Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2011, Article ID 468716,18 pages doi:10.1155/2011/468716

Research Article

On the Convergence of Implicit Iterative Processes for Asymptotically Pseudocontractive Mappings in the Intermediate Sense

Xiaolong Qin,1,2 Jong Kyu Kim,3 and Tianze Wang1

1 School of Mathematics and Information Sciences,

North China University of Water Resources and Electric Power, Zhengzhou 450011, China

2 Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

3 Department of Mathematics, Kyungnam University, Masan 631-701, Republic of Korea

Correspondence should be addressed to Jong Kyu Kim, jongkyuk@kyungnam.ac.kr Received 20 November 2010; Accepted 9 February 2011 Academic Editor: Narcisa C. Apreutesei

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.

An implicit iterative process is considered. Strong and weak convergence theorems of common fixed points of a finite family of asymptotically pseudocontractive mappings in the intermediate sense are established in a real Hilbert space.

1. Introduction and Preliminaries

Throughout this paper, we always assume that H is a real Hilbert space with the inner product (•, ■) and the norm || ■ ||. Let C be a nonempty closed convex subset of H and T : C ^ C a mapping. We denote F (T) by the fixed point of the mapping T.

Recall that T is said to be uniformly L-lipschitz if there exists a positive constant L such

||T"x - T"y|| < L||x - y\\, Vx,y e C, n > 1. (1.1)

T is said to be nonexpansive if

Tx - Ty|| < ||x - y||, Nx,y e C.

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\\, Vx,y e C, n > 1. (1.3)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. It is known that if C is a nonempty bounded closed convex subset of a Hilbert space space H, then every asymptotically nonexpansive mapping on C has a fixed point. Since 1972, a host of authors have been studing strong and weak convergence problems of the iterative processes for such a class of mappings.

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

limsupsup (WTnx - TnyW - Wx - yW) < 0. (1.4)

n ^to x,yeC

Putting

iIn = max] 0, sup (WTnx - TnyW - Wx - y^ 1, (1.5)

[ x,yeC J

we see that £n ^ 0 as n ^ to. Then, (1.4) is reduced to the following:

||Tnx - Tny|| < ||x - y|| + in, Vx,y e C, n > 1. (1.6)

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk [2] (see also Bruck et al. [3]) as a generalization of the class of asymptotically nonexpansive mappings. It is known [4] that if C is a nonempty bounded closed convex subset of a Hilbert space space H, then every asymptotically nonexpansive mapping in the intermediate sense on C has a fixed point.

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

||Tx - Ty||2 < ||x - y||2 + k||(I - T)x - (I - T)y||2, Vx,y e C. (1.7)

For such a case, T is also said to be a K-strict pseudocontraction. The class of strict pseudocontractions was introduced by Browder and Petryshyn [5] in 1967. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction.

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

||T"x - Tny||2 < kn||x - y||2 + k||(I - Tn)x - (I - Tn)y||2, Vx,y e C, n > 1. (1.8)

For such a case, T is also said to be an asymptotically K-strict pseudocontraction. The class of asymptotically strict pseudocontractions is introduced by Qihou [6] in 1996. It is clear that every asymptotically nonexpansive mapping is an asymptotical 0-strict pseudocontraction.

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

limsupsup (||Tnx - Tny||2 - kn||x - y\\2 - k|| (I - Tn)x - (I - Tn)y\\2) < 0. (1.9)

n ^œ x,yeC

For such a case, T is also said to be an asymptotically K-strict pseudocontraction in the intermediate sense. Putting

In = max jo, sup ^||Tnx - Tny||2 - kn||x - y||2 - k|| (I - Tn)x - (I - Tn )y||2) J, (1.10)

we see that ¿,n ^ 0 as n ^ œ. Then, (1.9) is reduced to the following:

WTnx - Tny|2 < knWx - y|2 + kW(I - Tn)x - (I - Tn)y|2 + £n, Vx,y e C, n > 1. (1.11)

The class of asymptotically strict pseudocontractions in the intermediate sense was introduced by Sahu et al. [7] as a generalization of the class of asymptotically strict pseudocontractions, see [7] for more details. We also remark that if kn = 1 for each n > 1 and k = 0in (1.9), then the class of asymptotically K-strict pseudocontractions in the intermediate sense is reduced to the class of asymptotically nonexpansive mappings in the intermediate sense.

