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

Authors: Xiaolong Qin; Jong Kyu Kim; Tianze Wang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

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.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper, we always assume thatH 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 thatT is said to be uniformly L-lipschitz if there exists a positive constant L such that (1.1)‖Tnx-Tny‖≤L‖x-y‖,∀x,y∈C,n≥1.T is said to be nonexpansive if
(1.2)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.T is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that (1.3)‖Tnx-Tny‖≤kn‖x-y‖,∀x,y∈C,n≥1.
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: (1.4)limsup⁡n→∞sup⁡x,y∈C(‖Tnx-Tny‖-‖x-y‖)≤0.
Putting (1.5)ξn=max⁡{0,sup⁡x,y∈C(‖Tnx-Tny‖-‖x-y‖)},
we see that ξn→0 as n→∞. Then, (1.4) is reduced to the following: (1.6)‖Tnx-Tny‖≤‖x-y‖+ξn,∀x,y∈C,n≥1.
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 κ∈[0,1) such that (1.7)‖Tx-Ty‖2≤‖x-y‖2+κ‖(I-T)x-(I-T)y‖2,∀x,y∈C.
For such a case, T is also said to be a κ-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}⊂[1,∞) with kn→1 as n→∞ and a constant κ∈[0,1) such that (1.8)‖Tnx-Tny‖2≤kn‖x-y‖2+κ‖(I-Tn)x-(I-Tn)y‖2,∀x,y∈C,n≥1.
For such a case, T is also said to be an asymptotically κ-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}⊂[1,∞) with kn→1 as n→∞ and a constant κ∈[0,1) such that (1.9)limsup⁡n→∞sup⁡x,y∈C(‖Tnx-Tny‖2-kn‖x-y‖2-κ‖(I-Tn)x-(I-Tn)y‖2)≤0.
For such a case, T is also said to be an asymptotically κ-strict pseudocontraction in the intermediate sense. Putting (1.10)ξn=max⁡{0,sup⁡x,y∈C(‖Tnx-Tny‖2-kn‖x-y‖2-κ‖(I-Tn)x-(I-Tn)y‖2)},
we see that ξn→0 as n→∞. Then, (1.9) is reduced to the following: (1.11)‖Tnx-Tny‖2≤kn‖x-y‖2+κ‖(I-Tn)x-(I-Tn)y‖2+ξn,∀x,y∈C,n≥1.
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 κ=0 in (1.9), then the class of asymptotically κ-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 (1.12)〈Tx-Ty,x-y〉≤‖x-y‖2,∀x,y∈C.
It is easy to see that (1.12) is equivalent to (1.13)‖Tx-Ty‖2≤‖x-y‖2+‖(I-T)x-(I-T)y‖2,∀x,y∈C.T is said to be asymptotically pseudocontractive if there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that (1.14)〈Tx-Ty,x-y〉≤kn+12‖x-y‖2,∀x,y∈C.
It is easy to see that (1.14) is equivalent to (1.15)‖Tnx-Tny‖2≤kn‖x-y‖2+‖(I-Tn)x-(I-Tn)y‖2,∀x,y∈C,n≥1.
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}⊂[1,∞) with kn→1 as n→∞ such that (1.16)limsup⁡n→∞sup⁡x,y∈C(‖Tnx-Tny‖2-kn‖x-y‖2-‖(I-Tn)x-(I-Tn)y‖2)≤0.
It is easy to see that (1.16) is equivalent to (1.17)limsup⁡n→∞sup⁡x,y∈C(〈Tnx-Tny,x-y〉-kn+12‖x-y‖2)≤0.
Put (1.18)ξn=max⁡{0,sup⁡x,y∈C(‖Tnx-Tny‖2-kn‖x-y‖2-‖(I-Tn)x-(I-Tn)y‖2)}.
Then, (1.16) is reduced to the following (1.19)‖Tnx-Tny‖2≤kn‖x-y‖2+‖(I-Tn)x-(I-Tn)y‖2+ξn,∀n≥1,x,y∈C.
It is easy to see that (1.19) is equivalent to (1.20)〈Tnx-Tny,x-y〉≤kn+12‖x-y‖2+ξn2,∀n≥1,x,y∈C.
