Iterative schemes for approximating solution of nonlinear operators in Banach spaces

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R E S E A R C H

Open Access

Iterative schemes for approximating solution

of nonlinear operators in Banach spaces

Siwaporn Saewan

1*

_{, Preedaporn Kanjanasamranwong}

1*

_{, Poom Kumam}

2*

_{and Yeol Je Cho}

3*

*_{Correspondence:}

[email protected]; [email protected]; [email protected]; [email protected]; [email protected]

1_{Department of Mathematics and}

Statistics, Faculty of Science, Thaksin University, Phatthalung, 93110, Thailand

2_{Department of Mathematics,}

Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangmod, Thrungkru, Bangkok, 10140, Thailand

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

Abstract

The purpose of this paper is to present a new modified Halpern-Mann type iterative scheme by using the generalizedf-projection operator for finding a common element in the set of zeroes of a system of maximal monotone operators, the set of fixed points of a totally quasi-φ-asymptotically nonexpansive mapping and the set of solutions of a system of generalized Ky Fan’s inequalities in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Furthermore, we show that our proposed iterative scheme converges strongly to a common element of the sets mentioned above.

MSC: 47H05; 47H09; 47H10

Keywords: generalizedf-projection operators; system of generalized Ky Fan’s inequalities; totally quasi-φ-asymptotically nonexpansive mapping; variational inequalities; maximal monotone operators

1 Introduction

In , Ky Fan’s inequalities were first introduced by Fan []. The study concerning Ky Fan’s inequalities, fixed points of nonlinear mappings and their approximation algorithms constitutes a topic of intensive research efforts. Many well-known problems arising in various branches of science can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of informa-tion implies the existence of a convex set in which the required soluinforma-tion lies.

Many authors have considered a family of nonexpansive mappings to show the existence of fixed points and related topics. Especially, the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonex-pansive mappings and the problem of finding an optimal point that minimizes a given cost function over the set of common fixed points of a family of nonexpansive mappings.

Solving the convex feasibility problem for a system of generalized Ky Fan’s inequal-ities is very general in the sense that it includes, as special cases, optimization prob-lems, equilibrium probprob-lems, variational inequality probprob-lems, minimax problems. More-over, the generalized Ky Fan’s inequality was shown in [] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization prob-lems, optimization probprob-lems, vector equilibrium probprob-lems, Nash equilibria in noncooper-ative games. In other words, the generalized Ky Fan’s inequality and equilibrium problem are a uniﬁed model for several problems arising in physics, engineering, science, optimiza-tion, economics and related topics.

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One of the most interesting and important problems in the theory of maximal mono-tone operators is to ﬁnd a zero point of maximal monomono-tone operators. This problem con-tains the convex minimization problem and the variational inequality problem. A popular method for approximating this problem is called the proximal point algorithm introduced by Martinet [] in a Hilbert space. In , Rockafellar [] extended the knowledge of Martinet [] and proved weak convergence of the proximal point algorithm. The proxi-mal point algorithm of Rockafellar [] is a successful algorithm for ﬁnding a zero point of maximal monotone operators. Thereafter, many papers have shown convergence the-orems of the proximal point algorithm in various spaces (see [–]).

A pointx∈Cis aﬁxed pointofSprovidedSx=x. We denote byF(S) the ﬁxed point set of

S, that is,F(S) ={x∈C:Sx=x}. A pointpinCis called anasymptotic ﬁxed pointofS[] ifCcontains a sequence{xn}which converges weakly topsuch thatlim_n_→∞xn–Sxn= .

The set of asymptotic fixed points ofSis denoted byF(S). Recently, Halpern and Mann iterative algorithms have been considered for approximations of common fixed points by many authors. For example, in , Saewan and Kumam [] introduced a modified Mann iterative scheme by using the generalizedf-projection method for approximating a com-mon fixed point of a countable family of relatively quasi-nonexpansive mappings. Chang

et al.[] considered a modiﬁed Halpern iterative scheme for approximating a common ﬁxed point for a totally quasi-φ-asymptotically nonexpansive mapping. Recently, Liet al.

[] introduced a hybrid iterative scheme for approximation of a ﬁxed point of relatively nonexpansive mappings by using the properties of generalizedf-projection operators in a uniformly smooth real Banach space, which is also uniformly convex, and proved some strong convergence theorems for the hybrid iterative scheme.

On the other hand, Ofoedu and Shehu [] extended the algorithm of Liet al.[] to prove strong convergence theorems for a common solution of the set of solutions of a system of Ky Fan’s inequalities and the set of common fixed points of a pair of relatively quasi-nonexpansive mappings in a Banach space by using the generalizedf-projection operator. Changet al.[] extended and improved the results of Qin and Su [] to obtain strong convergence theorems for finding a common element of the set of solutions for a generalized Ky Fan’s inequality, the set of solutions for a variational inequality problem and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space.

Motivated and inspired by the work mentioned above, in this paper, we introduce a new hybrid iterative scheme of the generalizedf-projection operator based on the Halpern-Mann type iterative scheme for ﬁnding a common element of the set of zeroes of a system of maximal monotone operators, the set of ﬁxed points of a totally quasi-φ-asymptotically nonexpansive mapping and the set of solutions of a system of generalized Ky Fan’s in-equalities in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

2 Preliminaries

A Banach spaceEwith the norm · is calledstrictly convexifx_+y<  for allx,y∈U

withx=y, whereU={x∈E:x= }is the unit sphere ofE. A Banach spaceEis called

smoothif the limit

lim

t→

x+ty–x

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exists for eachx,y∈U. It is also calleduniformly smoothif the limit exists uniformly for allx,y∈U. In this paper, we denote the strong convergence and weak convergence of a sequence{xn}byxn→xandxnx, respectively.

