Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 859795,11 pages doi:10.1155/2011/859795

Research Article

A Weak Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces

Xiaolong Qin,1 Sun Young Cho,2 and Shin Min Kang3

1 School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

2 Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

3 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Correspondence should be addressed to Shin Min Kang, smkang@gnu.ac.kr Received 13 December 2010; Accepted 1 February 2011 Academic Editor: Yeol J. Cho

Copyright © 2011 Xiaolong Qin 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.

The modified Ishikawa iterative process is investigated for the class of total asymptotically pseudocontractive mappings. A weak convergence theorem of fixed points is established in the framework of Hilbert spaces.

1. Introduction and Preliminaries

Throughout this paper, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by (•, ■) and || ■ ||. ^ and ^ are denoted by strong convergence and weak convergence, respectively. Let C be a nonempty closed convex subset of H and T : C ^ C a mapping. In this paper, we denote the fixed point set of T by F(T).

T is said to be a contraction if there exists a constant a e (0,1) such that

||Tx - Ty|| < a||x - y||, Vx,y e C. (1.1)

Banach contraction principle guarantees that every contractive mapping defined on complete metric spaces has a unique fixed point. T is said to be a weak contraction if

||Xx - Ty\\ < ||x - y|| - f(||x - y\\), Nx,y e C,

where y : [0, œ) ^ [0, œ) is a continuous and nondecreasing function such that y is positive on (0, œ), y(0) = 0, and limt^œy(t) = œ. We remark that the class of weak contractions was introduced by Alber and Guerre-Delabriere [1]. In 2001, Rhoades [2] showed that every weak contraction defined on complete metric spaces has a unique fixed point.

T is said to be nonexpansive if

||Tx - Ty|| < ||x - y||, Vx,y e C. (1.3)

T is said to be asymptotically nonexpansive if there exists a sequence {kn} c [1, to) with kn ^ 1 as n ^ to such that

||Tnx - Tny|| < kn||x - y||, Vn > 1, x,y e C. (1.4)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [3] as a generalization of the class of nonexpansive mappings. They proved that if C is a nonempty closed convex bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on C, then T has a fixed point.

T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

limsupsup (||Tnx - Tny|| - ||x - y||) < 0. (1 5)

n ^to x,yeC

Observe that if we define

In = max-j 0, sup (||Tnx - Tny|| - ||x - y||) \, (1.6)

then In ^ 0 as n ^ to. It follows that (1.5) is reduced to

Tnx - Tny^ < Hx - y! + ¿.n, Vn > 1, x,y e C.

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [4] (see also [5]). It is known [6] that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is asymptotically nonexpansive in the intermediate sense, then T has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous; see [5, 7].

T is said to be total asymptotically nonexpansive if

Tnx - TnyH < ||x - y|| + ¡in$(||x - y||) + ¿n, Vn > 1, x,y e Cr (1.8)

where $ : [0, to) ^ [0, to) is a continuous and strictly increasing function with 0(0) = 0 and {^n} and [¿,n] are nonnegative real sequences such that ¡in ^ 0 and ln ^ 0 as n ^ to. The class of mapping was introduced by Alber et al. [8]. From the definition, we see that

the class of total asymptotically nonexpansive mappings includes the class of asymptotically nonexpansive mappings and the class of asymptotically nonexpansive mappings in the intermediate sense as special cases; see [9,10] for more details.

T is said to be strictly pseudocontractive if there exists a constant k e [0,1) such that

Tx - Ty\\ < ||x - y||2 + k||(I - T)x - (I - T)y|2, Vx,y e C (1.9)

The class of strict pseudocontractions was introduced by Browder and Petryshyn [11] in a real Hilbert space. In 2007, Marino and Xu [12] obtained a weak convergence theorem for the class of strictly pseudocontractive mappings; see [12] for more details.

T is said to be an asymptotically strict pseudocontraction if there exist a constant k e [0,1) and a sequence {kn} c [1, œ) with kn ^ 1 as n ^ œ such that

Tnx - Tny|2 < knWx - y|2 + k^(I - Tn)x - (I - Tn)y|2, Vn > 1, x,y e C. (1.10)

The class of asymptotically strict pseudocontractions was introduced by Qihou [13] in 1996. Kim and Xu [14] proved that the class of asymptotically strict pseudocontractions is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [14] for more details.

T is said to be an asymptotically strict pseudocontraction in the intermediate sense if there exist a constant k e [0,1) and a sequence {kn} c [1, œ) with kn ^ 1 as n ^ œ such that

limsupsup (||Tnx - Tny||2 - kn||x - y||2 - k||(I - Tn)x - (I - Tn)y||2) < 0. (L11)

n ^œ x,yeC

In = ma^0, sup (||Tnx - Tny||2 - kn||x - y||2 - k|| (I - Tn)x - (I - Tn)y||2) (1.12)

It follows that In ^ 0 as n ^ œ. Then, (1.11) is reduced to the following:

Tnx - Tny|2 < knWx - y|2 + k^(I - Tn)x - (I - Tn)y|2 + t,n, Vn > 1, x,y e C. (1.13)

The class of mappings was introduced by Sahu et al. [15]. They proved that the class of asymptotically strict pseudocontractions in the intermediate sense is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [15] for more details.