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 by Qin et al. [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 {T1,T2,…,TN} with {αn} a real sequence in (0,1) and an initial point x0∈C: (1.21)x1=α1x0+(1-α1)T1x1,x2=α2x1+(1-α2)T2x2,⋮xN=αNxN-1+(1-αN)TNxN,xN+1=αN+1xN+(1-αN+1)T1xN+1,⋮
which can be written in the following compact form (1.22)xn=αnxn-1+(1-αn)Tnxn,∀n≥1,
where Tn=Tn(modN) (here the modN takes values in {1,2,…,N}).They obtained the following weak convergence theorem.Theorem XO.
LetH 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=⋂i=1NF(Ti)≠∅. Let {xn} be defined by (1.22). If {αn} is chosen so that αn→0 as n→∞, then {xn} converges weakly to a common fixed point of the family of {Ti}i=1N.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.
LetH be a real Hilbert space and let C be a nonempty closed convex subset of H. Let {Ti}i=1N be N strictly pseudocontractive self-maps of C such that F=⋂i=1NF(Ti)≠∅. Let x0∈C and let {αn} be a sequence in (0,1) such that αn→0 as n→∞. Then, the sequence {xn} defined by (1.22) converges weakly to a common fixed point of the mappings {Ti}i=1N.In 2008, Qin et al. [20] considered the following implicit iterative process for a finite family of asymptotically strict pseudocontractions: (1.23)x1=α1x0+(1-α1)T1x1,x2=α2x1+(1-α2)T2x2,⋮xN=αNxN-1+(1-αN)TNxN,xN+1=αN+1xN+(1-αN+1)T12xN+1,⋮x2N=α2Nx2N-1+(1-α2N)TN2x2N,x2N+1=α2N+1x2N+(1-α2N+1)T13x2N+1,⋮
where x0 is an initial value, {αn} is a sequence in (0,1). Since for each n≥1, it can be written as n=(h-1)N+i, where i=i(n)∈{1,2,…,N}, h=h(n)≥1 is a positive integer and h(n)→∞ as n→∞. Hence, the above table can be rewritten in the following compact form: (1.24)xn=αnxn-1+(1-αn)Ti(n)h(n)xn,∀n≥1.
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], Górnicki [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 spaceX is said to satisfy Opial's condition [26] if, for each sequence {xn} in X, the convergence xn→x weakly implies that (1.25)liminf⁡n→∞‖xn-x‖<liminf⁡n→∞‖xn-y‖,∀y∈E(y≠x).Recall that a mappingT:C→C is semicompact if any sequence {xn} in C satisfying limn→∞∥xn-Txn∥=0 has a convergent subsequence.Lemma 1.1 (see [27]).
In a real Hilbert space, the following inequality holds(1.26)‖ax+(1-a)y‖2=a‖x‖2+(1-a)‖y‖2-a(1-a)‖x-y‖2,∀a∈[0,1],x,y∈H.Lemma 1.2 (see [28]).
Let{an}, {bn}, and {cn} be three nonnegative sequences satisfying the following condition:
(1.27)an+1≤(1+bn)an+cn,∀n≥n0,
where n0 is some nonnegative integer, ∑n=1∞bn<∞ and ∑n=1∞cn<∞. Then, the limit limn→∞an exists.


## 2. Main Results

Theorem 2.1.
LetC 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}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Let ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-k(n,i)∥x-y∥2-∥(I-Tin)x-(I-Tin)y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(n,i):1≤i≤N}. Then, {xn} converges weakly to some point in ℱ.Proof.
First, we show that the sequence{xn} generated in the implicit iterative process (1.24) is well defined. Define mappings Rn:C→C by
(2.1)Rn(x)=αnxn-1+(1-αn)Ti(n)h(n)x,∀x∈C,n≥1.
Notice that
(2.2)‖Rn(x)-Rn(y)‖=‖(αnxn-1+(1-αn)Ti(n)h(n)x)-(αnxn-1+(1-αn)Ti(n)h(n)y)‖≤(1-αn)L‖x-y‖≤(1-a)L‖x-y‖,∀x,y∈C.
From the restriction (a), we see that Rn is a contraction for each n≥1. By Banach contraction principle, we see that there exists a unique fixed point xn∈C such that
(2.3)xn=αnxn-1+(1-αn)Ti(n)h(n)xn,∀n≥1.