LetEbe a real Banach space with the dual spaceE*and letCbe a nonempty closed and convex subset ofE. A mappingS:C→Cis said to be:

() nonexpansiveif

Sx–Sy ≤ x–y

for allx,y∈C;

() quasi-nonexpansiveifF(S)=∅and

Sx–y ≤ x–y

for allx∈Candy∈F(S);

() asymptotically nonexpansiveif there exists a sequence{kn} ⊂[,∞)withkn→as n→ ∞such that

Snx–Sny≤knx–y

for allx,y∈C;

() asymptotically quasi-nonexpansiveifF(S)=∅and there exists a sequence

{kn} ⊂[,∞)withkn→asn→ ∞such that

Snx–y≤knx–y

for allx∈Candy∈F(S);

() totally asymptotically nonexpansiveif there exist nonnegative real sequences{νn},

{μn}withνn→,μn→asn→ ∞and a strictly increasing continuous function ψ:R+→R+withψ() = such that

_Sn_x_–_Sn_y_≤_x_–_y₊_μ nψ

x–y+νn

for allx,y∈Candn≥.

A mappingS:C→C is said to beuniformly L-Lipschitz continuousif there exists a constantL>  such that

Snx–Sny≤Lx–y ()

for allx,y∈C. A mappingS:C→Cis said to beclosedif, for any sequence{xn} ⊂Csuch thatlimn→∞xn=xandlimn→∞Sxn=y, we haveSx=y.

Thenormalized duality mapping J:E→E*is deﬁned by

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for allx∈E. IfEis a Hilbert space, thenJ=I, whereIis the identity mapping. Consider the functionalφ:E×E→Rdeﬁned by

φ(x,y) =x– x,Jy+y, ()

whereJis the normalized duality mapping and·,·denotes the duality pairing ofEandE*. IfEis a Hilbert space, thenφ(y,x) =y–x_{. It is obvious from the deﬁnition of}_φ_that

y–x≤φ(y,x)≤y+x ()

for allx,y∈E.

A mappingS:C→Cis said to be:

() relatively nonexpansive[, ] ifFˆ(S) =F(S)and

φ(p,Sx)≤φ(p,x)

for allx∈Candp∈F(S);

() relatively asymptotically nonexpansive[] ifFˆ(S) =F(S)=∅and there exists a sequence{kn} ⊂[,∞)withkn→asn→ ∞such that

φp,Snx≤knφ(p,x)

for allx∈C,p∈F(S)andn≥; () φ-nonexpansive[, ] if

φ(Sx,Sy)≤φ(x,y)

for allx,y∈C;

() quasi-φ-nonexpansive[, ] ifF(S)=∅and

φ(p,Sx)≤φ(p,x)

for allx∈Candp∈F(S);

() asymptoticallyφ-nonexpansive[] if there exists a sequence{kn} ⊂[,∞)with

kn→asn→ ∞such that

φSnx,Sny≤knφ(x,y)

for allx,y∈Candn≥;

() quasi-φ-asymptotically nonexpansive[] ifF(S)=∅and there exists a sequence

{kn} ⊂[,∞)withkn→asn→ ∞such that

φp,Snx≤knφ(p,x)

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() totally quasi-φ-asymptotically nonexpansiveifF(S)=∅and there exist nonnegative real sequences{νn},{μn}withνn→,μn→asn→ ∞and a strictly increasing

continuous functionψ:R+→R+withψ() = such that

φp,Snx≤φ(p,x) +νnψ

φ(p,x)+μn

for allx∈C,p∈F(S)andn≥.

Lemma [] Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.Let S:C→C be a closed and totally quasi-φ-asymptotically nonexpansive mapping with nonnegative real sequences {νn}and{μn}withνn→andμn→as n→ ∞,respectively,and a strictly increasing continuous functionζ :R+_→_R+_with_ζ_{() = .}_If_μ

= ,then the set F(S)of ﬁxed points

of S is a closed convex subset of C.

Alber [] introduced that the generalized projectionΠC :E →C is a mapping that

assigns to an arbitrary pointx∈E the minimum point of the functionalφ(x,y), that is,

ΠCx=x¯, wherex¯is the solution of the minimization problem

φ(x¯,x) =inf

y∈Cφ(y,x). ()

The existence and uniqueness of the operator ΠC follows from the properties of the

functionalφ(y,x) and strict monotonicity of the mappingJ(see, for example, [–]). IfE is a Hilbert space, thenφ(x,y) =x–y_and_Π

C becomes the metric projection PC:H→C. IfCis a nonempty closed and convex subset of a Hilbert spaceH, thenPCis

nonexpansive.

Remark  The basic properties of a Banach space Erelated to the normalized duality mappingJare as follows (see []):

() IfEis an arbitrary Banach space, thenJis monotone and bounded; () IfEis a strictly convex Banach space, thenJis strictly monotone;

() IfEis a smooth Banach space, thenJis single-valued and semicontinuous; () IfEis a uniformly smooth Banach space, thenJis uniformly norm-to-norm

continuous on each bounded subset ofE;

() IfEis a reﬂexive smooth and strictly convex Banach space, then the normalized duality mappingJis single-valued, one-to-one and onto;

() IfEis a reﬂexive strictly convex and smooth Banach space andJis the duality mapping fromEintoE*_{, then}_J–_{is also single-valued, bijective and is also the} duality mapping fromE*_into_E_{, and thus}_JJ–₌_I

E*andJ–J=IE;

() IfEis a uniformly smooth Banach space, thenEis smooth and reﬂexive; () Eis a uniformly smooth Banach space if and only ifE*_{is uniformly convex;} () IfEis a reﬂexive and strictly convex Banach space, thenJ–_is

norm-weak*_-continuous.

