# Strong and Weak Convergence Theorems for an Infinite Family of Lipschitzian Pseudocontraction Mappings in Banach Spaces

Authors: Shih-sen Chang; Xiong Rui Wang; H. W. Joseph Lee; Chi Kin Chan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The purpose of this paper is to study the strong and weak convergence theorems of
the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings
in Banach spaces.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper, we assume thatE is a real Banach space, E* is the dual space of E, C is a nonempty closed convex subset of E, ℛ+ is the set of nonnegative real numbers, and J:E→2E* is the normalized duality mapping defined by (1.1)J(x)={f∈E*:〈x,f〉=‖x‖⋅‖f‖,‖x‖=‖f‖},x∈E.
Let T:C→C be a mapping. We use F(T) to denote the set of fixed points of T. We also use “→” to stand for strong convergence and “⇀” for weak convergence. For a given sequence {xn}⊂C, let Wω(xn) denote the weakω-limit set, that is, (1.2)Wω(xn)={z∈C:thereexistsasubsequence{xni}⊂{xn}suchthatxni⇀z}.Definition 1.1.
(1)  A mappingT:C→C is said to be pseudocontraction [1], if for any x,y∈C, there exists j(x-y)∈J(x-y) such that
(1.3)〈Tx-Ty,j(x-y)〉≤‖x-y‖2.
It is well known that [1] the condition (1.3) is equivalent to the following: (1.4)‖x-y‖≤‖x-y+s[(I-Tx)-(I-Ty)]‖,
for all s>0 and all x,y∈C.
(2)T:C→C is said to be strongly pseudocontractive, if there exists k∈(0,1) such that(1.5)〈Tx-Ty,j(x-y)〉≤k‖x-y‖2,
for each x,y∈C and for some j(x-y)∈J(x-y).
(3)T:C→C is said to be strictly pseudocontractive in the terminology of Browder and Petryshyn [1], if there exists λ>0 such that(1.6)〈Tx-Ty,j(x-y)〉≤‖x-y‖2-λ‖(I-T)x-(I-T)y‖2,
for every x,y∈C and for some j(x-y)∈J(x-y).
In this case, we sayT is a λ-strictly pseudocontractive mapping.
(4)T:C→C is said to be L-Lipschitzian, if there existsL>0such that(1.7)‖Tx-Ty‖≤L‖x-y‖,∀x,y∈C.Remark 1.2.
It is easy to see that ifT:C→C is a λ-strictly pseudocontractive mapping, then it is a (1+λ)/λ-Lipschitzian mapping.
In fact, it follows from (1.6) that for any x,y∈C,(1.8)λ‖(I-T)x-(I-T)y‖2≤〈(I-T)x-(I-T)y,j(x-y)〉≤‖(I-T)x-(I-T)y‖‖x-y‖.
Simplifying it, we have
(1.9)‖(I-T)x-(I-T)y‖≤1λ‖x-y‖,
that is,
(1.10)‖Tx-Ty‖≤1+λλ‖x-y‖,∀x,y∈C.Lemma 1.3 (see [2, Theorem 13.1] or [3]).
LetE be a real Banach space, C be a nonempty closed convex subset of E, and T:C→C be a continuous strongly pseudocontractive mapping. Then T has a unique fixed point in C.Remark 1.4.
LetE be a real Banach space, C be a nonempty closed convex subset of E and T:C→C be a Lipschitzian pseudocontraction mapping. For every given u∈C and s∈(0,1), define a mapping Us:C→C by
(1.11)Usx=su+(1-s)Tx,x∈C.
It is easy to see that Us is a continuous strongly pseudocontraction mapping. By using Lemma 1.3, there exists a unique fixed point xs∈C of Us such that
(1.12)xs=su+(1-s)Txs.