T is said to be asymptotically pseudocontractive if there exists a sequence {kn} c [1, œ) with kn ^ 1 as n ^ œ such that

(Tnx - Tny,x - y) < kn||x - y||2, Vn > 1, x,y e C. (1.14)

It is not hard to see that (1.14) is equivalent to

Tnx - Tny\\2 < (2kn - 1)||x - y||2 + ||x - y - (Tnx - Tny)\\2, Vn > 1, x,y e C. (1.15)

The class of asymptotically pseudocontractive mapping was introduced by Schu [16] (see also [17]). In [18], Rhoades gave an example to showed that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings; see [18] for more details. Zhou [19] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping which is also uniformly asymptotically regular has a fixed point.

T is said to be an asymptotically pseudocontractive mapping in the intermediate sense if there exists a sequence {kn} c [1, to) with kn ^ 1 as n ^ to such

limsupsup ((Tnx - Tny,x - y) - kn||x - y||2j < 0. (1.16)

n ^to x,yeC

tn = ma^0, sup ({Tnx - Tny,x - y) - kn||x - y||2) |. (1.17)

It follows that i,n ^ 0 as n ^ to. Then, (1.16) is reduced to the following:

(Tnx - Tny,x - y) < knWx - y|2 + t,n, Vn > 1, x,y e C. (1.18)

It is easy to see that (1.18) is equivalent to

Tnx - Tny||2 < (2kn - 1) ||x - y||2 + ||x - y - (Tnx - Tny) ||2 + 2&, Vn > 1, x,y e C.

(1.19)

The class of asymptotically pseudocontractive mappings in the intermediate sense was introduced by Qin et al. [20]. Weak convergence theorems of fixed points were established based on iterative methods; see [20] for more details.

In this paper, we introduce the following mapping.

Definition 1.1. Recall that T : C ^ C is said to be total asymptotically pseudocontractive if there exist sequences (yn} c [0, to) and [¿,n] c [0, to) with yn ^ 0 and ¿,n ^ 0as n ^ to such that

(Tnx - Tny,x - y) < ||x - y||2 + ¡in4>(Hx - y||) + tn, Vn > 1, x,y e Cr (1.20)

where 0 : [0, to) ^ [0, to) is a continuous and strictly increasing function with 0(0) = 0.

It is easy to see that (1.20) is equivalent to the following:

Tnx - Tny\\2 < \\x - y\\2 + 2^4>(\\x - y\\) + \\x - y - (Tnx - Tny) \\2 +

(1.21)

Vn > 1, x,y e C.

Remark 1.2. If $(X) = I2, then (1.20) is reduced to

(Tnx - Tny,x - y) < (1 + fin)\\x - y\\2 + in, Vn > 1, x,y e C. (1.22)

Remark1.3. Put

In = max

(1.23)

If = I2, then the class of total asymptotically pseudocontractive mappings is reduced to the class of asymptotically pseudocontractive mappings in the intermediate sense.

Recall that the modified Ishikawa iterative process which was introduced by Schu [16] generates a sequence {xn} in the following manner:

where T : C ^ C is a mapping, xi is an initial value, and {an} and {pn} are real sequences in

If pn = 0 for each n > 1, then the modified Ishikawa iterative process (1.24) is reduced to the following modified Mann iterative process:

The purpose of this paper is to consider total asymptotically pseudocontractive mappings based on the modified Ishikawa iterative process. Weak convergence theorems are established in real Hilbert spaces.

In order to prove our main results, we also need the following lemmas.

Lemma 1.4. In a real Hilbert space, the following inequality holds:

x1 e C, yn = pnTnxn + (1 - pn)xn, xn+1 = anTnyn + (1 - an)xn, Vn > 1,

(1.24)

[0,1].

x1 e C, xn+1 = anTnxn + (1 - an)xn, Vn > 1.

(1.25)

ax + (1 - a)y\\2 = a||x||2 + (1 - a)\\y\\2 - a(1 - a)\\x - y\\2, Va e [0,1], x,y e C. (1.26)

Lemma 1.5 (see [21]). Let {rn}, {sn}, and {tn} be three nonnegative sequences satisfying the following condition:

rn+1 < (1 + Sn)rn + tn, Vn > n0, (1.27)

where n0 is some nonnegative integer. If £to=1 sn< to and £to=1 tn < to, then limn ^^rn exists. 2. Main Results

Now, we are ready to give our main results.