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have

 =x– x,Jy+y

=x– xJy+y

=x–y.

That is,x=y. This implies thatx,Jy=x₌_Jy_{. From the deﬁnition of}_J_{, one has}

Jx=Jy. Therefore, we havex=y(see [, , ] for more details).

In , Wu and Huang [] introduced a new generalizedf-projection operator in a Banach space. They extended the deﬁnition of the generalized projection operators intro-duced by Abler [] and proved some properties of the generalizedf-projection operator. Consider the functionalG:C×E*→R∪ {+∞}deﬁned by

G(y,) =y– y,++ ρf(y) ()

for all (y,)∈C×E*_{, where} _ρ_{is a positive number and}_f _:_C_→_R_{∪ {}₊_∞} _{is proper,} convex and lower semicontinuous. From the deﬁnition ofG, Wu and Huang [] proved the following properties:

() G(y,)is convex and continuous with respect to whenyis ﬁxed;

() G(y,)is convex and lower semicontinuous with respect toywhenis ﬁxed.

Deﬁnition  LetEbe a real Banach space with its dual spaceE*_{and let}_C_{be a nonempty} closed and convex subset ofE. We say thatπ_Cf :E*_→_C_{is a}_{generalized f -projection} op-eratorif

π_Cf=

u∈C:G(u,) =inf

y∈CG(y,),∀∈E

*_.

Recall that a Banach spaceEhas the Kadec-Klee property [, , ] if for any sequence

{xn} ⊂Eandx∈Ewithxnxandxn → x, we havexn–x → asn→ ∞. It is well

known that ifEis a uniformly convex Banach space, thenEhas the Kadec-Klee property.

Lemma [] Let E be a real reﬂexive Banach space with its dual space E*_{and let C be a}

nonempty closed and convex subset of E.The following statements hold: () π_Cfis a nonempty,closed and convex subset ofCfor all∈E*_; () IfEis smooth,then for all∈E*_,_x_∈_πf

C if and only if

x–y,–Jx+ρf(y) –ρf(x)≥

for ally∈C;

() IfEis strictly convex andf:C→R∪ {+∞}is positive homogeneous(i.e.,f(tx) =tf(x)

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Recently, Fanet al.[] showed that the condition,f is positive homogeneous, which appears in [, Lemma .(iii)], can be removed.

Lemma [] Let E be a real reﬂexive Banach space with its dual space E*_{and let C be a}

nonempty closed and convex subset of E.If E is strictly convex,thenπ_Cf is single-valued.

Recall thatJis a single-valued mapping whenEis a smooth Banach space. There exists a unique element∈E*_{such that}₌_Jx_{, where}_x_∈_E_{. This substitution in () gives the} following:

G(y,Jx) =y– y,Jx+x+ ρf(y). ()

Now, we consider the second generalized f projection operator in a Banach space (see []).

Deﬁnition  LetEbe a real smooth Banach space and letCbe a nonempty closed and convex subset ofE. We say thatΠ_Cf :E→C_{is a}_{generalized f -projection operator}_if

Π_Cfx=

u∈C:G(u,Jx) =inf

y∈CG(y,Jx),∀x∈E

.

Lemma [] Let E be a Banach space and let f :E→R∪ {+∞}be a lower semicontin-uous and convex function.Then there exist x*_∈_E*_and_α_∈_R_{such that}

f(x)≥x,x*+α

for all x∈E.

Lemma [] Let E be a reﬂexive smooth Banach space and let C be a nonempty closed and convex subset of E.The following statements hold:

() Π_Cfxis a nonempty closed and convex subset ofCfor allx∈E; () For allx∈E,xˆ∈Π_Cfxif and only if

ˆx–y,Jx–Jxˆ+ρf(y) –ρf(xˆ)≥

for ally∈C;

() IfEis strictly convex,thenΠ_Cf is a single-valued mapping.

Lemma [] Let E be a real reﬂexive smooth Banach space and let C be a nonempty closed and convex subset of E.Then,for any x∈E andxˆ∈Π_Cfx,

φ(y,xˆ) +G(xˆ,Jx)≤G(y,Jx)

for all y∈C.

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Remark  LetEbe a uniformly convex and uniformly smooth Banach space andf(y) =  for ally∈E. Then Lemma  reduces to the property of the generalized projection operator considered by Alber [].

Iff(y)≥ for ally∈Candf() = , then the deﬁnition of a totally quasi-φ -asymptot-ically nonexpansiveSis equivalent to the following:

ForF(S)=∅and there exist nonnegative real sequences{νn},{μn}withνn→,μn→

asn→ ∞, respectively, and a strictly increasing continuous functionψ:R+_→_R+ _with

ψ() =  such that

Gp,Snx≤G(p,x) +νnψG(p,x) +μn

for allx∈C,p∈F(S) andn≥.

Letθ be a bifunction fromC×CtoR, whereRdenotes the set of real numbers.The equilibrium problem(for short, (EP)) is to ﬁndxˆ∈Csuch that

θ(xˆ,y)≥ ()

for ally∈C. The set of solutions of (EP) () is denoted byEP(θ).

