Implicit iteration scheme for two phi hemicontractive operators in arbitrary Banach spaces

R E S E A R C H

Open Access

Implicit iteration scheme for two

phi-hemicontractive operators in arbitrary

Banach spaces

Guiwen Lv

1*

_{, Arif Rafiq}

2

_{and Zhiqun Xue}

1

*_{Correspondence:}

[email protected]

1_{Department of Mathematics and}

Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, P.R. China

Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to characterize conditions for the convergence of the implicit Ishikawa iterative scheme with errors in the sense of Agarwalet al.(J. Math. Anal. Appl. 272:435-447, 2002) to a common fixed point of two

φ

-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

MSC: Primary 47H09; 47J25

Keywords: implicit iterative scheme;

φ

-hemicontractive mappings; Banach spaces

1 Introduction and preliminaries

LetKbe a nonempty subset of an arbitrary Banach spaceEandE∗be its dual space. The symbolsTandF(T) stand for the self-map ofKand the set of fixed points ofT. We denote byJthe normalized duality mapping fromEto E∗_{defined by}

J(x) =f∗∈E∗:x,f∗=x=f∗.

Definition .[–] (i)Tis said to be strongly pseudocontractive if there exists a constant t>  such that for eachx,y∈K, there existsj(x–y)∈J(x–y) satisfying

Tx–Ty,j(x–y)≤ tx–y

_.

(ii)T is said to be strictly hemicontractive ifF(T)=∅and if there exists a constantt>  such that for eachx∈Kandq∈F(T), there existsj(x–q)∈J(x–q) satisfying

Tx–Tq,j(x–q)≤ tx–q

_.

(iii)Tis said to beφ-strongly pseudocontractive if there exists a strictly increasing func-tionφ: [,∞)→[,∞) withφ() =  such that for eachx,y∈K, there existsj(x–y)∈ J(x–y) satisfying

Tx–Ty,j(x–y)≤ x–y–φx–y.

(iv)T is said to beφ-hemicontractive ifF(T)=∅and if there exists a strictly increasing functionφ: [,∞)→[,∞) withφ() =  such that for eachx∈K andq∈F(T), there

existsj(x–q)∈J(x–q) satisfying

Tx–Tq,j(x–q)≤ x–q–φx–q.

Clearly, each strictly hemicontractive operator isφ-hemicontractive.

Chidume [] established that the Mann iteration sequence converges strongly to the unique fixed point ofTin the case thatTis a Lipschitz strongly pseudo-contractive map-ping from a bounded, closed, convex subset ofLp (orlp) into itself. Afterwards, several

authors have generalized this result of Chidume in various directions [–].

In , Xu and Ori [] introduced the following implicit iteration process for a fi-nite family of nonexpansive mappings{Ti:i∈I}(hereI={, , . . . ,N}), with{αn}a real

sequence in (, ), and an initial pointx∈K:

x= ( –α)x+αTx,

x= ( –α)x+αTx,

.. .

xN = ( –αN)xN–+αNTNxN,

xN+= ( –αN+)xN+αN+TN+xN+,

.. .

which can be written in the following compact form:

xn= ( –αn)xn–+αnTnxn for alln≥, (XO)

whereTn=Tn(modN)(here themodNfunction takes values inI). Xu and Ori [] proved

that the process converges weakly to a common fixed point of a finite family in a Hilbert space. They remarked further that it is yet unclear what assumptions on the mappings and the parameters {αn} are sufficient to guarantee the strong convergence of the

se-quence{xn}.

In [], Osilike proved the following results.

Theorem .[, Theorem ] Let E be a real Banach space and K be a nonempty closed

convex subset of E.Let{Ti:i∈I}be N strictly pseudocontractive self-mappings of K with

F= N_i=F(Ti)=∅.Let{αn}∞n=be a real sequence satisfying the conditions:

(i)  <αn< ,

(ii) ∞

n=

( –αn) =∞,

(iii) ∞

n=

For arbitrary x∈K,define the sequence{xn}by the implicit iteration process(XO).Then

{xn}converges strongly to a common fixed point of the mappings{Ti:i∈I}if and only if

lim_n→∞infd(xn,F) = .

