An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces

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R E S E A R C H

Open Access

An implicit method for ﬁnding a common

ﬁxed point of a representation of

nonexpansive mappings in Banach spaces

Nawab Hussain

1

_{, Mahmood Lashkarizadeh Bami}

2

_{and Ebrahim Soori}

3*

*_{Correspondence: [email protected];}

[email protected] 3_{Department of Mathematics,}

Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran Full list of author information is available at the end of the article

Abstract

We introduce an implicit method for finding an element of the set of common fixed points of a representation of nonexpansive mappings. Then we prove the strong convergence of the proposed implicit scheme to the common fixed point of a representation of nonexpansive mappings.

MSC: 90C33; 47H10

Keywords: ﬁxed point; nonexpansive mapping; representation; semigroup; sunny nonexpansive retraction

1 Introduction

LetCbe a nonempty closed and convex subset of a Banach spaceEandE∗ be the dual space ofE. Let·,·denote the pairing betweenEandE∗. The normalized duality mapping

J:E→E∗is deﬁned by

J(x) =f∈E∗:x,f=x=f

for allx∈E. In the sequel, we usejto denote the single-valued normalized duality map-ping. LetU={x∈E:x= }.Eis said to be smooth or to have a Gâteaux diﬀerentiable norm if the limit

lim

t→

x+ty–x t

exists for eachx,y∈U.E is said to have a uniformly Gâteaux differentiable norm if for eachy∈U, the limit is attained uniformly for allx∈U.Eis said to be uniformly smooth or is said to have a uniformly Féchet differentiable norm if the limit is attained uniformly forx,y∈U. It is known that if the norm ofE is uniformly Gâteaux differentiable, then the duality mappingJis single-valued and uniformly norm to weak∗continuous on each bounded subset ofE. A Banach spaceEis smooth if the duality mappingJofEis single-valued. We know that ifEis smooth, thenJ is norm to weak-star continuous; for more details, see [].

LetCbe a nonempty closed and convex subset of a Banach spaceE. A mappingT ofC

into itself is called nonexpansive ifTx–Ty ≤ x–yfor allx,y∈C, and a mappingf is anα-contraction onEiff(x) –f(y) ≤αx–y,x,y∈Esuch that ≤α< .

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In this paper, motivated by Lashkarizadeh Bami and Soori [] and Hussain and Taka-hashi [], we introduce the following general implicit algorithm for finding a common element of the set of fixed points of a representationS={Tt:t∈S}of a semigroupSas nonexpansive mappings fromCinto itself, with respect to a left regular sequence of means defined on an appropriate subspace of bounded real-valued functions of the semigroup. On the other hand, our goal is to prove that there exists a sunny nonexpansive retractionP

ofContoFix(S) andx∈Csuch that the following sequence{zn}converges strongly toPx:

zn=nf(zn) + ( –n)Tμnzn (n∈N).

2 Preliminaries

LetS be a semigroup. We denote byB(S) the Banach space of all bounded real-valued functions deﬁned onSwith supremum norm. For eachs∈Sandf∈B(S), we deﬁnelsand

rsinB(S) by

(lsf)(t) =f(st), (rsf)(t) =f(ts) (t∈S).

LetXbe a subspace ofB(S) containing , and letX∗be its topological dual. An elementμ ofX∗is said to be a mean onXifμ=μ() = . We often writeμt(f(t)) instead ofμ(f) forμ∈X∗ andf ∈X. LetXbe left invariant (resp. right invariant),i.e.,ls(X)⊂X(resp.

rs(X)⊂X) for eachs∈S. A meanμonXis said to be left invariant (resp. right invariant) ifμ(lsf) =μ(f) (resp.μ(rsf) =μ(f)) for eachs∈Sandf ∈X.Xis said to be left (resp. right) amenable ifXhas a left (resp. right) invariant mean.Xis amenable ifXis both left and right amenable. As is well known,B(S) is amenable whenSis a commutative semigroup (see p. of []). A net{μα}of means onXis said to be left regular if

lim α l

∗

sμα–μα= 

for eachs∈S, wherel∗_s is the adjoint operator ofls.

Letf be a function of the semigroupSinto a reﬂexive Banach spaceEsuch that the weak closure of{f(t) :t∈S}is weakly compact, and letXbe a subspace ofB(S) containing all the functionst→ f(t),x∗withx∗∈E∗. We know from [] that for anyμ∈X∗, there exists a unique elementfμinEsuch thatfμ,x∗=μtf(t),x∗for allx∗∈E∗. We denote suchfμ

byf(t) dμ(t). Moreover, ifμis a mean onX, then from [],f(t) dμ(t)∈co{f(t) :t∈S}. LetCbe a nonempty closed and convex subset ofE. Then a familyS={Ts:s∈S}of mappings fromCinto itself is said to be a representation ofSas a nonexpansive mapping onCinto itself ifSsatisﬁes the following:

() Tstx=TsTtxfor alls,t∈Sandx∈C;

() for everys∈S, the mappingTs:C→Cis nonexpansive.

We denote byFix(S) the set of common ﬁxed points ofS, that is,Fix(S) =_s_∈_S{x∈C:

Tsx=x}.