The concept of pseudocontractive mappings is closely related to accretive operators. It is known thatT is pseudocontractive if and only if I-T is accretive, where I is the identity mapping. The importance of accretive mappings is from their connection with theory of solutions for nonlinear evolution equations in Banach spaces. Many kinds of equations, for example, Heat, wave, or Schrödinger equations can be modeled in terms of an initial value problem:(1.13)dudt=Tu-u,u(0)=u0,
where T is a pseudocontractive mapping in an appropriate Banach space.
In order to approximate a fixed point of Lipschitzian pseudocontractive mapping, in 1974, Ishikawa introduced a new iteration (it is calledIshikawa iteration). Since then, a question of whether or not the Ishikawa iteration can be replaced by the simpler Mann iteration has remained open. Recently Chidume and Mutangadura [4] solved this problem by constructing an example of a Lipschitzian pseudocontractive mapping with a unique fixed point for which every Mann-type iteration fails to converge.
Inspired by the implicit iteration introduced by Xu and Ori [5] for a finite family of nonexpansive mappings in a Hilbert space, Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9] proposed and studied convergence theorems for an implicit iteration process for a finite or infinite family of continuous pseudocontractive mappings.
The purpose of this paper is to study the strong and weak convergence problems of the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banach spaces. The results presented in this paper extend and improve some recent results of Xu and Ori [5], Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9].
For this purpose, we first recall some concepts and conclusions.
A Banach spaceE is said to be uniformly convex, if for each ɛ>0, there exists a δ>0 such that for any x,y∈E with ∥x∥, ∥y∥≤1 and ∥x-y∥≥ɛ, ∥x+y∥≤2(1-δ) holds.  The modulus of convexity of E is defined by(1.14)δE(ɛ)=inf⁡{1-‖x+y2‖:‖x‖,‖y‖≤1,‖x-y‖≥ɛ},∀ɛ∈[0,2].
Concerning the modulus of convexity ofE, Goebel and Kirk [10] proved the following result.Lemma 1.5 (see [10, Lemma 10.1]).
LetE be  a uniformly convex Banach space with a modulus of convexity δE. Then δE:[0,2]→[0,1] is continuous, increasing, δE(0)=0, δE(t)>0 for t∈(0,2] and
(1.15)‖cu+(1-c)v‖≤1-2min⁡{c,1-c}δE(‖u-v‖),
for all c∈[0,1], and u,v∈E with ∥u∥,∥v∥≤1.
A Banach spaceE is said to satisfy the Opial condition, if for any sequence {xn}⊂E with xn⇀x, then the following inequality holds:(1.16)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖,
for any y∈E with y≠x.Lemma 1.6 (Zhou [8]).
LetE be a real reflexive Banach space with Opial condition. Let C be a nonempty closed convex subset of E and T:C→C be a continuous pseudocontractive mapping. Then I-T is demiclosed at zero, that is, for any sequence {xn}⊂E, if xn⇀y and ∥(I-T)xn∥→0, then (I-T)y=0.Lemma 1.7 (Chang [11]).
LetJ:E→2E* be the normalized duality mapping, then for any x,y∈E,
(1.17)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).Definition 1.8 (see [12]).
Let{Tn}:C→E be a family of mappings with ⋂n=1∞F(Tn)≠∅. We say {Tn} satisfies the AKTT-condition, if for each bounded subset B of C the following holds:
(1.18)∑n=1∞sup⁡z∈B‖Tn+1z-Tnz‖<∞.Lemma 1.9 (see [12]).
Suppose that the family of mappings{Tn}:C→C satisfies the AKTT-condition. Then for each y∈C, {Tny} converges strongly to a point in C. Moreover, let T:C→C be the mapping defined by
(1.19)Ty=lim⁡n→∞Tny,∀y∈C.
Then, for each bounded subset B⊂C, limn→∞supz∈B∥Tz-Tnz∥=0.


## 2. Main Results

Theorem 2.1.