Theorem 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C ^ C a uniformly L-Lipschitz and total asymptotically pseudocontractive mapping as defined in (1.20). Assume that F (T) is nonempty and there exist positive constants M and M* such that 0(1) < M*12 for all 1 > M. Let {xn} be a sequence generated in the following manner:

x1 e C,

yn = PnTnXn + (1 - pn)xm (2.1)

xn+i = anTnyn + (1 - an)xn, Vn > 1,

where {an} and {fin} are sequences in (0,1). Assume that the following restrictions are satisfied:

(a) £TO=1 yn< to and%TO=1 In < to,

(b) a < an < fin < b for some a> 0 and some b e (0,L-2[V 1 + L2 - 1]).

Then, the sequence {xn} generated in (2.1) converges weakly to fixed point of T.

Proof. Fix x* e F(T). Since 0 is an increasing function, it results that 0(1) < 0(M) if 1 < M and 0(1) < M*12 if 1 > M. In either case, we can obtain that

0(||xn - x*||) < 0(M) + M*\\xn - x*\\. (2.2)

In view of Lemma 1.4, we see from (2.2) that

|| yn - x*||2 = ||^n(Tnxn - x*) + (1 - fin) (xn - x*)||2

n„Tnxn - x*||2 + (1 - 6n)\\xn - x*|2 - fin(1 - fin)\\Tnxn - xnf

= fin\\Tnxn - x*||2 + (1 - fin)\\xn - x*|2 - fin( 1 - fin)\\Tnxn - x,

< fin(\\xn - x*||2 + 2yn0(\\xn - x*||) + 2^n + \\xn - Tnxn\\2)

+ (1 - fin)\\xn - x* ||2 - fin (1 - fin)\\Tnxn - xnf

< (1 + 2fin^nM*)\\xn - x*||2 + fin\\Tnxn - xnf + 2finyn0(M) + 2finln

< qn\\xn - x*|2 + fin\\Tnxn - xn||2 + 2finyn0(M) + 2finln,

where qn = 1 + 2^nM* for each n > 1. Notice from Lemma 1.4 that

\\yn - Xnyn||2 = ||^n(TnXn - Tnyn) + (1 - pn){Xn - Tnyn) ||2

= mrnXn - Tnyn||2 + (1 -pn) ||xn - Tnyn||2 - pn (1 -pn)\\TnXn - Xn \\2 (2.4) < pn L2\\Xn - TnXn\\2 + (1 - pn)||Xn - Tnyn||2 - pn (1 - pn)\\TnXn - Xn \\2.

Since ty is an increasing function, it results that < if X < M and < M*I2 if

X> M. In either case, we can obtain that

||yn - x*||) < 4>(M)+ M*||yn - X*||2. (2.5)

This implies from (2.3) and (2.4) that

|Tnyn - X*|2 < ||yn - X*||2 + ||yn - X* || ) + 2^n + ||yn - Tnyn|2

< qn^yn - X*!2 + Wyn - Tnyn|2 + 2^n<p(M) + 2ln

2n^ ^ii2 a /1 „ a U2T2 a \ iit^x _ n2

< q2n\\Xn - x*\\2 - pn( 1 - qnpn - p2nL2 - pn)\\TnXn - x, + 2pn + (1 - pn) ||Xn - Tnyn|2,

where pn = qnpn^n$(M) + qnpnln + pn$(M) + ln for each n > 1. It follows that

\\Xn+i - x*\\ = Han(Tnyn - x*) + (1 - an)(Xn - x*)||

= an^yn - x* 12 + (1 - an)\\Xn - x*\\2 - an(1 - an)WTnyn - Xn|2 (2.7)

< q2n\\Xn - xl2 - anpn( 1 - qnpn - p2nL2 - pn)\\TnXn - Xn\\2 + 2anfn-

From the restriction (b), we see that there exists n0 such that

1 - qnpn - p2nL2 - pn > 1 - ^ ^ > 0, Vn > n0. (2.8)

It follows from (2.7) that

\\Xn+1 - xl2 < (1 + (qn + 1)2frM*)\\Xn - x*\\2 + 2anpn, Vn > m. (2.9)

Notice that ^°°=1(qn + 1)2ynM* < oo and ^°°=1 pn < o. In view of Lemma 1.5, we see that limn^o||xn - x*|| exists. For any n > n0, we see that

a2(1 - 2b - L2b2) 2

^-b-||TnXn - Xn||2

(2.10)

< (qn + l)2ynM*\\Xn - x*\\2 + \\%n - x*\\2 - ||x„+1 - x*\\2 + 2anpn, from which it follows that

lim \Tnxn - xn\ = 0. (2.11)

Note that

||Xn+1 - Xn|| < an(\TnVn - TnXn\\ + ||TnXn - Xn||)

< an(L\yn - Xn\\ + |TnXn - Xn|) (2.12)

< an( 1 + pnL) \\Tnx n - xn \ .