T is said to be pseudocontractive if

('Tx - Ty,x - y) < ||x - y||2, Vx,y e C. (1.12)

It is easy to see that (1.12) is equivalent to

||Tx - Ty||2 < ||x - y||2 + ||(I - T)x - (I - T)y||2, Vx,y e C. (1.13)

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

(Tx - Ty,x - y) < ^ ||x - y||2, Vx,ytC. (1-14)

It is easy to see that (1.14) is equivalent to

WTnx - Tny|2 < knWx - y|2 + W(I - Tn)x - (I - Tn)y|2, Vx,y e C, n > 1. (1.15)

We remark that the class of asymptotically pseudocontractive mappings was introduced by Schu [8] in 1991. For an asymptotically pseudocontractive mapping T, Zhou [9] proved that if

T is also uniformly Lipschitz and uniformly asymptotically regular, then T enjoys a nonempty fixed point set. Moreover, F (T) is closed and convex.

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 that

limsupsup (||Tnx - Tny||2 - kn||x - yf - \\(I - Tn)x - (I - Tn)y||2) < 0. (1.16)

n ^to x,yeC

It is easy to see that (1.16) is equivalent to

limsupsup ({T"x - Tny,x - y) - —r— ||x - y||2^ <0. (1.17)

n ^to x,yeC\ 2 /

In = max jo, supc(||Tnx - Tny||2 - kn||x - y||2 - || (I - Tn)x - (I - Tn)y||^ J. (1.18)

Then, (1.16) is reduced to the following

WTnx - Tny|2 < knWx - y|2 + W(I - Tn)x - (I - Tn)y|2 + ^n, Vn > 1, x,y e C. (1.19) It is easy to see that (1.19) is equivalent to

(T"x - T"y,x - y) < ||x ~ 3/||2 + Y' V«>1, x,ytC. (1.20)

The class of asymptotically pseudocontractive mappings in the intermediate sense which includes the class of asymptotically pseudocontractive mappings and the class of asymptotically strict pseudocontractions in the intermediate sense as special cases was introduced byQin etal. [10].

In 2001, Xu and Ori [11], in the framework of Hilbert spaces, introduced the following implicit iteration process for a finite family of nonexpansive mappings {Ti,T2,...,TN} with {an} a real sequence in (0,1) and an initial point x0 e C:

x1 = a1x0 + (1 - a1)T1 x1, x2 = a2x1 + (1 - a2)T2 x2,

(1.21)

which can be written in the following compact form

x„ = a„x„-1 + (1 - a„)T„x„, V„ > 1, (1.22)

where T„ = T„(modN) (here the mod N takes values in {1,2,.. .,N}). They obtained the following weak convergence theorem.

Theorem XO. Let H be a real Hilbert space, C a nonempty closed convex subset of H, and T : C ^ C be a finite family of nonexpansive mappings such that F = f|N=1 F(T) / 0- Let {x„} be defined by (1.22). If {a„} is chosen so that a„ ^ 0 as „ ^ <x>, then {x„} converges weakly to a common fixed point of the family of {Ti

Subsequently, fixed point problems based on implicit iterative processes have been considered by many authors, see [9, 12-23]. In 2004, Osilike [18] considered the implicit iterative process (1.22) for a finite family of strictly pseudocontractive mappings. To be more precise, he proved the following theorem.

Theorem O. Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let {Ti}N=1 be N strictly pseudocontractive self-maps of C such that F = f|Ni F(Tj) = 0. Let x0 e C and let {a„} be a sequence in (0,1) such that a„ ^ 0 as „ ^ <x>. Then, the sequence {x„} defined by (1.22) converges weakly to a common fixed point of the mappings {Ti}N=1.

In 2008, Qin et al. [20] considered the following implicit iterative process for a finite family of asymptotically strict pseudocontractions:

x1 = a1x0 + (1 - a1)T1x1, x2 = a2x1 + (1 - a2)T2x2,

Xn = aNxN-1 + (1 - &N)TnXn, xn+1 = №n+1Xn + (1 - aN+1 )Tj2Xn+!, (1.23)

x2N = «2Nx2N-1 + (1 - a2N)TNx2N, x2N+1 = «2N+1 x2N + (1 - &2N+1)T1x2N+1,

where x0 is an initial value, {a„} is a sequence in (0,1). Since for each „ > 1, it can be written as „ = (h - 1)N + i, where i = i(„) e {1,2,...,N}, h = h(„) > 1 is a positive integer and

h(n) —> to as n —> to. Hence, the above table can be rewritten in the following compact form:

Xn = anXn-i + (1 - an)Tinxn, Vn > 1. (1.24)

A weak convergence theorem of the implicit iterative process (1.24) for a finite family of asymptotically strict pseudocontractions was established.