It is well known thatαn=  –  n

,( –αn)=∞. Hence the results of Osilike [] need

to be improved.

The purpose of this paper is to characterize conditions for the convergence of the im-plicit Ishikawa iterative scheme with errors in the sense of Agarwalet al.[] to a common fixed point of twoφ-hemicontractive mappings in a nonempty convex subset of an arbi-trary Banach space. Our studying results improve and generalize most results in recent literature [, , –, ].

2 Main results

The following results are now well known.

Lemma .[] For all x,y∈E and j(x+y)∈J(x+y),

x+y≤ x+y,j(x+y).

Lemma .[] Let{θn}be a sequence of nonnegative real numbers,and let{λn}be a real

sequence satisfying

≤λn≤,

∞

n=

λn=∞.

Suppose that there exists a strictly increasing functionφ: [,∞)→[,∞)withφ() = .If there exists a positive integer nsuch that

θ_n+ ≤θ_n–λnφ(θn+) +σn+γn

for all n≥n,withσn≥,∀n∈N,σn= (λn)and

∞

n=

γn<∞,thenlimn→∞θn= .

Now we prove our main results.

Theorem . Let K be a nonempty convex subset of an arbitrary Banach space E,and let

T,S:K→K be two uniformly continuous with F(T)∩F(S)=∅andφ-hemicontractive mappings. Suppose that {un}∞n= and{vn}∞n= are bounded sequences in K and {an}∞n=,

{bn}∞n=,{cn}∞n=,{an}n=∞ ,{bn}∞n=and{cn}∞n=are sequences in[, ]satisfying conditions (i) an+bn+cn=an+bn+cn= ,

(ii) limn→∞bn=limn→∞cn=limn→∞bn= , (iii) ∞_n=c_n<∞,and

(iv) ∞_n=b_n=∞.

For any x∈K,define the sequence{xn}∞n=inductively as follows:

yn=anxn–+bnSxn+cnun,

xn=anxn–+bnTyn+cnvn, n≥.

Then the following conditions are equivalent:

(a) {xn}∞n=converges strongly to the common fixed pointqofTandS, (b) limn→∞Tyn=q,

(c) {Tyn}∞n=is bounded.

Proof SinceT andSareφ-hemicontractive, then the common fixed point ofF(T)∩F(S) is unique. Suppose thatpandqare all common fixed points ofT andS, then

p–q=p–q,j(p–q)=Tp–Tq,j(p–q)≤ p–q–φp–q<p–q,

which is a contradiction. So, we denote the unique fixed pointq.

Suppose thatlimn→∞xn=q. Then (ii) and the uniform continuity ofT andSyield that

lim

n→∞yn=nlim→∞[anxn–+bnSxn+cnun] =q,

which implies thatlimn→∞Tyn=q. Therefore{Tyn}∞n=is bounded.

Put

M=x–q+sup n≥

Tyn–q+sup n≥

un–q+sup n≥

vn–q.

Obviously,M<∞. It is clear thatx–p ≤M. Letxn––p ≤M. Next we prove that

xn–p ≤M.

Consider

xn–q=anxn–+bnTyn+cnvn–q

=a_n(xn––q) +bn(Tyn–q) +cn(vn–q)

≤ –b_nxn––q+bnTyn–q+cnvn–q

≤ –b_nM+bnTyn–q+cnvn–q

= –b_nx–q+sup n≥

Tyn–q+sup n≥

un–q+sup n≥

vn–q

+b_nTyn–q+cnvn–q

≤ x–q+

 –b_nsup

n≥

Tyn–q+bnTyn–q

+sup

n≥

un–q+

 –b_nsup

n≥

vn–q+bnvn–q

≤ x–q+

 –b_nsup

n≥

Tyn–q+bnsup n≥

Tyn–q

+sup

n≥

un–q+

 –b_nsup

n≥

vn–q+bnsup n≥

vn–q

=x–q+sup n≥

Tyn–q+sup n≥

un–q+sup n≥

vn–q

=M.