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 kn=max{kn,i:1≤i≤N}. In view of the assumption, we see that ∑n=1∞(kn-1)<∞. Fixing p∈ℱ, we see from Lemma 1.1 that
(2.4)‖xn-p‖2=αn‖xn-1-p‖2+(1-αn)‖Ti(n)h(n)xn-p‖2-αn(1-αn)‖Ti(n)h(n)xn-xn-1‖2≤αn‖xn-1-p‖2+(1-αn)(kh(n)‖xn-p‖2+‖Ti(n)h(n)xn-xn‖2+ξn)-αn(1-αn)‖Ti(n)h(n)xn-xn-1‖2≤αn‖xn-1-p‖2+(1-αn)kh(n)‖xn-p‖2+(1-αn)‖Ti(n)h(n)xn-xn‖2-αn(1-αn)‖Ti(n)h(n)xn-xn-1‖2+ξn≤αn‖xn-1-p‖2+(1-αn)kh(n)‖xn-p‖2-(1-αn)2αn‖Ti(n)h(n)xn-xn-1‖2+ξn.
From the restriction (a), we see that there exists some n0 such that
(2.5)(1-αn)kh(n)<Q<1,∀n≥n0,
where Q=(1-a)(1+a/(2(1-a))). It follows that
(2.6)‖xn-p‖2≤αn1-(1-αn)kh(n)‖xn-1-p‖2+ξn1-(1-αn)kh(n)≤(1+kh(n)-11-Q)‖xn-1-p‖2+ξn1-Q,∀n≥n0.
In view of the restriction (b), we obtain from Lemma 1.2 that limn→∞∥xn-p∥ exists. Hence, the sequence {xn} is bounded. Reconsidering (2.4), we see from the restriction (a) that
(2.7)(1-b)2a‖Ti(n)h(n)xn-xn-1‖2≤αn(‖xn-1-p‖2-‖xn-p‖2)+(kh(n)-1)‖xn-p‖2+ξn.
This implies that
(2.8)lim⁡n→∞‖Ti(n)h(n)xn-xn-1‖=0.
Notice that
(2.9)‖xn-xn-1‖≤‖Ti(n)h(n)xn-xn-1‖.
It follows from (2.8) that
(2.10)lim⁡n→∞‖xn-xn-1‖=0.
Observe that
(2.11)‖xn-1-Ti(n)h(n)xn-1‖≤‖xn-1-Ti(n)h(n)xn‖+‖Ti(n)h(n)xn-Ti(n)h(n)xn-1‖≤‖xn-1-Ti(n)h(n)xn‖+L‖xn-xn-1‖.
In view of (2.8), and (2.10), we obtain that
(2.12)lim⁡n→∞‖xn-1-Ti(n)h(n)xn-1‖=0.
Since for any positive integer n>N, it can be written as n=(h(n)-1)N+i(n), where i(n)∈{1,2,…,N}. Observe that
(2.13)‖xn-1-Tnxn-1‖≤‖xn-1-Ti(n)h(n)xn-1‖+‖Ti(n)h(n)xn-1-Tnxn-1‖≤‖xn-1-Ti(n)h(n)xn-1‖+L‖Ti(n)h(n)-1xn-1-xn-1‖≤‖xn-1-Ti(n)h(n)xn-1‖+L(‖Ti(n)h(n)-1xn-1-Ti(n-N)h(n)-1xn-N‖+‖Ti(n-N)h(n)-1xn-N-x(n-N)-1‖+‖x(n-N)-1-xn-1‖Ti(n-N)h(n)-1).
Since for each n>N, n=(n-N)(modN), 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,
(2.14)h(n-N)=h(n)-1,i(n-N)=i(n).
Notice that
(2.15)‖Ti(n)h(n)-1xn-1-Ti(n-N)h(n)-1xn-N‖=‖Ti(n)h(n)-1xn-1-Ti(n)h(n)-1xn-N‖≤L‖xn-1-xn-N‖,‖Ti(n-N)h(n)-1xn-N-x(n-N)-1‖=‖Ti(n-N)h(n-N)xn-N-x(n-N)-1‖.
Substituting (2.15) into (2.13), we arrive at
(2.16)‖xn-1-Tnxn-1‖≤‖xn-1-Ti(n)h(n)xn-1‖+L(L‖xn-xn-N‖+‖Ti(n-N)h(n-N)xn-N-x(n-N)-1‖+‖x(n-N)-1-xn-1‖).