We remark that the implicit iterative process (1.24) has been used to study the class of asymptotically pseudocontractive mappings by Osilike and Akuchu [19]. They obtained strong convergence of the implicit iterative process (1.24), however, there is no weak convergence theorem.

In this paper, motivated by the above results, we reconsider the implicit iterative process (1.24) for asymptotically pseudocontractive mappings in the intermediate sense. Strong and weak convergence theorems of common fixed points of a finite family of asymptotically pseudocontractive mappings in the intermediate sense are established. The results presented in this paper mainly improve and extend the corresponding results announced in Chang et al. [24], Chidume and Shahzad [13], Gornicki [25], Osilike [18], Qin et al. [20], Xu and Ori [11], and Zhou and Chang [23].

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

Recall that a space X is said to satisfy Opial's condition [26] if, for each sequence {xn} in X, the convergence xn — x weakly implies that

liminf||xn - x|| < liminf||xn - y||, Vy e E (y/x). (1-25)

n —> TO n —> TO

Recall that a mapping T : C — C is semicompact if any sequence {xn} in C satisfying limn—TO||xn - Txn || = 0 has a convergent subsequence-

Lemma 1.1 (see [27]). In a real Hilbert space, the following inequality holds

\\ax + (1 - a)y\\2 = a||x||2 + (1 - a^yf - a(1 - a)\\x - y\2, Va e [0,1], x,y e H. (1-26)

Lemma 1.2 (see [28]). Let {an}, {bn}, and {cn} be three nonnegative sequences satisfying the following condition:

an+1 < (1 + bn)an + cn, Vn > n0, (1.27)

where n0 is some nonnegative integer, ^TO=i bn < to and ^TO=i cn < to. Then, the limit limn—TO an exists.

2. Main Results

Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping in the intermediate sense with the sequence [k„ri} c [1, to) such that £ TO=1(k„,i - 1) < to for each 1 < i < N, where N > 1 is some positive integer. Let £„) = max{0,sup C(||X„x - T„y\\2 - k(„i)\\x - y\\2 -

\\(I - T„)x - (I - T„)y\\2)} for each 1 < i < N. Assume that the common fixed point set F = Di=i F(Ti) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L< a < a„ < b < 1, where L = max{Li : 1 < i < N}, for all „ > 1;

(b)STO=11„ < to, where '£,„ = max^^ : 1 < i < N}. Then, {x„} converges weakly to some point in F.

Proof. First, we show that the sequence {x„} generated in the implicit iterative process (1.24) is well defined. Define mappings R„ : C ^ C by

Rn(x) = anxn-1 + (1 - a^T^nx, Vx e C, n > 1. (2.1)

Notice that

\\R„(x) - R„{y) | = ka„x„-1 + (1 - a„)Th((t„) x) - {a„x„-1 + (1 - a^T^y)

< (1 - a„)L\\x - y\\ (2.2)

< (1 - a)L\\x - y\\, Nx,y e C.

From the restriction (a), we see that R„ is a contraction for each „ > 1. By Banach contraction principle, we see that there exists a unique fixed point x„ e C such that

x„ = a„x„-1 + (1 - a„)Th(„„)x„, V„ > 1. (2.3)

This shows that the implicit iterative process (1.24) is well defined for uniformly Lipschitz continuous and asymptotically pseudocontractive mappings in the intermediate sense. Let k„ = max{k„7i : 1 < i < N}. In view of the assumption, we see that ^TO=1(k„ - 1) < to. Fixing p eF, we see from Lemma 1.1 that

22 \\x„ - p\\2 = a„\\x„-1 - p\\2 + (1 - a„)\\Th„'))x„ - p\\ - a„(1 - a^lr^x„ - x„-1\

< a„\\x„-1 - p\\2 + (1 - a„)^ kh(„)\\x„ - p\\2 + l^x - x„\ +1„

\ ) \\2

- a„ (1 - a„) \\ Ti(„) x„ - x„-1 \\

< a„||x„_i -p\\2 + (1 - an)kh(n)\\xn -+ (1 - an)||rhn'J)Xn - Xn

\ ) \\2

- an (1 - an) \\ T.(n) Xn - xn-1 \\ +

< an \ Xn-1 - p\2 + (1 - an)kh(n)\\Xn - p\2 - (1 - an )2an\\Thnn)Xn - Xn-J + in-

2„ hTh(n).