LetE be a uniformly convex Banach space with a modulus of convexity δE, and C be a nonempty closed convex subset of E. Let {Tn}:C→C be a family of Ln-Lipschitzian and pseudocontractive mappings with L:=supn≥1Ln<∞ and ℱ:=⋂n≥1F(Tn)≠∅. Let {xn} be the sequence defined by
(2.1)x1∈C,xn=αnxn-1+(1-αn)Tnxn,n≥1,
where {αn} is a sequence in [0,1]. If the following conditions are satisfied: (i)
limsupn→∞αn<1;(ii)
there exists a compact subsetK⊂E such that ⋃n=1∞Tn(C)⊂K;(iii)
{Tn} satisfies the AKTT-condition, and F(T)⊂ℱ, where T:C→C is the mapping defined by (1.19).Then xn converges strongly to some point p∈ℱProof.
First, we note that, by Remark1.4, the method is well defined. So, we can divide the proof in three steps. (I)
For eachp∈ℱ the limit limn→∞∥xn-p∥ exists.In fact, since{Tn} is pseudocontractive, for each p∈ℱ, we have
(2.2)‖xn-p‖2=〈xn-p,j(xn-p)〉=αn〈xn-1-p,j(xn-p)〉+(1-αn)〈Tnxn-p,j(xn-p)〉≤αn‖xn-1-p‖‖xn-p‖+(1-αn)‖xn-p‖2,∀n≥1.
Simplifying, we have that
(2.3)‖xn-p‖≤‖xn-1-p‖,∀n≥1.
Consequently, the limit limn→∞∥xn-p∥ exists, and so the sequence {xn} is bounded.(II)
Now, we prove thatlimn→∞∥xn-Tnxn∥=0.In fact, by virtue of (2.1) and (1.4), we have
(2.4)‖xn-p‖≤‖xn-p+1-αn2αn(xn-Tnxn)‖=‖xn-p+1-αn2(xn-1-Tnxn)‖=‖αnxn-1+(1-αn)Tnxn-p+1-αn2(xn-1-Tnxn)‖=‖xn-1+xn2-p‖=‖xn-1-p‖⋅‖xn-1-p2‖xn-1-p‖+xn-p2‖xn-1-p‖‖.
Letting u=(xn-1-p)/∥xn-1-p∥  and v=(xn-p)/∥xn-1-p∥, from (2.3), we know that ∥u∥=1, ∥v∥≤1. It follows from (2.4) and Lemma 1.5 that
(2.5)‖xn-p‖≤‖xn-1-p‖{1-δE(‖xn-1-xn‖‖xn-1-p‖)}.
Simplifying, we have that
(2.6)‖xn-1-p‖{δE(‖xn-1-xn‖‖xn-1-p‖)}≤‖xn-1-p‖-‖xn-p‖.
This implies that
(2.7)∑n=1∞‖xn-1-p‖{δE(‖xn-1-xn‖‖xn-1-p‖)}≤‖x0-p‖.
Letting limn→∞∥xn-p∥=r, if r=0, the conclusion of Theorem 2.1 is proved. If r>0, it follows from the property of modulus of convexity δE that ∥xn-1-xn∥→0(n→∞). Therefore, from (2.1) and the condition (i), we have that
(2.8)‖xn-1-Tnxn‖=11-αn‖xn-xn-1‖⟶0(asn⟶∞).
In view of (2.1) and (2.8), we have
(2.9)‖xn-Tnxn‖=αn‖xn-1-Tnxn‖⟶0(asn⟶∞).(III)
Now, we prove that{xn} converges strongly to some point in ℱ.In fact, it follows from (2.9) and condition (ii) that there exists a subsequence {xni}⊂{xn} such that ∥xni-Tnixni∥→0 (as ni→∞), Tnixni→p and xni→p (some point in C). Furthermore, by Lemma 1.9, we have Tnip→Tp. consequently, we have
(2.10)‖p-Tp‖≤‖p-xni‖+‖xni-Tnip‖+‖Tnip-Tp‖≤‖p-xni‖+‖xni-Tnixni‖+‖Tnixni-Tnip‖+‖Tnip-Tp‖≤(1+L)‖p-xni‖+‖xni-Tnixni‖+‖Tnip-Tp‖⟶0.