In view of (2.8), (2.10), and (2.12), we obtain from (2.16) that
(2.17)lim⁡n→∞‖xn-1-Tnxn-1‖=0.
Notice that
(2.18)‖xn-Tnxn‖≤‖xn-xn-1‖+‖xn-1-Tnxn-1‖+‖Tnxn-1-Tnxn‖≤(1+L)‖xn-xn-1‖+‖xn-1-Tnxn-1‖.
From (2.10) and (2.17), we arrive at
(2.19)lim⁡n→∞‖xn-Tnxn‖=0.
Notice that
(2.20)‖xn-Tn+jxn‖≤‖xn-xn+j‖+‖xn+j-Tn+jxn+j‖+‖Tn+jxn+j-Tn+jxn‖≤(1+L)‖xn-xn+j‖+‖xn+j-Tn+jxn+j‖,∀j∈{1,2,…,N}.
It follows from (2.10) and (2.19) that
(2.21)lim⁡n→∞‖xn-Tn+jxn‖=0,∀j∈{1,2,…,N}.
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that
(2.22)lim⁡n→∞‖xn-Trxn‖=0,∀r∈{1,2,…,N}.
Since the sequence {xn} is bounded, we see that there exists a subsequence {xni}⊂{xn} such that {xni} converges weakly to a point x̅∈C. Choose α∈(0,1/(1+L)) and define yα,m,r=(1-α)x̅+αTrmx̅ for arbitrary but fixed m≥1. Notice that
(2.23)‖xni-Trmxni‖≤‖xni-Trxni‖+‖Trxni-Tr2xni‖+⋯+‖Trm-1xni-Trmxni‖≤[1+(m-1)L]‖xni-Trxni‖,∀r∈{1,2,…,N}.
It follows from (2.22) that
(2.24)lim⁡i→∞‖xni-Trmxni‖=0,∀r∈{1,2,…,N}.
Note that
(2.25)〈x̅-yα,m,r,yα,m,r-Trmyα,m,r〉=〈x̅-xni,yα,m,r-Trmyα,m〉+〈xni-yα,m,r,yα,m,r-Trmyα,m,r〉=〈x̅-xni,yα,m,r-Trmyα,m,r〉+〈xni-yα,m,r,Trmxni-Trmyα,m,r〉-〈xni-yα,m,r,xni-yα,m,r〉+〈xni-yα,m,r,xni-Trmxn〉≤〈x̅-xni,yα,m,r-Trmyα,m,r〉+1+km2‖xni-yα,m,r‖2+ξm2-‖xni-yα,m,r‖2+‖xni-yα,m,r‖‖xni-Trmxn‖=〈x̅-xni,yα,m,r-Trmyα,m,r〉+km-12‖xni-yα,m,r‖2+ξm2+‖xni-yα,m,r‖‖xni-Trmxni‖,∀r∈{1,2,…,N}.
Since xni⇀x̅ and (2.24), we arrive at
(2.26)〈x̅-yα,m,yα,m-Trmyα,m〉≤km-12‖xni-yα,m,r‖2+ξm2,∀r∈{1,2,…,N}.
On the other hand, we have(2.27)〈x̅-yα,m,r,(x̅-Trmx̅)-(yα,m,r-Trmyα,m,r)〉≤(1+L)‖x̅-yα,m,r‖2=(1+L)α2‖x̅-Trmx̅‖2,∀r∈{1,2,…,N}.
Notice that
(2.28)‖x̅-Trmx̅‖2=〈x̅-Trmx̅,x̅-Trmx̅〉=1α〈x̅-yα,m,r,x̅-Trmx̅〉=1α〈x̅-yα,m,r,(x̅-Trmx̅)-(yα,m,r-Trmyα,m,r)〉+1α〈x̅-yα,m,r,yα,m,r-Trmyα,m,r〉,∀r∈{1,2,…,N}.
Substituting (2.26) and (2.27) into (2.28), we arrive at
(2.29)α[1-(1+L)α]‖x̅-Trmx̅‖2≤km-12‖xni-yα,m,r‖2+ξm2,∀r∈{1,2,…,N},m≥1.
