# Implicit Mann Type Iteration Method Involving Strictly Hemicontractive Mappings in Banach Spaces

Authors: Arif Rafiq; Shin Min Kang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We proved that the modified implicit Mann iteration process can be applied to approximate the fixed point of strictly hemicontractive mappings in smooth Banach spaces.


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## Body

## 1.  Introduction

LetK be a nonempty subset of an arbitrary Banach space X and  let X* be its dual space. The symbols D(T) and F(T) stand for the domain and the set of fixed points of T (for a single-valued mapping T:X→X,x∈X is called a fixed point of T iff Tx=x). We denote by J the normalized duality mapping from X to 2X* defined by
(1.1)J(x)={f*∈X*:〈x,f*〉=‖x‖2=‖f*‖2},x∈X,
where 〈·,·〉 denotes the duality pairing. In a smooth Banach space, J is singlevalued (we denoted by j).Remark 1.1.
(1)X is called uniformly smooth if X* is uniformly convex.
(2) In a uniformly smooth Banach space,J is uniformly continuous on bounded subsets of X.LetT:D(T)⊂X→X be a mapping.Definition 1.2.
The mappingT is called Lipshitz if there exists a constant L>0 such that(1.2)‖Tx-Ty‖≤L‖x-y‖,
for allx,y∈D(T). If L=1, then T is called nonexpansive and if 0≤L<1, then T is called contractive.Definition 1.3 (see [1, 2]).
(1) The mappingT is said to be pseudocontractive if(1.3)‖x-y‖≤‖x-y+r((I-T)x-(I-T)y)‖,
for all x,y∈D(T) and r>0.
(2) The mappingT is said to be strongly pseudocontractive if there exists a constant t>1 such that(1.4)‖x-y‖≤‖(1+r)(x-y)-rt(Tx-Ty)‖,
for all x,y∈D(T) and r>0.
(3) The mappingT is said to be local strongly pseudocontractive if for each x∈D(T) there exists a constant t>1 such that(1.5)‖x-y‖≤‖(1+r)(x-y)-rt(Tx-Ty)‖,
for all y∈D(T) and r>0.
(4) The mappingT is said to be strictly hemicontractive if F(T)≠∅ and if there exists a constant t>1 such that(1.6)‖x-q‖≤‖(1+r)(x-q)-rt(Tx-q)‖,
for all x∈D(T),  q∈F(T) and r>0.Clearly, each strongly pseudocontractive mapping is local strongly pseudocontractive.Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudocontractive mapping from a bounded closed convex subset of Lp (or lp) into itself. Schu [3] generalized the result in [1] to both uniformly continuous strongly pseudocontractive mappings and real smooth Banach spaces. Park [4] extended the result in [1] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [5] proved that the Mann and Ishikawa iteration methods may exhibit different behaviors for different classes of nonlinear mappings. Afterwards, several generalizations have been made in various directions (see, e.g., [6–13]).In 2001, Xu and Ori [14] introduced the following implicit iteration process for a finite family of nonexpansive mappings {Ti:i∈I} (here I={1,2,…,N}) with {αn} a real sequence in (0,1) and an initial point x0∈K:(1.7)x1=(1-α1)x0+α1T1x1,x2=(1-α2)x1+α2T2x2,⋮xN=(1-αN)xN-1+αNTNxN,xN+1=(1-αN+1)xN+αN+1TN+1xN+1,⋮
which can be written in the following compact form:
(1.8)xn=(1-αn)xn-1+αnTnxn,n≥1,
where Tn=Tn(modN) (here the modN function takes values in I). Xu and Ori [14] proved the weak convergence of this process to a common fixed point of the finite family defined in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mappings and/or the parameters {αn} are sufficient to guarantee the strong convergence of the sequence {xn}.In [11], Osilike proved the following results.Theorem 1.4.
LetX be a real Banach space and let   K be a nonempty closed convex subset of  X. Let  {Ti:i∈I} be  N strictly pseudocontractive mappings from  K to  K with  ℱ=⋂i=1NF(Ti)≠∅. Let  {αn} be a real sequence satisfying the following conditions: (i)
0<αn<1,(ii)
∑n=1∞(1-αn)=∞,(iii)
∑n=1∞(1-αn)2<∞.
From arbitraryx0∈K, define the sequence {xn} by the implicit iteration process (1.8). Then {xn} converges strongly to a common fixed point of the mappings {Ti:i∈I} if and only if  liminfn→∞d(xn,ℱ)=0.Remark 1.5.
One can easily see that forαn=1-1/n1/2,  ∑n=1∞(1-αn)2=∞. Hence the results of Osilike [11] are needed to be improved.LetK be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and let  T:K→K be a continuous strictly hemicontractive mapping. We proved that the implicit Mann type iteration method converges strongly to a unique fixed point of T.The results presented in this paper extend and improve the corresponding results particularly in [1, 3, 4, 7, 8, 10, 11, 13, 15].