vn Xn-1 \ + ^

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

(1 - an)kh(n) <Q< 1, Vn > no, (2.5)

where Q = (1 - a)(1 + a/(2(1 - a))). It follows that

\x„ ~p\\2 < z—n a" -||x„_i -p\\2 +

1 - (1 - an)kh(n) 1 - (1 - an)kh(n)

In view of the restriction (b), we obtain from Lemma 1.2 that limn^œ||xn - p|| exists. Hence, the sequence (xn} is bounded. Reconsidering (2.4), we see from the restriction (a) that

(1 - b)2a||xh((\))Xn - Xn-11| < a^||xn-1 - p|2 - ||xn - p|2) + (kh(n) - 1)|x? - p|2 + in-

This implies that

lim Xn? Xn - Xn-11 = 0. (2.8)

n ^œll l(n)

Notice that

It follows from (2.8) that

Observe that

xn Xn-1 y < \\Ti(n) Xn xn—1 \\ - (2.9)

lim ||Xn - Xn-1 y = 0- (2.10)

I rrh(n) Il . Il T-h(n) II ||xh(n) rrh(n)

Xn-1 - z(n) Xn-1 \\ < \\Xn-1 - Ti(n) xn\\ + \\Ti(n) Xn - z(n) Xn-1 \

(2.11)

i(n) \ + | - |

< \\Xn-1 Ti(n) Xn\\ + L||xn Xn-1

In view of (2.8), and (2.10), we obtain that

lim||x„_i - xn-i\\ = 0. (2.12)

n ^<»11 l(n) II

Since for any positive integer n > N, it can be written as n = (h(n) - 1)N + i(n), where i(n) e {1,2,...,N}. Observe that

||xn-1 — TnXn-1|| — ||xn-1 — Ti(n) Xn-1 || + \\Thn) Xn-1 — TnXn-1 \\

c || X 1 - Th(n) x j + L ||Th(n)-1 x 1 - x j

— ||Xn-1 Ti(n) Xn-1 m "T Ti(n) Xn-1 Xn-1 m

— ||xn-1 - Tjnn)Xn-1|| (2.13)

r/^h(n)-1 rph(n)-1 m UTh(n)-1 n

+ h n1i(n) Xn-1 - Ti(n-N)Xn-N || + H^in-N) Xn-N - X(n-N)-11|

+ w X(n-N)-1 - Xn-1 w) .

Since for each n> N, n = (n - N)(mod N), on the other hand, we obtain from n = (h(n) -1)N + i(n) that n - N = ((h(n) - 1) - 1)N + i(n) = (h(n - N) - 1)N + i(n - N). That is,

h(n - N) = h(n) - 1, i(n - N) = i(n). (2.14)

Notice that

\Th(n)-1X Th(n)-1 x II _ \\Th(n)-l x _Th(n)-l x |

l^Hn) Xn-1 ii(n-N)Xn-N || _ I| t(n) Xn-1 1i(n) Xn-N |

< Lyxn-i - Xn-N ||, (2.15)

\Th(n)-1 || _ \\Th(n-N)

Ti(n-N)Xn-N - X(n-N)-1 || _ || Ti(n-N) Xn-N - X(n-N)-1

Substituting (2.15) into (2.13), we arrive at

||xn-1 - TnXn-11 — ||Xn-1 - Th(n) Xn-1 W

+ L(L||Xn - Xn-N || + UnN Xn-N - X(n-N)-11| + ¡X(n-N)-1 - Xn-1 •

(2.16)

In view of (2.8), (2.10), and (2.12), we obtain from (2.16) that

lim ||Xn-1 - TnXn-1|| _

(2.17)

Notice that

\\x„ - TnXn || < \\Xn - Xn-l\\ + ||Xn-1 - TnXn-tW + ||T„X„_1 - TnXn < (1 + L)\\Xn - Xn-1 \\ + \\Xn-1 - TnXn-1 \\.