This implies that p=Tp, that is, p∈F(T)⊂ℱ. Since xni→p and the limit limn→∞∥xn-p∥ exists, we have xn→p.
This  completes the proof of Theorem2.1.Theorem 2.2.
LetE be a uniformly convex Banach space satisfying the Opial condition. Let C be a nonempty closed convex subset of E and {Tn}:C→C be a family of Ln-Lipschitzian pseudocontractive mappings with L:=supn≥1Ln<∞ and ℱ:=⋂n≥1F(Tn)≠∅. Let {xn} be the sequence defined by (2.1) and {αn} be a sequence in (0, 1). If the following conditions are satisfied: (i)
limsupn→∞αn<1,(ii)
for any bounded subsetB of C(2.11)lim⁡n→∞sup⁡z∈B‖TmTnz-Tnz‖=0,foreachm≥1.Then the sequence {xn} converges weakly to some point u∈ℱ.Proof.
By the same method as given in the proof of Theorem2.1, we can prove that the sequence {xn} is bounded and
(2.12)lim⁡n→∞‖xn-p‖exists,foreachp∈F;lim⁡n→∞‖xn-Tnxn‖=0.
Now, we prove that(2.13)lim⁡n→∞‖Tmxn-xn‖=0,foreachm≥1.
Indeed, for eachm≥1, we have(2.14)‖Tmxn-xn‖≤‖Tmxn-TmTnxn‖+‖TmTnxn-Tnxn‖+‖Tnxn-xn‖≤(1+L)‖Tnxn-xn‖+‖TmTnxn-Tnxn‖≤(1+L)‖Tnxn-xn‖+sup⁡z∈{xn}‖TmTnz-Tnz‖.
By (2.12) and condition (ii), we have
(2.15)lim⁡n→∞‖Tmxn-xn‖=0,foreachm≥1.
The conclusion of (2.13) is proved.
Finally, we prove that{xn} converges weakly to some point u∈ℱ.
In fact, sinceE is uniformly convex, and so it is reflexive. Again since {xn}⊂C is bounded, there exists a subsequence {xni}⊂{xn} such that xni⇀u. Hence from (2.13), for any m>1, we have(2.16)‖Tmxni-xni‖⟶0(asni⟶∞).
By virtue of Lemma 1.6, u∈F(Tm), for all m≥1. This implies that
(2.17)u∈F:=⋂n≥F(Tn)∩Wω(xn).
Next, we prove thatWω(xn) is a singleton. Let us suppose, to the contrary, that if there exists a subsequence {xnj}⊂{xn} such that xnj⇀q∈Wω(xn) and q≠u. By the same method as given above we can also prove that q∈ℱ:=⋂n≥1F(Tn)∩Wω(xn). Taking p=u and p=q in (2.12). We know that the following limits(2.18)lim⁡n→∞‖xn-u‖,lim⁡n→∞‖xn-q‖
exist. Since E satisfies the Opial condition, we have
(2.19)lim⁡n→∞‖xn-u‖=limsup⁡ni→∞‖xni-u‖<limsup⁡ni→∞‖xni-q‖=lim⁡n→∞‖xn-q‖=limsup⁡nj→∞‖xnj-q‖<limsup⁡nj→∞‖xnj-u‖=lim⁡n→∞‖xn-u‖.
This is a contradiction, which shows that q=u. Hence,
(2.20)Wω(xn)={u}⊂F:=⋂n≥1F(Tn).
This implies that xn⇀u.
This completes the proof of Theorem2.2.In the next lemma, we propose a sequence of mappings that satisfy condition (iii) in Theorem2.1. Moreover, we apply this lemma to obtain a corollary of our main Theorem 2.1.LetE be a Banach space and C be a nonempty closed convex subset of E. From Definition 1.1(3), we know that if T:C→C is a λ-strictly pseudocontractive mapping, then it is a ((1+λ)/λ)-Lipschitzian pseudocontractive mapping.On the other hand, by the same proof as given in [12] we can prove the following result.Lemma 2.3 (see [12] or [9]).