## 2.  Preliminaries

We need the following results.Lemma 2.1 (see [4]).
LetX be a smooth Banach space. Suppose that one of the following holds: (a)
J is uniformly continuous on any bounded subsets of X,(b)
〈x-y,j(x)-j(y)〉≤∥x-y∥2 for all x,y in X,(c)
for any bounded subsetD of X, there is a function c:[0,∞)→[0,∞) such that(2.1)Re〈x-y,j(x)-j(y)〉≤c(‖x-y‖),
for allx,y∈D, where c satisfies limt→0+(c(t)/t)=0.
Then for anyϵ>0 and any bounded subset K, there exists δ>0 such that(2.2)‖sx+(1-s)y‖2≤(1-2s)‖y‖2+2sRe〈x,j(y)〉+2sϵ,
for all x,y∈K and s∈[0,δ].Remark 2.2.
(1) IfX is uniformly smooth, then (a) in Lemma 2.1 holds.
(2) IfX is a Hilbert space, then (b) in Lemma 2.1 holds.Lemma 2.3 (see [8]).
LetT:D(T)⊂X→X be a mapping with F(T)≠∅. Then T is strictly hemicontractive if and only if there exists a constant t>1 such that for all x∈D(T) and q∈F(T), there exists j(x-q)∈J(x-q) satisfying(2.3)Re〈x-Tx,j(x-q)〉≥(1-1t)‖x-q‖2.Lemma 2.4 (see [10]).
LetX be an arbitrary normed linear space and let   T:D(T)⊂X→X be a mapping. (a)
IfT is a local strongly pseudocontractive mapping and F(T)≠∅, then F(T) is a singleton and T is strictly hemicontractive.(b)
IfT is strictly hemicontractive, then F(T) is a singleton.Lemma 2.5 (see [10]).
Let{θn},{σn}, and  {ωn} be nonnegative real sequences and let   ϵ′>0 be a constant satisfying
(2.4)σn+1≤(1-θn)σn+ϵ′θn+ωn,n≥1,
where ∑n=1∞θn=∞,  θn≤1 for all n≥1 and ∑n=1∞ωn<∞. Then limsupn→∞σn≤ϵ′.


