - Xn))|| < c. (2.25)

Combining (2.23) with (2.25), we obtain

nlimJ|an(Ti(PTi)n-1PCTn - p + Jn(un - Xn)) + (1 - an) ( Xn - p + Yn(un - Xn)) || = c. (2.26) Hence, applying Lemma 1.3, we find

nlim||Ti(PTi)n-1Pan - Xn || = 0. (2.27)

Note that

||Xn -p|| < ||T1(PT1)n-1Pan - p|| + ||T1(PT1)n-1Pan - Xn|| < MK -p|| (2.28) which yields that

c < liminf||on - p|| < limsup|on - p|| < c. (2.29)

That is, limn- p|| = c. This implies that

linmiinf||^^T1(PT1)n-1yn - p) + (1 - ßn)(yn - p)|| > c. (2.30)

Similarly, we have

(2.31)

(T1(PT1)n-1yn - p) + (1 - pn)(y)n - p

< pn\\T1(PT1)n-1yn - v|| + (1 - pn) II (;yn - v) II < kn\\yn - p\\, limsup||^^T1(PT1)n-1yn - p) + (1 - pn)(yn - p)\\ < c. (2.32)

Combining (2.30) with (2.32), we obtain

nlim\\p^T1 (PT1)n-1yn - p) + (1 - pn)(yn - p)\\ = c. (2.33)

On the other hand, we have

\\T1(PT1)n-1yn - p\\ < kn\\yn - p\\, (2.34)

limsup\\T1(PT1)n-1yn - p\\ < c. (2.35)

n ^<x> 11 11

Hence, using (2.32), (2.33), (2.35), and Lemma 1.3, we find

nlim \\T1(PT1)n-1yn - yn\\ = 0. (2.36)

Note that from (2.36), we have

\K - p\\ = \\(1 - pn)yn + pnT1(PT1 )n-1yn - p\\

< (1 - Pn) \\yn - p\\ + pn\\T1(PT1)n-1yn - p\\

11 11 (2.37)

< (1 - Pn)\\yn - p\\ + Pn\\T1(PT1)n-1yn - yn\\ + pn\\yn - p\\ = \\yn - p\

which yields that

c < liminf\yn - p\\ < limsup\yn - p\\ < c. (2.38)

n^^ nV y

That is, limn^®||yn - p|| = c.

Again, limn—x,Hyn - p|| = c means that

linminf\\an(T2(PT2)n-1P6n - p + jn(vn - Xn)) + (1 - a'n)(xn - p + Y(vn - Xn))\\ > c. (2.39)

Discrete Dynamics in Nature and Society By using (2.39) and (2.3), we obtain

a'n(T2(PT2)n 1P6fl - p + inV - Xn)) + (1 - a'n) (xn - p + in(Vn - Xn)) ||

< «n||T2(PT2)n-1P6n - p|| + (1 - tt'n) ||Xn - p|| + iW(Vn - Xn)\\

< a'Jnpn - p! + (1 - a'n)!xn - p! + i\\(vn - Xn)W

< ann^n - p! + i1 - a'n)!Xn - p! + inr

< l2n HXn - p! + inr-

Therefore, we have

lim sup || a'n (T2(PT2)n 1P6n - p + in (Vn - Xn)) + (1 - a'n) (Xn - p + i (Vn - Xn)) || < C. (2.41)

Combining (2.39) with (2.41), we obtain

Km \\an(T2(PT2)n-1P6n - p + i(vn - Xn)) + (1 - a'n)(xn - p + i(vn - x„))|| = c. (2.42)

On the other hand, we have

||T2(PT2)n-1P6n - p + Yn(Vn - Xn)|| < ||T2(PT2)n-1P6n - p|| + Yn\Vn - Xn\\

< hpn- p! + Ynr

which implies that

lim sup11T2(PT2)n-1 P6n - p + Y(vn - Xn) || < c. (2.44)

Notice that

Xn - p + Yn (Vn - Xn)|| < || Xn - p|| + Yn\\Vn - Xn\\ < ||Xn - p|| + Ynr, (2.45)

which implies that

limsup\xn - p + Yn (Vn - Xn) \\ < c. (2.46)