For solving the equilibrium problem for a bifunctionθ:C×C→R, let us assume that

θ satisﬁes the following conditions: (A) θ(x,x) = for allx∈C;

(A) θ is monotone,i.e.,θ(x,y) +θ(y,x)≤for allx,y∈C; (A) for allx,y,z∈C,

lim

t↓θ

tz+ ( –t)x,y≤θ(x,y);

(A) for allx∈C,y→θ(x,y)is convex and lower semicontinuous.

For example, letBbe a continuous and monotone operator ofCintoE*_{and deﬁne}

θ(x,y) =Bx,y–x

for allx,y∈C. Thenθ satisﬁes (A)-(A).

Lemma [] Let C be a closed convex subset of a smooth,strictly convex and reﬂexive Banach space E and letθ be a bifunction from C×C toRsatisfying the conditions(A)

-(A).Then,for any r> and x∈E,there exists z∈C such that

θ(z,y) +

ry–z,Jz–Jx ≥

for all y∈C.

Lemma [] Let C be a closed convex subset of a uniformly smooth,strictly convex and reﬂexive Banach space E and letθbe a bifunction from C×C toRsatisfying the conditions

(A)-(A).For all r> and x∈E,deﬁne a mapping Tθ

r :E→C as follows:

Tθ rx=

z∈C:θ(z,y) +

ry–z,Jz–Jx ≥,∀y∈C

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Then the following hold: () Tθ

r is single-valued;

() Tθ

r is a ﬁrmly nonexpansive-type mapping[],that is,for allx,y∈E,

T_rθx–T_rθy,JT_rθx–JT_rθy≤T_rθx–T_rθy,Jx–Jy;

() F(Tθ

r) =EP(θ);

() EP(θ)is closed and convex.

Lemma [] Let C be a closed convex subset of a smooth,strictly convex and reﬂexive Banach space E and letθ be a bifunction from C×C toRsatisfying the conditions(A)

-(A).Then,for any r> ,x∈E and q∈F(Tθ r),

φq,Tθ rx

+φTθ

rx,x

≤φ(q,x).

An operatorA⊂E×E*_{is said to be}_monotone_if

x–y,x*–y*≥

for all (x,x*_{), (}_y_,_y*₎_∈_A_{. A point}_z_∈_E_{is called a}_{zero point}_of_A_if

∈Az. ()

We denote the set of zeroes of the operatorAbyA–_{, that is,}

A– ={z∈E: ∈Az}.

A monotoneA⊂E×E*_{is said to be}_maximal_{if its graph}_G₍_A_{) =}_{₍_x_,_y*_{) :}_y*_∈_Ax_}_is not property contained in the graph of any other monotone operator. If Ais maximal monotone, then the solution setA–_{ is closed and convex.}

LetEbe a smooth strictly convex and reﬂexive Banach space, letCbe a nonempty closed convex subset of Eand letA⊂E×E* _{be a monotone operator satisfying}_D₍_A₎_⊂_C_⊂

J–₍

λ>R(J+λA)). Then theresolvent Jλ:C→D(A) ofAis deﬁned by

Jλx=z∈D(A) :Jx∈Jz+λAz,∀x∈C .

Jλis a single-valued mapping fromEtoD(A). On the other hand,Jλ= (J+λA)–_J_{for all}

λ> .

For anyλ> , theYosida approximation Aλ:C→E*ofAis deﬁned byAλx= Jx–JJλx

λ

for allx∈C. We know thatAλx∈A(Jλx) for allλ>  andx∈E. Since relatively quasi-nonexpansive mappings and quasi-φ-nonexpansive mappings are the same, we can see thatJλis a quasi-φ-nonexpansive mapping (see [, Theorem .]).

Lemma [] Let E be a smooth strictly convex and reﬂexive Banach space,let C be a nonempty closed convex subset of E and let A⊂E×E*be a monotone operator satisfying D(A)⊂C⊂J–(λ>R(J+λA)).For anyλ> ,let Jλand Aλbe the resolvent and the Yosida

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() φ(p,Jλx) +φ(Jλx,x)≤φ(p,x)for allx∈Candp∈A–__; () (Jλx,Aλx)∈Afor allx∈C;

() F(Jλ) =A–__.

Lemma [] Let E be a reﬂexive strictly convex and smooth Banach space.Then an operator A⊂E×E*is maximal monotone if and only if R(J+λA) =E*for allλ> .

3 Main result

Now, we give the main results in this paper.

Theorem  Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.For each i= , , . . . ,m,letθi be a bifunction from C×C toRsatisfying the conditions(A)-(A).Let Aj⊂E×E*be a maximal monotone operator satisfying D(Aj)⊂C and J

Aj

λj,n= (J+λj,nAj)

–_{J for all}_λ

j,n>  and j= , , . . . ,l.Let S:C→C be a closed and totally quasi-φ-asymptotically nonexpan-sive mapping with nonnegative real sequences{νn},{μn}withνn→,μn→as n→ ∞, respectively, and a strictly increasing continuous functionψ :R+_→_R+ _with _ψ_{() = .}

Let f :E→R+ _{be a convex and lower semicontinuous function with C}_⊂_int₍_D₍_f₎₎_and

f() = .Assume that S is uniformly L-Lipschitz continuous andF=F(S)∩(m_i_=EP(θi))∩

(l_j_=A–

j )=∅.For any initial point x∈E,deﬁne C=C and the sequence{xn}in C by ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=JλAll,n◦J Al–

λl–,n◦ · · · ◦J A

λ,nxn, un=Trθmm,n◦T

θm–

rm–,n◦ · · · ◦T θ

r,nzn, yn=J–(αnJx+βnJSnxn+γnJun),

Cn+={v∈Cn:G(v,Jyn)≤αnG(v,Jx) + ( –αn)G(v,Jxn) +ζn}, xn+=ΠCfn+x

()

for each n≥,where{αn},{βn}and{γn}are the sequences in(, )such thatαn+βn+γn= , ζn=νnsupq∈Fψ(G(q,Jxn)) +μnand for each i= , , . . . ,m,{ri,n} ⊂[d,∞)for some d> .