So, from the above discussion, we can conclude that the sequence{xn–q}n≥is bounded.

M>  satisfying

M=sup n≥

xn–q+sup n≥

Sxn–q+sup n≥

Tyn–q

+sup

n≥

un–q+sup n≥

vn–q. (.)

DenoteM=M+M. Obviously,M<∞. Letwn=Tyn–Txnfor eachn≥. The uniform

continuity ofTensures that

lim

n→∞wn= . (.)

Because

yn–xn = bn(Sxn–xn–) +bn(xn––Tyn) +cn(un–xn–) –cn(vn–xn–)

≤ bnSxn–xn–+bnxn––Tyn+cnun–xn–+cnvn–xn–

≤ M

bn+cn+bn+cn

→

asn→ ∞.

By virtue of Lemma . and (.), we infer that

xn–q =anxn–+bnTyn+cnvn–q 

=a_n(xn––q) +bn(Tyn–q) +cn(vn–q) 

≤ –b_nxn––q+ bn

Tyn–q,j(xn–q)

+ c_nvn–q,j(xn–q)

≤ –b_nxn––q+ bn

Tyn–Txn,j(xn–q)

+ b_nTxn–q,j(xn–q)

+ c_nvn–qxn–q

≤ –b_nxn––q+ bnTyn–Txnxn–q

+ b_nxn–q– bnφ

xn–q

+ Mc_n

≤ –b_nxn––q+ Mbnwn+ bnxn–q

– b_nφxn–q

+ Mc_n. (.)

Consider

xn–q =anxn–+bnTyn+cnvn–q 

=a_n(xn––q) +bn(Tyn–q) +cn(vn–q) 

≤a_nxn––q+bnTyn–q+cnvn–q

≤ xn––q+M

b_n+c_n, (.)

Substituting (.) in (.), we get

xn–q

≤ –b_n+ b_nxn––q+ Mbn

wn+M

b_n+c_n+ Mc_n

– b_nφxn–q

= +b_nxn––q+ Mbn

wn+M

b_n+c_n+ Mc_n

– b_nφxn–q

≤ xn––q+Mbn

Mb_n+ wn+ Mbn

+ Mc_n

– b_nφxn–q

=xn––q+bnln+ Mcn– bnφ

xn–q

, (.)

where

ln=M

Mb_n+ wn+ Mbn

→ (.)

asn→ ∞. Denote

θn=xn–q,

λn= bn,

σn=bnln,

γn= Mcn.

Condition (i) assures the existence of a rankn∈Nsuch thatλn= bn≤ for alln≥n.

Now, with the help of conditions (ii), (iii), (.), (.) and Lemma ., we obtain from (.) that

lim

n→∞xn–q= ,

completing the proof.

Using the method of proof in Theorem ., we have the following result.

Corollary . Let E,K,T,S,{un}∞n=,{vn}∞n=,{xn}∞n=and{yn}∞n= be as in Theorem..

Suppose that {an}∞n=,{bn}∞n=,{cn}∞n=,{an}∞n=,{bn}∞n= and{cn}∞n= are sequences in [, ]

satisfying conditions(i), (ii), (iv)and c_n=o(b_n).Then the conclusions of Theorem.hold.

Proof From the conditionc_n=o(b_n), setc_n=b_ntn, wherelimn→∞tn= . Substituting (.)

in (.), we get

xn–q

≤ –b_n+ b_nxn––q+ Mbn

wn+M

– b_nφxn–q

xn–q

= +b_nxn––q+ Mbn

wn+M

b_n+c_n+tn

– b_nφxn–q

≤ xn––q+Mbn

wn+M

b_n+ c_n+ tn

– b_nφxn–q

=xn––q– bnφ

xn–q

xn–q+bnln, (.)

where

l_n =Mwn+M

b_n+ c_n+ tn

→ (.)

asn→ ∞.