Using (2.42), (2.44), (2.46), and Lemma 1.3, we find

fim ||T2(PT2)n-1P6n - Xn|| = 0. (2.47)

Observe that

\\xn - p\\ < \\T2(PT2)n-1P6n - Xn || + ||T2(PT2)n-1P6„ - p || < ln\\6n - p\\ (2.48) which yields that

c < liminf\6n - p\\ < lim sup\6n - p\\ < c. (2 49)

That is, limn — - p\\ = c. This implies that

limmf^n (T2(PT2)n-1Xn - p) + (1 - pn ){xn - p) \\ > c. (2.50)

Similarly, we have

k(T2(PT2)n-1 Xn - p) + (1 - pn )(xn - p)\\

\ \ (2.51) < P'n\\T2(PT2)n-1Xn - p\\ + (1 - P'n)\\Xn - p\\ < ln\\Xn - p\\,

limsup\k (T2(PT2)n-1Xn - p) + (1 - p'n )(Xn - p)\ < c. (2.52)

n — <x> 11 11

Combining (2.50) with (2.52), we obtain

Hmjpn (T2(PT2)n-1Xn - p) + (1 - pn)(Xn - p)\\ = c. (2.53)

On the other hand, we have

(2.54)

||T2(PT2)"-1x„ - p|| < ln\\xn - p||, limsup||T2(PT2)n-1xn - p|| < c,

limsup\\xn - p\\ < c. (2.55)

n — rn

Hence, using (2.53), (2.54), (2.55), and Lemma 1.3, we find

Jim ||T2(PT2)n-1xn - xn\\ = 0. (2.56)

Discrete Dynamics in Nature and Society 15

In addition, from y„ = P((1 - a'n - j„)x„ + a'„T2(PT2)n-1P6„ + j„v„) and (2.47), we have

\\yn - x„|| = ||p((1 - an - y'n)x„ + a'nT2(PT2)n-1P6n + y'nv„) - x„||

< a'nWT2(PT2)n-1P6n - x„| + r„\\v„ - x„w

< ||T2(PT2)n-1P6n - x„| + Y„r. —> 0, (as n —> to).

Hence, from (2.36) and (2.57), we find

|T1(PT1)„-1 x„ - x„| < ||T1(PT1)n-1xn - T1 (PT1)n-1ynW

(2.57)

+ ||X1(PX1)n yn - yn| + IIyn - Xn\\ < kn\\y n Xn || + ||T1(PT1 )n-1 yn

—> 0, (as n —> to).

(2.58)

That is,

limJ|71(P71 )n-1xn - Xn || = 0. (2.59)

Since T1 and T2 are uniformly L1-Lipschitzian and uniformly L2-Lipschitzian, respectively, for some L1,L2 > 0, it follows from (2.56), (2.59), and Lemma 2.2 that

lim \\Xn - T1Xn|| = lim \\Xn - XzXnH = 0. (2.60)

n ^TO n ^to y '

This completes the proof. □

Theorem 2.4. Let E be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K ^ E be two asymptotically nonexpansive nonself-mappings of E with sequences {kn}, {ln} c [1, to) such that£ TO=1(kn-1) < to, 2TO=1(ln-1) < to, respectively, and X(T1)nX (T2) = 0. Suppose that {an}, {ßn}, {yn}, {a'n}, {ß'n}, {fn} are appropriate sequences in [0,1] satisfying an+ßn+jn = 1 = an+ßn +yn, and {un}, {vn} are bounded sequences in K such that £TO=1 Jn < to, %TO=1 j'n < to. Moreover, 0 < a < an,a'n,ßn,ß'n < b < 1 for all n > 1 and some a,b e (0,1). If one of T1 and T2 is completely continuous, then the sequence {Xn} defined by the recursion (1.7) converges strongly to some common fixed point of T1 and T2.

Proof. By Lemma 2.1, {Xn} is bounded. In addition, by Lemma 2.3; limn^TO\\Xn - T1Xn\ = limn^TO\\Xn - T2Xn\\ = 0; then {T1Xn} and {T2Xn} are also bounded. If T1 is completely continuous, there exists subsequence {T1 Xnj} of {T^n} such that T*iXnj ^ p as j ^ to. It follows from Lemma 2.3 that limj^TO\\Xnj - T1Xnj\\ = limj^TO\\Xnj - T2Xnj\\ = 0. So by the continuity of T1 and Lemma 1.4, we have limj^TO\\Xnj - p\\ = 0 and p e F(T1) n F(T2).