Letting m→∞ in (2.29), we see that Trmx̅→x̅ for each 1≤r≤N. Since Tr is uniformly Lr-Lipschitz, we can obtain that x̅=Trx̅ for each 1≤r≤N. This means that x̅∈F.
Next we show that{xn} converges weakly to x̅. Supposing the contrary, we see that there exists some subsequence {xnj} of {xn} such that {xnj} converges weakly to x*∈C, where x*≠x̅. Similarly, we can show x*∈ℱ. Notice that we have proved that limn→∞∥xn-p∥ exists for each p∈ℱ. Assume that limn→∞∥xn-x̅∥=d where d is a nonnegative number. By virtue of the Opial property of H, we see that
(2.30)d=liminf⁡ni→∞‖xni-x̅‖<liminf⁡ni→∞‖xni-x*‖=liminf⁡nj→∞‖xnj-x*‖<liminf⁡nj→∞‖xnj-x̅‖=d.
This is a contradiction. Hence x̅=x*. This completes the proof.For the class of asymptotically pseudocontractive mappings, we have, from Theorem2.1, the following results immediately.Corollary 2.2.
LetC 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 {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions 0<1-1/L<a≤αn≤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 Theorem2.1 the following results immediately.Corollary 2.3.
LetC 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 ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-∥x-y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(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 Theorem2.1 the following results immediately.Corollary 2.4.
LetC be a nonempty closed convex subset of a Hilbert space H. Let Ti:C→C be an asymptotically nonexpansive mapping with the sequence {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restriction 0<1-1/L<a≤αn≤b<1, where L=max{supn≥1{kn,i}: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.
LetC 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}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Let ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-k(n,i)∥x-y∥2-∥(I-Tin)x-(I-Tin)y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(n,i):1≤i≤N}. If one of {T1,T2,…,TN} is semicompact, then the sequence {xn} converges strongly to some point in ℱ.Proof.
Without loss of generality, we may assume thatT1 is semicompact. From (2.22), we see that there exists a subsequence {xni} of {xn} that converges strongly to x∈C. For each r∈{1,2,…,N}, we get that
(2.31)‖x-Trx‖≤‖x-xni‖+‖xni-Trxni‖+‖Trxni-Trx‖.
Since Tr is Lipschitz continuous, we obtain from (2.22) that x∈⋂r=1NF(Tr)=ℱ. In view of Theorem 2.1, we obtain that limn→∞∥xn-x∥ exists. Therefore, we can obtain the desired conclusion immediately.For the class of asymptotically pseudocontractive mappings, we have from Theorem2.5 the following results immediately.Corollary 2.6.
LetC 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 {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: 0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1. If one of {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 Theorem2.5 the following results immediately.Corollary 2.7.
LetC 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 ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-∥x-y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(n,i):1≤i≤N}. If one of {T1,T2,…,TN} is semicompact, then the sequence {xn} converges strongly to some point in ℱ.For the class of asymptotically nonexpansive mappings, we can conclude from Theorem2.5 the following results immediately.Corollary 2.8.
LetC be a nonempty closed convex subset of a Hilbert space H. Let Ti:C→C be an asymptotically nonexpansive mapping with the sequence {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restriction: 0<1-1/L<a≤αn≤b<1, where L=max{supn≥1{kn,i}:1≤i≤N}, for all n≥1. If one of  {T1,T2,…,TN} is semicompact, then the sequence {xn} converges strongly to some point in ℱ.In 2005, Chidume and Shahzad [13] introduced the following conception. Recall that a family {Ti}i=1N:C→C with ℱ=⋂i=1NF(Ti)≠∅ is said to satisfy Condition (B) on C if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(m)>0 for all m∈(0,∞) such that for all x∈C(2.32)max⁡1≤i≤N{‖x-Tix‖}≥f(d(x,F)).Next, we give strong convergence theorems with the help of Condition(B).Theorem 2.9.
LetC 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}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Let ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-k(n,i)∥x-y∥2-∥(I-Tin)x-(I-Tin)y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(n,i):1≤i≤N}. If {T1,T2,…,TN} satisfies Condition (B), then the sequence {xn} converges strongly to some point in ℱ.Proof.