(2.18)

From (2.10) and (2.17), we arrive at

lim \\Xn - TnXn\\ = 0. (2.19)

Notice that

|Xn Tn+;Xn|| < \\Xn Xn+j y + 11 Xn+j Tn+jXn+j y + || Tn+jXn+j Tn+jXn |

(2.20)

< (1 + L)|| xn xn+j1 + ||xn+j Tn+jxn+j Vj e (1,2, ...,N}. It follows from (2.10) and (2.19) that

lim ||xn-Tn+;Xn|| = 0, Vj e(1,2,...,N}. (2.21)

Note that any subsequence of a convergent number sequence converges to the same limit. It follows that

lim \\Xn - TrXn\\ = 0, Vr e{1,2,...,N}. (2.22)

Since the sequence (xn} is bounded, we see that there exists a subsequence (xni} c (xn} such that {x„J converges weakly to a point x G C. Choose a G (0,1/(1 + L)) and define y a,m,r = (1 ~ <x)x + aT]"x for arbitrary but fixed m > 1. Notice that

Wxni Tf1 xni y < Wxni Trxni y + ¡Trx„, Tr xni|| + '■'+ ||Tr xni Tf1 xni

< [1 + (m - 1)L]Wxni - Trxni W, Vr e (1,2,...,N}. It follows from (2.22) that

(2.23)

lim \\Xni - TTnX„i \\ = 0, Vr e{1,2,...,N}. (2.24)

Note that

(x — yarm,r/ ya,m,r ~ Ty1 yafm,r)

— (x — Xyii, yarm,r ~ ya,m} + (p^rii ~ ya,m,r/ ya,m,r ~ Tr ya,m,r)

— (x — Xni/ yarm,r ~ ya.fmfr} + (p^rii ~ y'a,mfr/ Tj^Xrii ~ V#,m,r)

— {xn, — ya,m,r, xn — ya,m,r} + {xni — ya,m,r, xn — Tr Xn)

</r-r 1/ -T"v \ i 1+fc"'llr -1/ + (2'25)

¿fa/m/r Lr }jctrm,rf ' 2 y a,m,r \ \ ' 2

— ||Xnj — yafmfr || + ||xnj — yafmfr || ||xnj — T^xn ||

_ /— tw \ — 1 11 112 ^m

— — ya,m,r ~ i-r ya,m,r / + 2 ll*^1* — yafmfr \ \ ^

+ ||xn - ya^,r\Hxni - rmxnt ||, Vr e {1,2,...,N}. Since x,H — x and (2.24), we arrive at

(x-ya,m,ya,m-Trlya,m) < "3hx,m,r\\2 + y, Vr € {1, 2,. . ., N}. (2.26)

On the other hand, we have

(x yarvirTf (x Ty x) Tj- y a ah,)') }

< (1 + L)p-yw||2 = (1 + L)tr2||x-X;"x||2, Vr G {1,2.....N).

Notice that

11X J.X11 — ^X J.Xf X i'y X

1 _ _ _

= — ^X — y a ,¡11,1 ' x — T; X)

l _ _ _

= — yafmfr> (x — Tr x) — (j/djn,! ~ T^- y<t,fiij ) )

+ - (X - ya,m,r, ya,m,r ~ Vlya,m,r), Vr G {1, 2, . . ., N].

(2.27)

(2.28)

Substituting (2.26) and (2.27) into (2.28), we arrive at

a[l - (1 + L)a]||x-X;"x||2 < ^^\\x,u - yw||2 + Vr e {1,2.....N], m > 1.

(2.29)

Letting m -H- cjj in (2.29), we see that V"x x for each 1 < r < N. Since Tr is uniformly Lr-Lipschitz, we can obtain that x = T, x for each 1 < r < N. This means that x e F.

Next we show that {x„} converges weakly to x. Supposing the contrary, we see that there exists some subsequence [xnj} of {xn} such that [xnj} converges weakly to x* e C, where x* fx. Similarly, we can show x* G F- Notice that we have proved that lim^^Jxn-p\\ exists for each p e F- Assume that lim,,^.^ ||x„ -x\\=d where d is a nonnegative number. By virtue of the Opial property of H, we see that

d = liminf||x„; -x|| < liminf||x„; -x*||

ni TO ni ^TO

(2.30)

= liminf x,, -x* <liminf x,, -x = d.

This is a contradiction. Hence x = x*. This completes the proof. □

For the class of asymptotically pseudocontractive mappings, we have, from Theorem 2.1, the following results immediately.

Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti: C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {knii} c [1, to) such that£ TO=1(kn,i-1) < to for each 1 < i < N,whereN > 1 is some positive integer. Assume that the common fixed point set F = DNi FT) is nonempty. Let {xn}TO=o be a sequence generated in (1.24). Assume that the control sequence {an} in [0,1] satisfies the following restrictions 0 < 1 - 1/L < a < an < b < 1, where L = max{Li : 1 < i < N},for all n > 1. Then, {xn} converges weakly to some point in F.

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.1 the following results immediately.

Corollary 2.3. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically nonexpansive mapping in the intermediate sense for each 1 < i < N, where N > 1 is some positive integer. Let £(n/i) = max{0, supxyeC(||T"x - T^yW2 - ||x - y\\2)} for each 1 < i < N. Assume that the common fixed

point set F = Pi¿N1 F(Ti) is nonempty. Let {xn}TO=0 be a sequence generated in (1.24). Assume that the control sequence {an} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L < a < an < b < 1, where L = max{Li : 1 < i < N}, for all n > 1;

(b)STO=1 In < to, where In = max{£(n,i) : 1 < i < N}. Then {xn} converges weakly to some point in F.

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.1 the following results immediately.

Corollary 2.4. Let C be a nonempty closed convex subset of a Hilbert space H. LetTi : C ^ C be an asymptotically nonexpansive mapping with the sequence {kn/i} c [1, to) such that XTOU (knri-1) < to for each 1 < i < N, where N > 1 is some positive integer. Assume that the common fixed point set F = nN=1 F(Ti) is nonempty. Let {xn}TO=0 be a sequence generated in (1.24). Assume that the control

sequence {an} in [0,1] satisfies the following restriction 0 < 1 - 1/L < a < an < b < 1, where L = max{supn>1 {fcn,t} : 1 < i < N},for all n > 1. Then, {xn} converges weakly to some point in F.

Next, we give strong convergence theorems with the help of semicompactness.

Theorem 2.5. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping in the intermediate sense with the sequence {kn/i} c [1, to) such that £ TO=1(kn,i - 1) < to for each 1 < i < N, where N > 1 is some positive integer. Let Ini) = max{0,supxyeC(||T"x - T"y||2 - k(ni)\\x - y\\2 -\\(I - Tn)x - (I - T^yf)} for each 1 < i < N. Assume that the common fixed point set F = n^i F(Ti) is nonempty. Let {xn}TO=0 be a sequence generated in (1.24). Assume that the control sequence {an} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L < a < an < b < 1, where L = max{Li : 1 < i < N}, for all n > 1;

(b) ETO=1 In < to, where In = max{£ni) : 1 < i < N}.

If one of {T1,T2,...,TN} is semicompact, then the sequence {xn} converges strongly to some point in

Proof. Without loss of generality, we may assume that T1 is semicompact. From (2.22), we see that there exists a subsequence {xni} of {xn} that converges strongly to x e C. For each r e {1,2,. ..,N}, we get that

\\x - Trx\\ < \\x - xm|| + Hxni - Trxni || + \\Trxnt - Trx\\. (2.31)

Since Tr is Lipschitz continuous, we obtain from (2.22) that x e nN=1 F(Tr) = F. In view of Theorem 2.1, we obtain that limn^TO\\xn - x|| exists. Therefore, we can obtain the desired conclusion immediately. □

For the class of asymptotically pseudocontractive mappings, we have from Theorem 2.5 the following results immediately.

Corollary 2.6. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {knii} c [1, to) such that^ TO=1(kn,i-1) < to for each 1 < i < N,whereN > 1 is some positive integer. Assume that the common fixed point set F = n¿N1 F(Ti) is nonempty. Let {xn}TO=0 be a sequence generated in (1.24). Assume that the control sequence {an} in [0,1] satisfies the following restrictions: 0 < 1-1/L < a < an < b < 1,whereL = max{Li : 1 < i < N}, for all n > 1. Ifoneof {T1,T2,...,TN} is semicompact, then the sequence {xn} converges strongly to some point in FF.

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.5 the following results immediately.

Corollary 2.7. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically nonexpansive mapping in the intermediate sense for each 1 < i < N, where N > 1 is some positive integer. Let = max{0, supxyeC(||T™x - T^y! - ||x - y||2)} for each 1 < i < N. Assume that the common fixed

point set F = Pi ii F(Ti) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L< a < a„ < b < 1, where L = max{Li : 1 < i < N}, for all „ > 1;

(b) ETO=1l„ < to, where '£,„ = max^^ : 1 < i < N}.