## 3.  Main Results

We now prove our main results.Lemma 3.1.
LetX be a smooth Banach space. Suppose that one of the following holds:(a)
J is uniformly continuous on any bounded subsets of X,(b)
〈x-y,j(x)-j(y)〉≤∥x-y∥2 for all x,y in X,(c)
for any bounded subsetD of X, there is a function c:[0,∞)→[0,∞) such that(3.1)Re〈x-y,j(x)-j(y)〉≤c(‖x-y‖)
for allx,y∈D, where c satisfies limt→0+c((t)/t)=0.
Then for anyϵ>0 and any bounded subset K, there exists δ>0 such that(3.2)‖αx+βy+γz‖2≤(1-2α)‖x‖2+2αβ1-αRe〈y,j(x)〉+2αγ1-αRe〈z,j(x)〉+2ϵα
for all x,y,z∈K and α,β,γ∈[0,δ];α+β+γ=1.Proof.
Forα,β,γ∈[0,δ];α+β+γ=1, by using (2.2), consider
(3.3)‖αx+βy+γz‖2=‖αx+(1-α)(β1-αy+γ1-αz)‖2≤(1-2α)‖x‖2+2ϵα+2αRe〈β1-αy+γ1-αz,j(x)〉=(1-2α)‖x‖2+2ϵα+2αβ1-αRe〈y,j(x)〉+2αγ1-αRe〈z,j(x)〉.
This completes the proof.Theorem 3.2.
LetXbe a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let  K be a nonempty closed bounded convex subset of  X and let   T:K→K be a continuous strictly hemicontractive mapping. Let  {αn},{βn} and  {γn} be real sequences in  [0,1] satisfying conditions (iv)
αn+βn+γn=1,
foralln≥1,(v)
∑n=1∞αn=∞,(vi)
∑n=1∞βn<∞ and ∑n=1∞γn<∞.
For a sequence{vn} in K, suppose that {xn} is the sequence generated from an arbitrary x0∈K by(3.4)xn=αnxn-1+βnTxn+γnTvn,n≥1,
satisfying ∑n=1∞∥vn-xn∥<∞.
Then the sequence{xn} converges strongly to a unique fixed point q of  T.Proof.
By [2, Corollary 1], T has a unique fixed point q in K. It follows from Lemma 2.4 that F(T) is a singleton. That is, F(T)={q} for some q∈K.
SetM=1+diamK. It is easy to verify that(3.5)M=sup⁡n≥1‖xn-q‖+sup⁡n≥1‖Txn-q‖+sup⁡n≥1‖Tvn-q‖.
Also
(3.6)‖vn-q‖2≤‖vn-xn‖2+‖xn-q‖2+2‖vn-xn‖‖xn-q‖≤‖vn-xn‖2+‖xn-q‖2+2M‖vn-xn‖.
Consider
(3.7)‖xn-q‖2=‖αnxn-1+βnTxn+γnTvn-q‖2=‖αn(xn-1-q)+βn(Txn-q)+γn(Tvn-q)‖2≤αn‖xn-1-q‖2+βn‖Txn-q‖2+γn‖Tvn-q‖2≤‖xn-1-q‖2+M2(βn+γn),
where the first inequality holds by the convexity of ∥·∥2.
Now we putk=1/t, where t satisfies (2.3). Using (3.4) and Lemma 3.1, we infer that
(3.8)‖xn-q‖2=‖αnxn-1+βnTxn+γnTvn-q‖2=‖αn(xn-1-q)+βn(Txn-q)+γn(Tvn-q)‖2≤(1-2αn)‖xn-1-q‖2+2αnβn1-αnRe〈Txn-q,j(xn-1-q)〉+2αnγn1-αnRe〈Tvn-q,j(xn-1-q)〉+2ϵαn=(1-2αn)‖xn-1-q‖2+2αnβn1-αnRe〈Txn-q,j(xn-q)〉+2αnβn1-αnRe〈Txn-q,j(xn-1-q)-j(xn-q)〉+2αnγn1-αnRe〈Tvn-q,j(vn-q)〉+2αnγn1-αnRe〈Tvn-q,j(xn-1-q)-j(vn-q)〉+2ϵαn≤(1-2αn)‖xn-1-q‖2+2αnβn1-αnk‖xn-q‖2+2αnβn1-αn‖Txn-q‖‖j(xn-1-q)-j(xn-q)‖+2αnγn1-αnk‖vn-q‖2+2αnγn1-αn‖Tvn-q‖‖j(xn-1-q)-j(vn-q)‖+2ϵαn≤(1-2αn)‖xn-1-q‖2+2αnβn1-αnk‖xn-q‖2+2Mαnβn1-αnδn+2αnγn1-αnk‖vn-q‖2+2Mαnγn1-αnηn+2ϵαn≤(1-2αn)‖xn-1-q‖2+2αnβn1-αnk‖xn-q‖2+2αnγn1-αnk‖vn-q‖2+2Mαnmax⁡{δn,ηn}+2ϵαn,
where
(3.9)δn=‖j(xn-1-q)-j(xn-q)‖,ηn=‖j(xn-1-q)-j(vn-q)‖.
Also, we have
(3.10)‖xn-1-xn‖=‖xn-1-αnxn-1-βnTxn-γnTvn‖=‖βn(xn-1-Txn)+γn(xn-1-Tvn)‖≤βn‖xn-1-Txn‖+γn‖xn-1-Tvn‖≤2M(βn+γn)<∞
implies
(3.11)‖xn-1-xn‖⟶0,
as n→∞, and consequently