If,for each j= , , . . . ,l,lim infn→∞λj,n> ,limn→∞αn= andlim infn→∞βn< ,then the sequence{xn}converges strongly to a pointΠ_Ff x.

Proof We split the proof into ﬁve steps.

Step . We ﬁrst show thatCnis closed and convex for alln≥. From the deﬁnitions C=Cis closed and convex. Suppose thatCnis closed and convex for alln≥. For any b∈Cn, we know thatG(b,Jyn)≤G(b,Jxn) +ζnis equivalent to the following:

αnb,Jx+ ( –αn)b,Jxn– b,Jyn ≤αnx+ ( –αn)xn–yn+ζn.

Therefore,Cn+is closed and convex for alln≥.

Step . We show thatF⊂Cnfor alln≥. Now, we show by induction thatF⊂Cnfor

alln≥. It is obvious thatF⊂C=C. Suppose thatF⊂Cnfor somen≥. Deﬁneun= Km

nzn, whenKni=T θi ri,nT

θi–

ri–,n· · ·T θ

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whenΔjn:=J Aj λj,n◦J

Aj–

λj–,n◦ · · · ◦J A

λ,nfor allj= , , . . . ,lwithΔ



n=I. Letq∈F. Then we have

G(q,Jun) =G

q,JK_nmzn

≤G(q,Jzn)

=Gq,JΔl_nxn

≤G(q,Jxn). ()

SinceSis a totally quasi-φ-asymptotically nonexpansive mapping, from () we have

G(q,Jyn)

=Gq,αnJx+βnJSnxn+γnJun

=q– αnq,Jx– βn

q,JSnxn

– γnq,Jun

+αnJx+βnJSnxn+γnJun+ ρf(q) ≤ q– αnq,Jx– βn

q,JSn_x n

– γnq,Jun

+αnJxn+βnJSnxn



+γnJun+ ρf(q)

=αnG(q,Jx) +βnG

q,JSnxn

+γnG(q,Jun)

≤αnG(q,Jx) +βn

G(q,Jxn) +νnψ

G(q,Jxn)

+μn

+γnG(q,Jun)

≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jun) +βn

νnψ

G(q,Jxn)

+μn

≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jun) +νnsup q∈F

ψG(q,Jxn)

+μn

≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jun) +ζn ≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jzn) +ζn ≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jxn) +ζn ≤αnG(q,Jx) + (βn+γn)G(q,Jxn) +ζn

=αnG(q,Jx) + ( –αn)G(q,Jxn) +ζn. ()

This shows thatq∈Cn+, which implies thatF⊂Cn+and soF⊂Cnfor alln≥ and the

sequence{xn}is well deﬁned.

Step . We show thatxn→p,yn→p,zn→pandun→pasn→ ∞. Sincef:E→Ris

a convex and lower semi-continuous function, from Lemma , we known that there exist

x*∈E*andα∈Rsuch that

f(x)≥x,x*+α

for allx∈E. Sincexn∈Cn⊂E, it follows that

G(xn,Jx) =xn– xn,Jx+x+ ρf(xn) ≥ xn– xn,Jx+x+ ρ

xn,x*

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=xn– xn,Jx–ρx*

+x+ ρα

≥ xn– xnJx–ρx*+x+ ρα

=xn–Jx–ρx* 

+x–Jx–ρx* 

+ ρα. ()

For allq∈Fandxn=ΠCfnx, we have

G(q,Jx)≥G(xn,Jx)≥

xn–Jx–ρx*



+x–Jx–ρx* 

+ ρα.

That is, {xn} is bounded and so are {G(xn,Jx)} and{yn}. By using the fact thatxn+=

Π_Cf

n+x∈Cn+⊂Cnandxn=Π

f

Cnx, it follows from Lemma  and () that

≤xn+–xn 

≤φ(xn+,xn)≤G(xn+,Jx) –G(xn,Jx). ()

This implies that{G(xn,Jx)}is nondecreasing and solimn→∞G(xn,Jx) exists. Takingn→

∞, we obtain

lim

n→∞φ(xn+,xn) = . ()

Since{xn}is bounded,E is reﬂexive andCnis closed and convex for alln≥. We can

assume thatxnp∈Cnasn→ ∞. From the fact thatxn=ΠCfnx, we get

G(xn,Jx)≤G(p,Jx) ()

for alln≥. Sincef is convex and lower semi-continuous, we have

lim inf

n→∞ G(xn,Jx) =lim infn→∞

xn– xn,Jx+x+ ρf(xn)

≥ p– p,Jx+x+ ρf(p)

=G(xn,Jx). ()

By () and (), we get

G(p,Jx)≤lim inf

n→∞ G(xn,Jx)≤lim supn→∞ G(xn,Jx)≤G(p,Jx).