Furthermore, by Lemma 2.1, we get that limn—TO\\xn - p\\ exists. Thus limn— TO\\xn - p\\ = 0. The proof is completed. □

The following result gives a strong convergence theorem for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying condition (A').

Theorem 2.5. Let E be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K — E be two asymptotically nonexpansive nonself-mappings of E with sequences {kn}, {ln} c [1, to) such that £TO=1(kn -1) < to, J^TO=1(ln-1) < to, respectively, and F(T1)nF(T2) = 0. Suppose that {an}, {fin}, {yn}, {a'n}, {fin}, Y} are appropriate sequences in [0,1] satisfying an +pn+jn = 1 = a'n +p'n+j'n, and {un}, {vn} arebounded sequences in K such that £TO=i Jn < to,%TO=i j'n < to. Moreover, 0 < a < an, a'n,pn,ftn < b < 1 for all n > 1 and some a,b e (0,1). Suppose that T1 and T2 satisfy condition (A'). Then, the sequence {xn} defined by the recursion (1.7) converges strongly to some common fixed point of T1 and T2.

Proof. By Lemma 2.1, we readily see that limn—TO\\xn -p\\ and so, limn— TOd(xn,F(T1) nF(T2)) exists for all p e F(T1) nF(T2). Also, by Lemma 2.3, limn ^TO\\T1xn - xn\\ = limn—TO\\T2xn - xn\\ = 0. It follows from condition (A') that

lim f (d(xn,F(T{) n F(T2))) < limf1 (\\xn - Tx\\ + \xn - T2xn\\)) = 0. (2.61)

n — to n — to\2 /

That is,

lim f (d(xn,F(T1) n F(T2))) = 0. (2.62)

n — TO x '

Since f : [0, to) — [0, to) is a nondecreasing function satisfying f (0) = 0, f (t) > 0 for all t e (0, to), therefore, we have

lim d(xn,F(T{) n F(T2)) = 0. (2.63)

n — TO v x

Now we can take a subsequence {xnj} of {xn} and sequence {yj} cF such that \\xnj -yj \\ < 2-j for all integers j > 1. Using the proof method of Tan and Xu [5], we have

||xnJ+1 - yj || < ||xnj - yj|| < 2-, (2.64)

and hence

||yj+1 - yj || < ||yj+1 - xn,+11 + |xn,+1 - yj I < 2-(j+V) + 2-j < 2-j+1. (2.65)

We get that {yj} is a Cauchy sequence in F and so it converges. Let yj — y. Since F is closed, therefore, y e F and then xnj — y. As limn—TO\\xn - p\\ exists, xn — y e F(T1) n F(T2). Thereby completing the proof. □

Remark 2.6. If j„ = y„ = p„ = p'„ = 0, then the iterative scheme (1.7) reduces to the iterative scheme (1.4) of [9]. Moreover, the condition (A') is weaker than both the compactness of K and the semicompactness of the asymptotically nonexpansive nonself-mappings T1, T2 : K ^ E. Also, the condition 0 < a < a„,a„ < b < 1 for all „ > 1 is weaker than the condition 0 < e < an,a„, < 1 - e, for all „ > 1 and some e e [0,1). Hence, Theorems 2.4 and 2.5 generalize Theorems 3.3 and 3.4 in [9], respectively.

In the next result, we prove the weak convergence of the iterative scheme (1.7) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying Opial's condition.

Theorem 2.7. Let E be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K ^ E be two asymptotically nonexpansive nonself-mappings of E with sequences {k„}, {l„} c [1, to) such that £TO=1(k„ -1) < to, m=i(l„-1) < to, respectively, and F(T1)nF(T2) = 0. Suppose that {a„}, {fi„}, {y„}, {a„}, {$'„}, {j„} are appropriate sequences in [0,1] satisfying a„+p„+y„ = 1 = a'„+p'n +j'n,and {u„}, {v„} are bounded sequences in K such that £TO=1 Y„ < to, %TO=1 Yn < to. Moreover, 0 < a < a„,al„,p„,pl„ < b < 1 for all „ > 1 and some a,b e (0,1). Suppose that T1 and T2 satisfy Opial's condition. Then, the sequence {x„} defined by the recursion (1.7) converges weakly to some common fixed point of T and T2.