(3.12)‖xn-1-vn‖≤‖xn-1-xn‖+‖xn-vn‖⟶0
as n→∞. Since J is uniformly continuous on any bounded subsets of X, we have
(3.13)δn,ηn⟶0asn⟶∞.
For any  given  ϵ>0 and the bounded subset K, there exists a δ>0 satisfying (2.2). Note that (3.13) and (vi) ensure that there exists an N such that
(3.14)βn,γn<min⁡{δ,ϵ8M2k},δn,ηn≤ϵ4M,n≥N.
Now substituting (3.6) in (3.8) to obtain(3.15)‖xn-q‖2≤(1-2αn)‖xn-1-q‖2+2kαn‖xn-q‖2+2Mαnmax⁡{δn,ηn}+2ϵαn+2αnγn1-αnk(‖vn-xn‖2+2M‖vn-xn‖),
by using (3.7), implies
(3.16)‖xn-q‖2≤(1-2(1-k)αn)‖xn-1-q‖2+2ϵαn+2M2kαn(βn+γn)+2Mαnmax⁡{δn,ηn}+2αnγn1-αnk(‖vn-xn‖2+2M‖vn-xn‖)≤(1-2(1-k)αn)‖xn-1-q‖2+3ϵαn+2αnγn1-αnk(‖vn-xn‖2+2M‖vn-xn‖)
for all n≥N.
Put(3.17)σn=‖xn-1-q‖2,θn=2(1-k)αn,ϵ′=3ϵ2(1-k),ωn=2αnγn1-αnk(‖vn-xn‖2+2M‖vn-xn‖),
and we have from (3.16)
(3.18)σn+1≤(1-θn)σn+ϵ′θn+ωn,n≥1.
For k<1/2, set δ=1/2(1-k)<1. Because αn≤δ, we imply 1-αn≥1-δ and 2(1-k)αn≤1. Now observe that ∑n=1∞θn=∞,θn≤1 for all n≥1 and ∑n=1∞ωn<∞. It follows from Lemma 2.5 that
(3.19)limsup⁡n→∞‖xn-q‖2≤ϵ′.
Letting ϵ′→0+, we obtain that limsupn→∞∥xn-q∥2=0, which implies that xn→q as n→∞. This completes the proof.Corollary 3.3.
LetXbe a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let  K be a nonempty closed bounded convex subset of  X and  let  T:K→K be a Lipschitz strictly hemicontractive mapping. Let  {αn},{βn} and  {γn} be real sequences in  [0,1] satisfying the conditions (iv)–(vi).
From arbitraryx0∈K, define the sequence {xn} by the implicit iteration process (3.4). Then the sequence {xn} converges strongly to a unique fixed point q of T.Corollary 3.4.
LetXbe a smooth Banach space satisfying any one of the Axioms  (a)–(c) of Lemma 3.1. Let  K be a nonempty closed bounded convex subset of  X and  let  T:K→K be a continuous strictly hemicontractive mapping. Suppose that  {αn} be a real sequence in  [0,1] satisfying the conditions  (v) and  limn→∞αn=0.
From arbitraryx0∈K, define the sequence {xn} by the implicit iteration process (1.8). Then the sequence {xn} converges strongly to a unique fixed point q of T.Corollary 3.5.
LetX  be a smooth Banach space satisfying any one of the Axioms  (a)–(c) of Lemma 3.1. Let  K be a nonempty closed bounded convex subset of  X and let   T:K→K be a Lipschitz strictly hemicontractive mapping. Suppose that  {αn} be a real sequence in  [0,1] satisfying the conditions  (v) and  limn→∞αn=0.
From arbitraryx0∈K, define the sequence {xn} by the implicit iteration process (1.8). Then the sequence {xn} converges strongly to a unique fixed point q of T.Remark 3.6.
Similar results can be found for the iteration processes involved error terms; we omit the details.Remark 3.7.
Theorem3.2 and Corollary 3.3 extend and improve  Theorem 1.4 in the following directions.
We do not need the assumptionliminfn→∞d(xn,ℱ)=0 as in Theorem 1.4.