In view of (2.11), we obtain that

lim \\xn+1 - xn\\ = 0. (2.13)

Note that

\\xn - Txn\\ < \\xn - xn+1 \\ + || xn+1 - T xn+11 + ||T xn+1 - T xn | + || T xn - Txn |

< (1 + L)\xn - xn+1\\ + ||xn+1 - Tn+!xn+1|| + L\\Tnxn - xn\.

(2.14)

Combining (2.11) and (2.13) yields that

lim \\Txn - xn\ = 0. (2.15)

Since {xn} is bounded, we see that there exists a subsequence (xni} c {xn} such that xni ^ x. Next, we claim that x e F(T). Choose a e (0,1/(1 + L)) and define ya,m = (1 - a)x + aTmx for arbitrary but fixed m > 1. From the assumption that T is uniformly L-Lipschitz, we see that

\ \xn T xn\\ < \\xn Txn\\ + Txn T xn\\ + *** + T xn T xn

II II II II (2.16)

< [1 + (m - 1)L]\\xn - Txn\\.

It follows from (2.15) that

lim ||x„ - Tmxn\\ — 0. (2.17)

Tt —^ rv-i V '

Since $ is an increasing function, it results that $(X) < $(M) if X < M and $(X) < M*X2 if X> M. In either case, we can obtain that

\\xn - y*mV) < $(M)+ M*\\xn - yam\ • (218)

This in turn implies that

(x — ya,m, ya,m - T ya,m) _ (x — xn, ya,m - T ya,m) + (xn - ya,m, ya,m - T ya,m)

— (x - xn, ya,m - T ya,m) + (xn - ya,m, T xn - T ya,m} — (xn - ya,m, xn - ya,m) + (xn - ya,m, xn - Tmxn)

< (x - xn, ya,m - T ya,m) + \\xn - ya,m \\) +

+ \\xn - ya,m \\ \ \xn - Tnlxn\

< (x — xn,ya,m - T ya,m) + ftm$(MM) + ftmM^ \\xn - ya,m\\ + £m

+ \\xn ya,m \\ \ \xn T xi

Since xn ^ x, we see from (2.17) that

(2.19)

(x - ya,m, ya,m - Tmya,m) < ftm$(M) + ftmM*\\xn - ya,m\\ + lm. (2.20) On the other hand, we have

(x - ym (x - Tmx) - (yaim - Tmya^ni)) < (1 + L) \\x - ycl^ni\\2 — (1 + L)a2p - Tmx\t (2.21)

Note that

\\x - Tmx\\2 — (x - Tmx,x - Tmx) 1 _ _ _

— - y«'m'x - Tmx) (2.22)

1 _ _ _ 1 _

— aix - ya,m, (x - T x) - (ya/m - T ya,m) ) + ^ (x - ya,m, ya,m - Tmya/m).

Substituting (2.20) and (2.21) into (2.22), we arrive at

p - Tmx\\2 < (1 + L)a\\x - Tmx\\2 + ftm$(M)+ \xn - yam\\ + lm (2.23)

This implies that

a[1 - (1 + L)a]p - Tmx\\2 < + ymM"\\xn - y^mf + m Vm > 1. (2.24)

Letting m —> to in (2.24), we see that Tmx — x. Since T is uniformly L-Lipschitz, we can obtain that x = TX.

Next, we prove that {xn} converges weakly to x. Suppose the contrary. Then, we see that there exists some subsequence {xnj} c {xn} such that {xnj} converges weakly to x e C, where x fx. It is not hard to see that that x e F (T). Put d = limn—TO\\xn - x\\. Since H enjoys Opial property, we see that

d = lim inf\\xni - x\\ < lim inf\\xni - x\\

i — to i — to

= lim inf \xnj - x\\ < lim inf\\xnj - x\\ (2.25)

j — to N j — to N ' II v '

= lim inf\\xni - x\ = d.

This derives a contradiction. It follows that xx = x. This completes the proof. □

Remark 2.2. Demiclosedness principle of the class of total asymptotically pseudocontractive mappings can be deduced from Theorem 2.1.

Remark 2.3. Since the class of total asymptotically pseudocontractive mappings includes the class of strict pseudocontractions, the class of asymptotically strict pseudocontractions, the class of pseudocontractive mappings, the class of asymptotically pseudocontractive mappings and the class of asymptotically pseudocontractive mappings in the intermediate sense as special cases, Theorem 2.1 improves the corresponding results in Marino and Xu [12], Kim and Xu [14], Sahu et al. [15], Schu [16], Zhou [19], and Qin et al. [20].

Remark 2.4. It is of interest to improve the main results of this paper to a Banach space. Acknowledgment

The authors thank the referees for useful comments and suggestions. References

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