That is,limn→∞G(xn,Jx) =G(p,Jx), which implies thatxn → pasn→ ∞. SinceE

has the Kadec-Klee property, we obtain

lim

n→∞xn=p. ()

We also have

lim

n→∞xn+=p. ()

From (), we get

lim

n→∞ζn=nlim→∞

νnsup q∈Fψ

G(q,Jxn)

+μn

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From () and (), we havelimn→∞xn–xn+= . SinceJis uniformly norm-to-norm continuous, it follows that

lim

n→∞Jxn–Jxn+= . ()

Moreover, sincexn+=ΠCfn+x∈Cn+⊂Cnand (), we have

G(xn+,Jyn)≤αnG(xn+,Jx) + ( –αn)G(xn+,Jxn) +ζn

is equivalent to the following:

φ(xn+,yn)≤αnφ(xn+,x) + ( –αn)φ(xn+,xn) +ζn. ()

Sincelimn→∞αn= , () and (), we have

lim

n→∞φ(xn+,yn) = . ()

By (), it follows that

yn → p ()

asn→ ∞. SinceJis uniformly norm-to-norm continuous, we obtain

Jyn → Jp ()

asn→ ∞. This implies that{Jyn}is bounded inE*, SinceE*is reﬂexive, we assume that

Jyny*∈E*asn→ ∞. In view ofJ(E) =E*, there existsy∈Esuch thatJy=y*. It follows

that

φ(xn+,yn) =xn+– xn+,Jyn+yn

=xn+– xn+,Jyn+Jyn. ()

Takinglim infn→∞on both sides of the equality above, since · is weak lower

semi-continuous, this yields that

≥ p– p,y*+y*

=p– p,Jy+Jy

=p– p,Jy+y

=φ(p,y). ()

From Remark ,p=y, which implies thaty*=Jp. It follows thatJynJp∈E*asn→ ∞.

From () and the Kadec-Klee property ofE*, we haveJyn→Jpasn→ ∞. Note that J–:E*→E is norm-weak*-continuous, that is,ynipasn→ ∞. From () and the

Kadec-Klee property ofE, we have

lim

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From (), we have

G(q,Jyn)≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jzn) +ζn ≤αnG(q,Jx) + ( –αn)G(q,Jxn) +ζn.

From (), (), () and the conditionslimn→∞αn= ,lim infn→∞βn< , it follows that

for anyq∈F,limn→∞φ(q,zn) =φ(q,p). Letzn=Δlnxnfor alln≥. From Lemma (), it

follows that for anyq∈F,

φ(zn,xn) =φ

Δl_nxn,xn

≤φ(q,xn) –φ

q,Δl_nxn

=φ(q,xn) –φ(q,zn).

Takingn→ ∞on both sides of the inequality above, we have

lim

n→∞φ(zn,xn) = .

From (), it follows that (xn–zn)→ asn→ ∞. Sincexn → pasn→ ∞, we have

zn → p ()

asn→ ∞. SinceJ is uniformly norm-to-norm continuous on bounded subsets ofE, it follows that

Jzn → Jp ()

asn→ ∞. This implies that{Jzn}is bounded inE*. SinceE*is reﬂexive, we can assume thatJznz*∈E*asn→ ∞. In view ofJ(E) =E*, there existsz∈Esuch thatJz=z*, and

so

φ(xn,zn) =xn– xn,Jzn+zn

=xn– xn,Jzn+Jzn. ()

Taking lim infn→∞ on both sides of the equality above, from the weak lower

semi-continuity of the norm · , it follows that

≥ p– p,z*+z*

=p– p,Jz+Jz

=p– p,Jz+z

=φ(p,z). ()

From Remark , we havep=z, which implies thatz*₌_Jp_{and so}_Jz

nJp∈E*asn→ ∞.

From () and the Kadec-Klee property ofE*_{, we have}_Jz

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norm-weak*_{-continuous, that is,}_z

np, from () and the Kadec-Klee property ofE, it

follows that

lim

n→∞zn=p. ()

From (), we have

G(q,Jyn)≤αnG(q,Jx) +βnG(q,Jxn) +γnG(q,Jun) +ζn ≤αnG(q,Jx) + ( –αn)G(q,Jxn) +ζn.

From (), (), () and the conditionslim_n_→∞αn= ,lim infn→∞βn< , it follows that

lim_n_→∞φ(q,un) =φ(q,p). From Lemma , it follows that for anyq∈Fandun=Knmzn, φ(un,xn) =φ

K_nmzn,xn

≤φ(q,zn) –φ

q,K_nmxn

=φ(q,zn) –φ(q,un).

Takingn→ ∞on both sides of the inequality above, we have

lim

n→∞φ(xn,un) = . ()

From (), we have

xn–un→ ()

asn→ ∞. Sincexn → p, we have

un → p ()

asn→ ∞, and so

Jun → Jp ()

asn→ ∞. That is, {Jun}is bounded inE*. SinceE*is reﬂexive, we can assume that

Junu*∈E*asn→ ∞. In view ofJ(E) =E*, there existsu∈Esuch thatJu=u*. It follows

that

φ(xn+,un) =xn+– xn+,Jun+un

=xn+– xn+,Jun+Jun. ()

Takinglim infn→∞on both sides of the equality above, since · is weak lower

semi-continuous, it follows that

≥ p– p,u*+u*

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=p– p,Ju+u

=φ(p,u). ()

From Remark , p=u, that is, u*₌_Jp_{. It follows that} _Ju

nJp∈E*. From () and

the Kadec-Klee property ofE*_{, we have}_Ju

n→Jpasn→ ∞. SinceJ– is norm-weak*

-continuous, that is,unpasn→ ∞. From () and the Kadec-Klee property ofE, we

have

lim

n→∞un=p. ()