Theorem .[] Let S be a semigroup,let C be a closed,convex subset of a reﬂexive Banach space E,S={Ts:s∈S}be a representation of S as a nonexpansive mapping from C into

itself such that weak closure of{Ttx:t∈S}is weakly compact for each x∈C,and let X be

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each x∈C and x∗∈E,andμbe a mean on X.If we write Tμx instead of

Ttxdμ(t),then

the following hold.

(i) Tμis a nonexpansive mapping fromCintoC. (ii) Tμx=xfor eachx∈Fix(S).

(iii) Tμx∈co{Ttx:t∈S}for eachx∈C.

(iv) IfXisrs-invariant for eachs∈Sandμis right invariant,thenTμTt=Tμfor each

t∈S.

Remark From Theorem .. in [], every uniformly convex Banach space is strictly

con-vex and reﬂexive.

LetDbe a subset ofB, whereBis a subset of a Banach spaceE, and letPbe a retraction ofBontoD, that is,Px=xfor eachx∈D. ThenPis said to be sunny if for eachx∈B

andt≥ withPx+t(x–Px)∈B,P(Px+t(x–Px)) =Px. A subsetDofBis said to be a sunny nonexpansive retract ofBif there exists a sunny nonexpansive retractionPofB

ontoD. We know that ifE is smooth andPis a retraction ofBontoD, thenPis sunny and nonexpansive if and only if for eachx∈Bandz∈D,x–Px,J(z–Px) ≤. For more details, see [].

Lemma .[] Let S be a semigroup,and let C be a compact convex subset of a real strictly convex and smooth Banach space E.Suppose thatS={Ts:s∈S}is a representation of S as

a nonexpansive mapping from C into itself.Let X be a left invariant subspace of B(S)such that∈X,and the function t→ Ttx,x∗is an element of X for each x∈C and x∗∈E∗.

Ifμis a left invariant mean on X,thenFix(Tμ) =TμC=Fix(S)and there exists a unique

sunny nonexpansive retraction from C ontoFix(S).

Throughout the rest of this paper, the open ball of radiusrcentered at  is denoted byBr. LetCbe a nonempty closed convex subset of a Banach spaceE. For>  and a mapping

T:C→C, we letF(T) be the set of-approximate ﬁxed points ofT,i.e.,F(T) ={x∈C:

x–Tx ≤}.

3 Main result

In this section, we deal with a strong convergence approximation scheme for ﬁnding a common element of the set of common ﬁxed points of a representation of nonexpansive mappings.

Theorem . Let S be a semigroup.Let C be a nonempty compact convex subset of a real

strictly convex and reﬂexive and smooth Banach space E.Suppose thatS={Ts:s∈S}is a

representation of S as a nonexpansive mapping from C into itself such thatFix(S)=∅.Let X be a left invariant subspace of B(S)such that∈X,and the function t→ Ttx,x∗is an

element of X for each x∈C and x∗∈E∗.Let{μn}be a left regular sequence of means on X.

Suppose that f is anα-contraction on C.Letnbe a sequence in(, )such thatlimnn= .

Then there exists a unique sunny nonexpansive retraction P of C ontoFix(S)and x∈C such that the following sequence{zn}generated by

zn=nf(zn) + ( –n)Tμnzn (n∈N) ()

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Proof By Proposition .. and Theorem .. in [], any compact subsetCof a reﬂexive Banach spaceEis weakly compact, and from Proposition .. in [], any closed convex subset of a weakly compact subsetCof a Banach spaceEis itself weakly compact, and by Proposition .. in [], any convex subsetCof a normed spaceEis weakly closed if and only ifCis closed. Therefore, weak closure of{Ttx:t∈S}is weakly compact for each

x∈C.

We divide the proof into ﬁve steps.

Step . The existence ofznwhich satisﬁes ().

This follows immediately from the fact that for everyn∈N, the mappingNngiven by

Nnx:=nf(x) + ( –n)Tμnx (x∈C)

is a contraction. To see this, putβn= ( +n(α– )), then ≤βn<  (n∈N). Then we have Nnx–Nny ≤nf(x) –f(y)+ ( –n)Tμnx–Tμny

≤nαx–y+ ( –n)x–y = +n(α– )

x–y=βnx–y.

Therefore, by the Banach contraction principle [], there exists a unique pointzn∈Csuch thatNnzn=zn.

Step .limn→∞zn–Ttzn=  for allt∈S.