If one of {T1,T2, ...,TN} is semicompact, then the sequence {x„} converges strongly to some point in

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.5 the following results immediately.

Corollary 2.8. Let C be a nonempty closed convex subset of a Hilbert space H. LetTi : C ^ C be an asymptotically nonexpansive mapping with the sequence {k„ri} c [1, to) suchthatetou (k„ri -1) < to for each 1 < i < N, where N > 1 is some positive integer. Assume that the common fixed point set F = nN=1 F(Ti) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restriction: 0 < 1 - 1/L < a < a„ < b < 1, where L = max{sup „>1{k„ri} : 1 < i < N},for all „ > 1. If one of {T1, T2,...,TN} is semicompact, then the sequence {x„} converges strongly to some point in F.

In 2005, Chidume and Shahzad [13] introduced the following conception. Recall that a family {Ti}N1 : C ^ C with F = H¿N1 F(Ti) = 0 is said to satisfy Condition (B) on C if there is a nondecreasing function f : [0, to) ^ [0, to) with f (0) = 0 and f (m) > 0 for all m e (0, to) such that for all x e C

max{\\x - Tix\\}> f (d(x, F)). (2.32)

Next, we give strong convergence theorems with the help of Condition (B).

Theorem 2.9. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping in the intermediate sense with the sequence {k„ri} c [1, to) such that ETO=1(k„,i - 1) < to for each 1 < i < N, where N > 1 is some positive integer. Let £(„¿) = max{0,supxyeC(\\T„x - T^yW2 - k(„ri)\\x - y\\2 -

\\(I - T„)x - (I - T„)y\\2)} for each 1 < i < N. Assume that the common fixed point set F = nN=1 F(Ti) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L < a < a„ < b < 1, where L = max{Li : 1 < i < N}, for all „ > 1;

(b)STO=11„ < to, where l„ = max^^ : 1 < i < N}.

If {T1, T2.....TN} satisfies Condition (B), then the sequence {x„} converges strongly to some point

Proof. In view of Condition (B), we obtain from (2.22) that f (d(x„, F)) ^ 0, which implies d(x„, F) ^ 0. Next, we show that the sequence {x„} is Cauchy. In view of (2.6), for any positive integers m, „, where m> „> „0, we obtain that

||*m - p\\ < B\\x„ - p\\ + B X ^ + J*-, (2.33)

i=„+1 Q Q

where B = exp{£TO=1((kh(„) - 1)/(1 - Q))}. It follows that

TO £ £

\\Xn-xm\\<\\xn-p\\ + \\xm-p\\<(l + B)\\xn-p\\ (2-34)

i=n+1 Q Q

It follows that {x„} is a Cauchy sequence in C,so {x„} converges strongly to some q £ C. Since Tr is Lipschitz for each r e {1,2,. ..,N}, we see that F is closed. This in turn implies that q £ F- This completes the proof. □

For the class of asymptotically pseudocontractive mappings, we have from Theorem 2.9 the following results immediately.

Corollary 2.10. Let C be a nonempty closed convex subset of a Hilbert space H. LetTi : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {knii} c [1, to) such that£ TO=1(k„,i-1) < to for each 1 < i < N,whereN > 1 is some positive integer. Assume that the common fixed point set F = D¿=i F(Ti) is nonempty. Let {x„}TO=o be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions: 0 < 1 - 1/L< a < a„ < b < 1, where L = max{Li : 1 < i < N}, for all n > 1. If {T1,T2,...,Tn } satisfies Condition (B), then the sequence {x„} converges strongly to some point in F.

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.9 the following results immediately.

Corollary 2.11. Let C be a nonempty closed convex subset of a Hilbert space H. Let Ti: C ^ C be a uniformly Li-Lipschitz continuous and asymptotically nonexpansive mapping in the intermediate sense for each 1 < i < N, where N > 1 is some positive integer. Let £(„ri) = max{0, supx yeC(\\T„x - T^yW2 - ||x - y||2)} for each 1 < i < N. Assume that the common fixed point

set F = flN F(Ti) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L< a < a„ < b < 1, where L = max{Li : 1 < i < N}, for all „ > 1;

(b)STO=1 £„ < to, where £„ = max^^ : 1 < i < N}.

If {T1, T2.....TN} satisfies Condition (B), then the sequence {x„} converges strongly to some point

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.9 the following results immediately.