Step . We show thatp∈F =F(S)∩(m_i_=EP(θi))∩(

l

j=A–j ). First, we show that p∈l_j_=A–

j . Letzn=Δlnxnfor eachn≥. Then, for anyq∈F, it follows that for each j= , , . . . ,l,

φ(q,zn) = φ

q,Δl_nxn

≤ φq,Δl_n–xn

≤ φq,Δl_n–xn

· · ·

≤ φq,Δj_nxn

. ()

By Lemma , for eachj= , , . . . ,m, we have

φΔj_nxn,xn

≤φ(q,xn) –φ

q,Δj_nxn

≤φ(q,xn) –φ(q,zn). ()

Sincexn→pandzn→pasn→ ∞, we getφ(Δjnxn,xn)→ asn→ ∞for allj= , , . . . ,m.

From (), it follows that

Δj_nxn–xn



→

asn→ ∞for allj= , , . . . ,m. Sincexn → pasn→ ∞, we also have

Δj_nxn→ p ()

asn→ ∞for allj= , , . . . ,m. This implies that for eachj= , , . . . ,m,{Δjnxn}is bounded.

SinceEis reﬂexive, without loss of generality, we can assume thatΔjnxnkasn→ ∞.

SinceCnis closed and convex for eachn≥, it is obvious thatk∈Cn. Again, since

φΔj_nxn,xn

=Δj_nxn



– Δj_nxn,Jxn

+xn,

takinglim infn→∞on both sides of the equality above, we have

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That is,k=pand it follows that for allj= , , . . . ,l,

Δj_nxnp ()

asn→ ∞. Thus, from (), () and the Kadec-Klee property, it follows that

lim

n→∞Δ

j

nxn=p ()

for allj= , , . . . ,m. We also have

lim

n→∞Δ

j–

n xn=p ()

for allj= , , . . . ,m, and so

lim

n→∞Δ

j

nxn–Δjn–xn=  ()

for allj= , , . . . ,m. SinceJis uniformly norm-to-norm continuous on bounded subsets ofEandlim infn→∞λj,n>  for eachj= , , . . . ,l, we have

lim

n→∞



λj,n

JΔj_nxn–JΔjn–xn= . ()

LetΔjnxn=Jλjj,nΔ j–

n xnfor eachj= , , . . . ,l. Then we have

lim

n→∞Aλj,nΔ j–

n xn= lim n→∞



λj,n

JΔj_nxn–JΔjn–xn= . ()

For any (w,w*₎_∈_G₍_A

j) and (Δjnxn,Aλj,nΔ j–

n xn)∈G(Aj) for eachj= , , . . . ,l, it follows from

the monotonicity ofAjthat for alln≥,

w–Δj_nxn,w*–Aλj,nΔ j–

n xn

≥

for allj= , , . . . ,l. Lettingn→ ∞in the inequality above, we getw–p,w*_≥_{ for all}

j= , , . . . ,l. SinceAjis maximal monotone for allj= , , . . . ,l, we obtainp∈

l j=A–j .

Next, we show thatp∈m_i_=EP(θi). For anyq∈Fandun=Knmzn, we observe that

φ(q,un) = φ

q,K_nmzn

≤ φq,K_nm–zn

≤ φq,K_nm–zn

· · ·

≤ φq,K_nizn

. ()

By Lemma , fori= , , . . . ,m, we have

φK_nizn,xn

≤φ(q,xn) –φ

q,K_nizn

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Sincexn→pandun→pasn→ ∞, we getφ(Knizn,xn)→ asn→ ∞for alli= , , . . . ,m.

From (), it follows that

_Ki

nzn–xn



→

asn→ ∞. Sincexn → pasn→ ∞, we also have

K_nizn→ p ()

asn→ ∞. Since{Ki

nzn}is bounded andEis reﬂexive, without loss of generality, we assume

thatKi

nznhasn→ ∞. SinceCnis closed and convex for eachn≥, it is obvious that h∈Cn. Again, since

φK_nizn,xn

=K_nizn– 

K_nizn,Jxn

+xn,

takinglim infn→∞on both sides of the equality above, we have

≥ h– h,Jp+p=φ(h,p). ()

That is,h=pand it follows that for alli= , , . . . ,m, it follows that

K_niznp ()

asn→ ∞. Thus, from (), () and the Kadec-Klee property, it follows that

lim

n→∞K i

nzn=p ()

for alli= , , . . . ,m. We also have

lim

n→∞K

i–

n zn=p ()

for alli= , , . . . ,m, and so

K_nizn–Kni–zn= 

for alli= , , . . . ,m. SinceJis uniformly norm-to-norm continuous, we obtain

lim

n→∞JK

i

nzn–JKni–zn= 

for alli= , , . . . ,m. Fromri,n>  for alli= , , . . . ,m, we have

JKi

nzn–JKni–zn

ri,n →



asn→ ∞and

θi

K_nizn,y

+ 

ri,n

y–K_nizn,JKnizn–JKni–zn

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for ally∈C. Thus, by (A), we have

y–K_nizn JKi

nzn–JKni–zn

ri,n ≥



ri,n

y–K_nizn,JKnizn–JKni–zn

≥–θi

K_nizn,y

≥θi

y,K_nizn

()

for ally∈CandKi

nzn→pasn→ ∞, and soθi(y,p)≤ for ally∈C. For anytwith  <t<

, deﬁneyt=ty+ ( –t)p. Thenyt∈C, which implies thatθi(yt,p)≤ for alli= , , . . . ,m.