It follows from Lemma . thatlimn→∞xn–q= .

Corollary . Let K be a nonempty convex subset of an arbitrary Banach space E,and

let T:K→K be a uniformly continuous andφ-hemicontractive mapping.Suppose that {un}∞n=and{vn}n=∞ are bounded sequences in K and{an}∞n=,{bn}∞n=,{cn}∞n=,{an}∞n=,{bn}∞n=

and{c_n}∞_n=are sequences in[, ]satisfying conditions (i) an+bn+cn=an+bn+cn= ,

(ii) limn→∞bn=limn→∞cn=limn→∞bn= , (iii) c_n=o(b_n),and

(iv) ∞_n=b_n=∞.

For any x∈K,define the sequence{xn}∞n=inductively as follows:

yn=anxn–+bnTxn+cnun,

xn=anxn–+bnTyn+cnvn, n≥.

Then the following conditions are equivalent:

(a) {xn}∞n=converges strongly to the unique fixed pointqofT, (b) lim_n→∞Tyn=q,

(c) {Tyn}∞n=is bounded.

Corollary . Let E,K,T,{un}∞n=,{vn}n=∞ ,{xn}∞n=and{yn}∞n=be as in Corollary..

Sup-pose that{an}∞n=,{bn}∞n=,{cn}∞n=,{an}∞n=,{bn}n=∞ and{cn}∞n=are sequences in[, ]

satis-fying conditions(i), (ii), (iv)and∞_n=c_n<∞.Then the conclusions of Corollary.hold.

Corollary . Let K be a nonempty convex subset of an arbitrary Banach space E,and let T,S:K→K be two uniformly continuous andφ-hemicontractive mappings.Suppose that {αn}∞n=,{βn}∞n=are any sequences in[, ]satisfying

(i) limn→∞βn=  =limn→∞αn, (ii) ∞_n=αn=∞.

For any x∈K,define the sequence{xn}∞n=inductively as follows:

yn= ( –βn)xn–+βnSxn,

Then the following conditions are equivalent:

(a) {xn}∞n=converges strongly to the common fixed pointqofTandS, (b) limn→∞Tyn=q,

(c) {Tyn}∞n=is bounded.

Corollary . Let K be a nonempty convex subset of an arbitrary Banach space E,and

let T:K→K be a uniformly continuous andφ-hemicontractive mapping.Suppose that {αn}∞n=,{βn}∞n=are any sequences in[, ]satisfying

(i) limn→∞βn=  =limn→∞αn, (ii) ∞_n=αn=∞.

For any x∈K,define the sequence{xn}∞n=inductively as follows:

yn= ( –βn)xn–+βnTxn,

xn= ( –αn)xn–+αnTyn, n≥.

Then the following conditions are equivalent:

(a) {xn}∞n=converges strongly to the unique fixed pointqofT, (b) limn→∞Tyn=q,

(c) {Tyn}∞n=is bounded.

Corollary . Let K be a nonempty convex subset of an arbitrary Banach space E,and

let T:K→K be a uniformly continuous andφ-hemicontractive mapping.Suppose that {αn}∞n=is any sequence in[, ]satisfying

(i) limn→∞αn= , (ii) ∞_n=αn=∞.

For any x∈K,define the sequence{xn}∞n=inductively as follows:

xn=αnxn–+ ( –αn)Txn, n≥.

Then the following conditions are equivalent:

(a) {xn}∞n=converges strongly to the unique fixed pointqofT, (b) lim_n→∞Txn=q,

(c) {Txn}∞n=is bounded.

All of the above results are also valid for Lipschitzφ-hemicontractive mappings.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in writing this paper, and read and approved the final manuscript.

Author details

1_{Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, P.R. China.}2_Hajvery

University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan.

Acknowledgements

The authors are grateful to the reviewers for valuable suggestions which helped to improve the manuscript.

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10.1186/1029-242X-2013-521