In view of Condition(B), we obtain from (2.22) that f(d(xn,ℱ))→0, which implies d(xn,ℱ)→0. Next, we show that the sequence {xn} is Cauchy. In view of (2.6), for any positive integers m,n, where m>n>n0, we obtain that
(2.33)‖xm-p‖≤B‖xn-p‖+B∑i=n+1∞ξi1-Q+ξm1-Q,
where B=exp{∑n=1∞((kh(n)-1)/(1-Q))}. It follows that
(2.34)‖xn-xm‖≤‖xn-p‖+‖xm-p‖≤(1+B)‖xn-p‖+B∑i=n+1∞ξi1-Q+ξm1-Q.
It follows that {xn} is a Cauchy sequence in C,so {xn} converges strongly to some q̅∈C. Since Tr is Lipschitz for each r∈{1,2,…,N}, we see that ℱ is closed. This in turn implies that q̅∈ℱ. This completes the proof.For the class of asymptotically pseudocontractive mappings, we have from Theorem2.9 the following results immediately.Corollary 2.10.
LetC 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 {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: 0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1. If {T1,T2,…,TN} satisfies Condition (B), then the sequence {xn} converges strongly to some point in ℱ.For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem2.9 the following results immediately.Corollary 2.11.
LetC 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 ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-∥x-y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(n,i):1≤i≤N}. If {T1,T2,…,TN} satisfies Condition (B), then the sequence {xn} converges strongly to some point in ℱ.For the class of asymptotically nonexpansive mappings, we can conclude from Theorem2.9 the following results immediately.Corollary 2.12.
LetC be a nonempty closed convex subset of a Hilbert space H. Let Ti:C→C be an asymptotically nonexpansive mapping with the sequence {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restriction: 0<1-1/L<a≤αn≤b<1, where L=max{supn≥1{kn,i}:1≤i≤N}, for all n≥1. If {T1,T2,…,TN} satisfies Condition (B), then the sequence {xn} converges strongly to some point in ℱ.Finally, we give the following strong convergence criteria.Theorem 2.13.
LetC 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}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Let ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-k(n,i)∥x-y∥2-∥(I-Tin)x-(I-Tin)y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(n,i):1≤i≤N}. Then, the sequence {xn} converges strongly to some point in ℱ  if and only if liminfn→∞d(xn,ℱ)=0.Proof.
The necessity is obvious. We only show the sufficiency. Assume that(2.35)liminf⁡n→∞d(xn,F)=0.
In view of Lemma 1.2, we can obtain from (2.6) that limn→∞d(xn,ℱ)=0. The desired results can be obtain from Theorem 2.9 immediately.For the class of asymptotically pseudocontractive mappings, we have from Theorem2.13 the following results immediately.Corollary 2.14.
LetC 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 {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions 0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1. Then, the sequence {xn} converges strongly to some point in ℱ if and only if  liminfn→∞d(xn,ℱ)=0.For the class of asymptotically nonexpansive mappings in the intermediate sense, we can obtain from Theorem2.13 the following results immediately.Corollary 2.15.
LetC 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 ξ(n,i)=max{0,supx,y∈C(∥Tinx-Tiny∥2-∥x-y∥2)} for each 1≤i≤N. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restrictions: (a)
0<1-1/L<a≤αn≤b<1, where L=max{Li:1≤i≤N}, for all n≥1;(b)
∑n=1∞ξn<∞, where ξn=max{ξ(n,i):1≤i≤N}. Then, the sequence {xn} converges strongly to some point in ℱ if and only if  liminfn→∞d(xn,ℱ)=0.For the class of asymptotically nonexpansive mappings, we can conclude from Theorem2.13 the following results immediately.Corollary 2.16.
LetC be a nonempty closed convex subset of a Hilbert space H. Let Ti:C→C be an asymptotically nonexpansive mapping with the sequence {kn,i}⊂[1,∞) such that ∑n=1∞(kn,i-1)<∞ for each 1≤i≤N, where N≥1 is some positive integer. Assume that the common fixed point set ℱ=⋂i=1NF(Ti) is nonempty. Let {xn}n=0∞ be a sequence generated in (1.24). Assume that the control sequence {αn} in [0,1] satisfies the following restriction: 0<1-1/L<a≤αn≤b<1, where L=max{supn≥1{kn,i}:1≤i≤N}, for all n≥1. Then, the sequence {xn} converges strongly to some point in ℱ if and only if  liminfn→∞d(xn,ℱ)=0.



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*Source: 468716-2011-04-17.xml*