Corollary 2.12. Let C be a nonempty closed convex subset of a Hilbert space H. LetTi: C ^ C be an asymptotically nonexpansive mapping with the sequence {k„ri} c [1, to) such that XTO=1 (k„i-1) < to for each 1 < i < N, where N > 1 is some positive integer. Assume that the common fixed point set F = nN=1 F(Ti) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restriction: 0 < 1 - 1/L < a < a„ < b < 1, where L = max^up^-Jk^} : 1 < i < N}, for all „ > 1. If {T1, T2,..., TN} satisfies Condition (B), then the sequence {x„} converges strongly to some point in F.

Finally, we give the following strong convergence criteria.

Theorem 2.13. Let C be a nonempty closed convex subset of a Hilbert space H. Let Xi : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping in the intermediate sense with the sequence [k„ri} c [1, to) such that £ TO=1(k„,i - 1) < to for each 1 < i < N, where N > 1 is some positive integer. Let l(„i) = max{0,sup C(||X„x - X„y||2 - k(„ri)\\x - y\\2 -\\(I - X„)x - (I - X„)y\\2)} for each 1 < i < N. Assume that the common fixed point set F = n^i F(Xi) is nonempty. Let {x„}TO=o be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L< a < a„ < b < 1, where L = max{Li : 1 < i < N}, for all „ > 1;

(b) ETO=11„ < to, where '£,„ = max{£(„,i) : 1 < i < N}.

Then, thesequence {x„} converges strongly to some point in F ifandonlyif liminf„ ^TOd(x„, F) = 0. Proof. The necessity is obvious. We only show the sufficiency. Assume that

liminf d(x„, F)= 0. (2.35)

In view of Lemma 1.2, we can obtain from (2.6) that lim„^TOd(x„, F) = 0. The desired results can be obtain from Theorem 2.9 immediately. □

For the class of asymptotically pseudocontractive mappings, we have from Theorem 2.13 the following results immediately.

Corollary 2.14. Let C be a nonempty closed convex subset of a Hilbert space H. LetXi : C ^ C be a uniformly Li-Lipschitz continuous and asymptotically pseudocontractive mapping with the sequence {k„i} c [1, to) such that^ TO=1(k„,i-1) < to for each 1 < i < N,whereN > 1 is some positive integer. Assume that the common fixed point set F = n¿N1 F(Xi) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions 0 < 1 - 1/L < a < a„ < b < 1, where L = max{Li : 1 < i < N},for all „ > 1. Then, the sequence {x„} converges strongly to some point in F if and only if lim inf„ ^TO>d(x„, F) = 0.

For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem 2.13 the following results immediately.

Corollary 2.15. Let C be a nonempty closed convex subset of a Hilbert space H. Let Xi: C ^ C be a uniformly Li-Lipschitz continuous and asymptotically nonexpansive mapping in the intermediate sense for each 1 < i < N, where N > 1 is some positive integer. Let £(„ri) = max{0, supx C(HXj„x - X„ty\\2 - \x - yH2)} for each 1 < i < N. Assume that the common fixed point

set F = n¿N1 F(Xi) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restrictions:

(a) 0 < 1 - 1/L < a < a„ < b < 1, where L = max{Li : 1 < i < N}, for all „ > 1;

(b)STO=11„ < to, where £„ = max{£(„,i) : 1 < i < N}.

Then, thesequence {x„} converges strongly to some point in F ifandonlyif liminf„ ^TOd(x„, F) = 0.

For the class of asymptotically nonexpansive mappings, we can conclude from Theorem 2.13 the following results immediately.

Corollary 2.16. Let C be a nonempty closed convex subset of a Hilbert space H. LetTi : C ^ C be an asymptotically nonexpansive mapping with the sequence [k„ri} c [1, to) such that ^TO^ (k„ri -1) < to for each 1 < i < N, where N > 1 is some positive integer. Assume that the common fixed point set F = f)j=i F(T) is nonempty. Let {x„}TO=0 be a sequence generated in (1.24). Assume that the control sequence {a„} in [0,1] satisfies the following restriction: 0 < 1 - 1/L < a < a„ < b < 1, where L = max{sup„>1 {k„ri} : 1 < i < N},for all „ > 1. Then, the sequence {x„} converges strongly to some point in F if and only if lim inf„ ^TOd(x„, F) = 0.

Acknowledgment

This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (2010-0016000).

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