Proof. Let p e F(Xi) n F(T2). By Lemma 2.1, we see that lim„- p\\ exists and {xn} bounded. Now we prove that {xn} has a unique weak subsequential limit in F(T1) n F(T2). Firstly, suppose that subsequences {xnk} and [xnj} of {xn} converge weakly to p1 and p2, respectively. By Lemma 2.3, we have limn^^\\xnk - T1xnk\\ = 0. And Lemma 1.4 guarantees that (I - T1)p1 = 0, that is., T1p1 = p1. Similarly, T2p1 = p1. Again in the same way, we can prove that p2 e F(T1) n F(T2).

Secondly, assume p1 = p2, then by Opial's condition, we have

lim IIxn - p1H = lim \\xnk - p1H < lim \\xnk - p2H

n ^^ k k

= lim \\xnj - p2\\ < lim \\xnk - p1 \ (2.66)

j ^^ 11 II k ^^

= lim \\xn - p1\,

which is a contradiction, hence, pi = p2. Then, {xn} converges weakly to a common fixed point of T1 and T2. This completes the proof. □

Remark 2.8. The above Theorem generalizes Theorem 3.5 of Wang [9].

3. Case of Two Nonself-Nonexpansive Mappings

Let T1,T2 : K ^ E be two nonexpansive nonself-mappings. Then, the iterative scheme (1.7) is written as follows:

x„+1 = P((1 - a„ - Y„)x„ + a„T1P(1 - p„)y„ + p„T1y„ + Y„u„),

y„ = ^(1 - a„ - y„)x„ + a'„T2P((1 - p'„)x„ + p'„T2X„) + y„v„), X1 e K,„ > 1.

Nothing prevents one from proving the results of the previous section for nonexpansive nonself-mappings case. Thus, one can easily prove the following.

Theorem 3.1. Let E be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K ^ E be two nonexpansive nonself-mappings of E with sequences {kn}, {l„} c [1, to) such that ^=1(k„ - 1) < to,^TO=1(l„ - 1) < <x>, respectively, and F(T1) n F(T2) = 0. Suppose that {a„}, {fi„}, {y„}, {a'„}, {fi„}, {j„} are appropriate sequences in [0,1] satisfying a„ + f}„ + j„ = 1 = a„ + f}'„ + j„, and {u„}, {v„} are bounded sequences in K such that £TO=i Y„ < to,£TO=i Y„ < to. Moreover, 0 < a < a„, a„, f}„, f>'n< b < 1 for all „ > 1 and some a,b e (0,1). Suppose that T1 and T2 satisfy condition (A'). Then, the sequence {x„} defined by the recursion (3.1) converges strongly to some common fixed point of T1 and T2.

Theorem 3.2. Let E be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1,T2 : K ^ E be two nonexpansive nonself-mappings of E with sequences {k„}, {l„} c [1, to) such that ^TO=1(k„ - 1) < to,^TO=1(l„ - 1) < to, respectively, and F(T1) n F(T2) / 0. Suppose that {a„}, {£>„}, {y„}, {a„}, {fi„}, {y„} are appropriate sequences in [0,1] satisfying a„ + f}„ + j„ = 1 = a„ + f}'„ + j„, and {u„}, {vn} are bounded sequences in K such that ^TO=1 Y„ < to, %TOU Y„ < to. Moreover, 0 < a < a„, al„,p„,p1^ < b < 1 for all „ > 1 and some a,b e (0,1). Suppose that T1 and T2 satisfy Opial's condition. Then, the sequence {x„} defined by the recursion (3.1) converges weakly to some common fixed point of T and T2.

Remark 3.3. If T1 = T2 = T and T is a nonexpansive nonself-mapping, then the iterative scheme (3.1) reduces to the iterative scheme (1.6) of Thianwan [11]. Then, Theorems 3.1-3.2 generalize Theorems 2.4 and 2.6 in [11], respectively.

Acknowledgment

The authors would like to thank the referees for their helpful comments. References

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