Considert∈Sand let> . By Lemma  in [], there existsδ>  such thatcoFδ(Tt) +

Bδ⊆F(Tt). By Corollary . in [], there also exists a natural numberNsuch that



N+  N

i=

T_ti_sy–Tt



N+  N

i=

T_ti_sy

≤δ ()

for alls∈Sandy∈C. Letp∈Fix(S) andMbe a positive number such thatsupy∈Cy ≤

M. Lett∈S, since{μn}is strongly left regular, there existsN∈Nsuch thatμn–l_t∗iμn ≤

δ

(M) forn≥Nandi= , , . . . ,N. Then we have

sup

y∈C Tμny–



N+  N

i=

T_ti_sydμn(s)

=sup

y∈C

sup x∗=

Tμny,x∗

–



N+  N

i=

Tti_sydμn(s),x∗

=sup

y∈C

sup x∗=



N+  N

i=

(μn)sTsy,x∗

– 

N+  N

i=

(μn)sT_ti_sy,x∗

≤ 

N+  N

i= sup

y∈C

sup x∗=

(μn)sTsy,x∗

–l∗_tiμn

s

Tsy,x∗

≤ max

i=,,...,N

μn–l∗_tiμnM+ p

≤ max

i=,,...,N

μn–l∗_tiμn(M)

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By Theorem . we have



N+  N

i=

Tti_sydμn(s)∈co



N+  N

i=

Tti(T_sy) :s∈S

. ()

It follows from ()-() that

Tμny∈co



N+  N

i=

T_ti_sy:s∈S

+Bδ

⊂coFδ(Tt) + Bδ⊂F(Tt)

for ally∈Candn≥N. Therefore,lim supn→∞supy∈CTt(Tμny) –Tμny ≤. Since>  is arbitrary, we have

lim sup

n→∞

sup

y∈C

Tt(Tμny) –Tμny= . ()

Lett∈Sand> , then there existsδ>  which satisﬁes (). TakeL= ( +α)M+

f(p) –p. Now, from the conditionlim_nn=  and from (), there exists a natural number

Nsuch thatTμny∈Fδ(Tt) for ally∈Candn<_δ_L_ for alln≥N. Sincep∈Fix(S), we

have

nf(zn) –Tμnzn≤nf(zn) –f(p)+f(p) –p+Tμnp–Tμnzn

≤n

αzn–p+f(p) –p+Azn–p

≤n

αzn–p+f(p) –p+zn–p

≤n

( +α)zn–p+f(p) –p ≤n

( +α)M+f(p) –p

=nL≤

δ 

for alln≥N. Observe that

zn=nf(zn) + ( –n)Tμnzn =Tμnzn+n

f(zn) –Tμnzn

∈Fδ(Tt) +Bδ



⊆Fδ(Tt) + Bδ

⊆F(Tt)

for alln≥N. This shows that

zn–Ttzn ≤ (n≥N).

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Step .S{zn} ⊂Fix(S), whereS{zn}denotes the set of strongly limit points of{zn}. Letz∈S{zn}, and let{znj}be a subsequence of{zn}such thatznj→z,

Ttz–z ≤ Ttz–Ttznj+Ttznj–znj+znj–z ≤znj–z+Ttznj–znj,

then by Step ,

Ttz–z ≤lim

j znj–z+limj Ttznj–znj= , thereforez∈Fix(S).

Step . There exists a unique sunny nonexpansive retractionPofC ontoFix(S) and

x∈Csuch that :=lim sup

n

x–Px,J(zn–Px)

≤. ()

By Lemma . there exists a unique sunny nonexpansive retractionPofContoFix(S). The Banach contraction mapping principle guarantees thatfPhas a unique ﬁxed point

x∈C. We show that

:=lim sup

n

x–Px,J(zn–Px)

≤.

Note that from the deﬁnition ofand the fact thatCis a compact subset ofE, we can select a subsequence{znj}of{zn}with the following properties:

(i) limjx–Px,J(znj–Px)=;

(ii) {znj}converges strongly to a pointz.

By Step , we havez∈Fix(S). SinceEis smooth, we have

=lim

j

x–Px,J(znj–Px)

=x–Px,J(z–Px)≤.

SincefPx=x, we have (f –I)Px=x–Px. From Theorem ..(v) in [], forx,y∈Eand

f ∈J(y),x–y≥(x–y,f). Therefore, for eachn∈N, we have n(α– )zn–Px

≥nαzn–Px+ ( –n)zn–Px 

–zn–Px ≥nf(zn) –f(Px)+ ( –n)Tμnzn–Px



–zn–Px ≥n

f(zn) –f(Px)+ ( –n)(Tμnzn–Px) – (zn–Px),J(zn–Px)

= –n

(f –I)Px,J(zn–Px)

= –n

x–Px,J(zn–Px)

.

Hence

zn–Px≤   –α

x–Px,J(zn–Px)

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Step .{zn}strongly converges toPx.

Indeed, from (), () andPx∈Fix(S), we conclude

lim sup

n

zn–Px≤ 

 –αlim supn

x–Px,J(zn–Px)

≤.

That is,zn→Px.

Remark . It would be an interesting problem to prove Theorem . for continuous

representations instead of nonexpansive.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.}2_{Department of}

Mathematics, University of Isfahan, Isfahan, Iran. 3_{Department of Mathematics, Lorestan University, P.O. Box 465,} Khoramabad, Lorestan, Iran.

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support. The authors would like to thank the referee of the paper for his helpful comments and invaluable suggestions. This research was supported by the Center of Excellence for Mathematics and the Office of Graduate Studies of the Lorestan University and the University of Isfahan.

Received: 31 August 2014 Accepted: 12 November 2014 Published:04 Dec 2014 References

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