## 4.  Applications for Multistep Implicit Iterations

LetK be a nonempty closed convex subset of a smooth Banach space X and let  T,T1,T2,…,Tp:K→K(p≥2) be a family of p+1 mappings.Algorithm 4.1.
For a givenx0∈K, compute the sequence {xn} by the implicit iteration process of arbitrary fixed order p≥2:(4.1)xn=αnxn-1+βnTxn+γnT1yn1,yni=βnixn-1+(1-βni)Ti+1yni+1,i=1,2,…,p-2,ynp-1=βnp-1xn-1+(1-βnp-1)Tpxn,n≥1,
which is called the multistep implicit iteration process, where {αn}, {βn},  {γn},  and {βni},i=1,2,…,p-1 are real sequences in [0,1] and αn+βn+γn=1, foralln≥1.Forp=3, we obtain the following three-step implicit iteration process.Algorithm 4.2.
For a givenx0∈K, compute the sequence {xn} by the iteration process
(4.2)xn=αnxn-1+βnTxn+γnT1yn1,yn1=βn1xn-1+(1-βn1)T2yn2,yn2=βn2xn-1+(1-βn2)T3xn,n≥1,
where {αn}, {βn},  {γn},  {βn1} and {βn2} are real sequences in [0,1] satisfying some certain conditions.
Forp=2, we obtain the following two-step implicit iteration process.Algorithm 4.3.
For a givenx0∈K, compute the sequence {xn} by the iteration process
(4.3)xn=αnxn-1+βnTxn+γnT1yn1,yn1=βn1xn-1+(1-βn1)T2xn,n≥1,
where {αn},  {βn},  {γn} and {βn1} are real sequences in [0,1] satisfying some certain conditions.
IfT1=T,  T2=I and βn1=0 in (4.3), we obtain the following implicit Mann iteration process.Algorithm 4.4.
For any givenx0∈K, compute the sequence {xn} by the iteration process
(4.4)xn=αnxn-1+(1-αn)Txn,n≥1,
where {αn} is a real sequence in [0,1] satisfying some certain conditions.Theorem 4.5.
LetXbe a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let  K be a nonempty closed bounded convex subset of  X and let   T,T1,T2,…,Tp:K→K(p≥2) be  p+1 mappings. Let  T,T1 be continuous strictly hemicontractive mappings. Let  {αn},  {βn},  {γn} and  {βni},i=1,2,…,p-1 be real sequences in  [0,1] satisfying the conditions  (iv)–(vi) and  ∑n=1∞(1-βn1)<∞. For arbitrary  x0∈K, define the sequence  {xn} by (4.1). Then  {xn} converges strongly to the common fixed point of  ⋂i=1pF(Ti)∩F(T)≠∅.Proof.
By applying Theorem3.2 under assumption that T and T1 are continuous strictly hemicontractive mappings, we obtain Theorem 4.5 which proves strong convergence of the iteration process defined by (4.1). Consider by taking T1=T and vn=yn1,
(4.5)‖vn-xn‖=‖yn1-xn‖=‖βn1xn-1+(1-βn1)T2yn2-xn‖=‖βn1(xn-1-xn)+(1-βn1)(T2yn2-xn)‖≤βn1‖xn-1-xn‖+(1-βn1)‖T2yn2-xn‖≤βn1‖xn-1-xn‖+M′(1-βn1).
From (4.5) and the condition ∑n=1∞(1-βn1)<∞, we obtain
(4.6)∑n=1∞‖vn-xn‖<∞.
This completes the proof.Corollary 4.6.
LetXbe a smooth Banach space satisfying any one of the Axioms  (a)–(c) of Lemma 3.1. Let  K be a nonempty closed bounded convex subset of  X and let   T,T1,T2,…,Tp:K→K(p≥2) be  p+1 mappings. Let  T,T1 be Lipschitz strictly hemicontractive mappings. Let  {αn},  {βn},   {γn} and  {βni}, i=1,2,…,p-1 be real sequences in  [0,1] satisfying the conditions  (iv)-(vi) and  ∑n=1∞(1-βn1)<∞. For arbitrary  x0∈K, define the sequence  {xn} by  (4.1). Then  {xn} converges strongly to the common fixed point of  ⋂i=1pF(Ti)∩F(T)≠∅.



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*Source: 294718-2012-06-18.xml*