LetE be a smooth Banach space, C be a closed convex subset of E. Let {Sn}:C→C be a family of λn-strictly pseudocontractive mappings with ℱ:=⋂n=1∞F(Sn)≠∅ and λ:=infn≥1λn>0. For each n≥1 define a mapping Tn:C→C by:
(2.21)Tnx=∑k=1nβnkSkx,x∈C,n≥1,
where {βnk} is sequence of nonnegative real numbers satisfying the following conditions: (i)
∑k=1nβnk=1, for all  n≥1;(ii)
βk:=limn→∞βnk>0, for all k≥1;(iii)
∑n=1∞∑k=1n|βn+1k-βnk|<∞.Then,  (1)
eachTn, n≥1 is a λ-strictly pseudocontractive mapping;(2)
{Tn} satisfies the AKTT-condition;(3)
ifT:C→C is the mapping defined by
(2.22)Tx=∑k=1∞βkSkx,x∈C.Then Tx=limn→∞Tnx and F(T)=⋂k=1∞F(Tn)=ℱ:=⋂n=1∞F(Sn).The following result can be obtained from Theorem2.1 and Lemma 2.3 immediately.Theorem 2.4.
LetE be a uniformly convex Banach space, C be a nonempty closed convex subset of E. Let {Sn}:C→C be a family of λn-strictly pseudocontractive mappings with ℱ:=⋂n=1∞F(Sn)≠∅ and λ:=infn≥1λn>0. For each n≥1 define a mapping Tn:C→C by
(2.23)Tnx=∑k=1nβnkSkx,x∈C,n≥1,
where {βnk} is a sequence of nonnegative real numbers satisfying the following conditions: (i)
∑k=1nβnk=1, for all n≥1;(ii)
βk:=limn→∞βnk>0, for all k≥1;(iii)
∑n=1∞∑k=1n|βn+1k-βnk|<∞.
Let{xn} be the sequence defined by(2.24)x1∈C,xn=αnxn-1+(1-αn)Tnxn,n≥1,
where {αn} is a sequence in [0,1]. If the following conditions are satisfied: (i)
limsupn→∞αn<1;(ii)
there exists a compact subsetK⊂E such that ⋃n=1∞Sn(C)⊂K. Then, {xn} converges strongly to some point p∈ℱ.Proof.
Since{Sn}:C→C is a family of λn-strictly pseudocontractive mappings with λ:=infn≥1λn>0. Therefore, {Sn} is a family of λ-strictly pseudocontractive mappings. By Remark 1.2, {Sn} is a family of (1+λ)/λ-Lipschitzian and strictly pseudocontractive mappings. Hence, by Lemma 2.3, {Tn} defined by (2.21) is a family of (1+λ)/λ-Lipschitzian, strictly pseudocontractive mappings with ⋂n=1∞F(Tn)=ℱ:=⋂n=1∞F(Sn)≠∅ and it has also the following properties: (1)
{Tn} satisfies the AKTT-condition;(2)
ifT:C→C is the mapping defined by (2.22), then Tx=limn→∞Tnx,x∈C and F(T)=ℱ:=⋂k=1∞F(Sn)=⋂n=1∞F(Tn). Hence, by Definition 1.1, {Tn} is also a family of (1+λ)/λ-Lipschitzian and pseudocontractive mappings having the properties (1) and (2) and ℱ:=⋂n=1∞F(Tn)≠∅. Therefore, {Tn} satisfies all the conditions in Theorem 2.1. By Theorem 2.1, the sequence {xn} converges strongly to some point p∈ℱ:⋂k=1∞F(Sn)=⋂n=1∞F(Tn).
This completes the proof of Theorem2.4.



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