Thus, from (A), it follows that

 =θi(yt,yt)≤tθi(yt,y) + ( –t)θi(yt,p)≤tθi(yt,y),

and soθi(yt,y)≥ for alli= , , . . . ,m. From (A), we haveθi(p,y)≥ for ally∈Cand i= , , . . . ,m, that is,p∈EP(θi) for alli= , , . . . ,m. This implies thatp∈

m i=EP(θi).

Finally, we show thatp∈F(S). Since{xn}is bounded, the mappingSis also bounded. Fromyn→pasn→ ∞and (), we have

JSnxn→ Jp ()

asn→ ∞. SinceJ–_:_E*_→_E_{is norm-weak}*_-continuous,

Snxnp ()

asn→ ∞.

On the other hand, in view of (), it follows that

Snxn–p=J

Snxn–Jp≤J

Snxn

–Jp= 

and soSn_x_{n →}_p_{. Since}_E_{has the Kadee-Klee property, we get}

Snxn→p ()

for alln≥. By using the triangle inequality, sinceSis uniformlyL-Lipschitz continuous, we get

Sn+xn–Snxn

≤Sn+xn–Sn+xn++Sn+xn+–xn++xn+–xn+xn–Snxn

≤(L+ )xn+–xn+Sn+xn+–xn++xn–Snxn. ()

SinceSn_x

n→pasn→ ∞, we getSn+xn→pasn→ ∞, and soSSnxn→pasn→ ∞. In

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Π_Cf

nxandpˆ∈F⊂Cn, we also have

G(xn,Jx)≤G(pˆ,Jx)

for alln≥. By the deﬁnitions ofGandf, we know that for anyx∈E,G(ξ,Jx) is convex and lower semicontinuous with respect toξ, and so

G(p,Jx)≤lim inf

n→∞ G(xn,Jx)≤lim sup_n_→∞ G(xn,Jx)≤G(pˆ,Jx).

From the deﬁnition ofΠ_Ff x, sincep∈F, we conclude thatpˆ=p=Π_Ffxandxn→pas

n→ ∞. This completes the proof.

Settingνn= (kn– ),μn=  andψ:R+→ in Theorem , we have the following result.

Corollary  Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.For each i= , , . . . ,m,letθi be a bifunction from C×C toRsatisfying the conditions(A)-(A).Let Aj⊂E×E*be a maximal monotone operator satisfying D(A)⊂C and J_λAj

j,n= (J+λj,nAj)

j,n> and j= , , . . . ,l.Let S:C→C be a closed and quasi-φ-asymptotically nonexpansive mapping.

Let f :E→R+ _{be a convex and lower semicontinuous function with C}_⊂_int₍_D₍_f₎₎_and

f() = .Assume that S is uniformly L-Lipschitz continuous andF=F(S)∩(m_i_=EP(θi))∩

(l_j_=A–

j )=∅.For an initial point x∈E,deﬁne C=C and the sequence{xn}in C by ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

λl–,n◦ · · · ◦J A

θm–

rm–,n◦ · · · ◦T θ

Cn+={v∈Cn:G(v,Jyn)≤αnG(q,Jx) + ( –αn)G(q,Jxn) +ζn}, xn+=ΠCfn+x

()

for all n≥,where{αn},{βn} and{γn}are the sequences in(, )withαn+βn+γn= , ζn=supq∈F(kn– )G(q,Jxn)and,for each i= , , , . . . ,m,{ri,n} ⊂[d,∞)for some d> .

Iflimn→∞αn= ,lim infn→∞βn< andlim infn→∞λj,n>  for all j= , , . . . ,l,then the sequence{xn}converges strongly to a pointΠ_Ff x.

Iff(x) =  for allx∈Ein Theorem , thenG(x,Jy) =φ(x,y) andΠ_Ff =Π_Fand so we have the following corollary.

Corollary  Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.For each i= , , . . . ,m,letθibe a bifunction from C×C toRsatisfying the conditions(A)-(A).Let Aj⊂E×E*be a max-imal monotone operator satisfying D(Aj)⊂C and J

Aj

λj,n= (J+λj,nAj)

j,n> and j=

, , . . . ,l.Let S:C→C be a closed and totally quasi-φ-asymptotically nonexpansive map-ping with nonnegative real sequences{νn},{μn}withνn→,μn→as n→ ∞, respec-tively,and a strictly increasing continuous functionψ:R+→R+ withψ() = .Assume that S is uniformly L-Lipschitz continuous andF=F(S)∩(m_i_=EP(θi))∩(

l

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For an initial point x∈E,deﬁne C=C and the sequence{xn}in C by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

λl–,n◦ · · · ◦J A

θm–

rm–,n◦ · · · ◦T θ

Cn+={v∈Cn:φ(v,yn)≤αnφ(v,x) + ( –αn)φ(v,xn) +ζn}, xn+=ΠCn+x

()

for all n≥,where{αn},{βn} and{γn}are the sequences in(, )withαn+βn+γn= , ζn=νnsupq∈Fψ(φ(q,xn)) +μnand,for each i= , , , . . . ,m,{ri,n} ⊂[d,∞)for some d> .

Iflimn→∞αn= ,lim infn→∞βn< andlim infn→∞λj,n> for each j= , , . . . ,l,then the sequence{xn}converges strongly to a pointΠ_Fx.

Settingνn= (kn– ),μn=  andψ(x) =xin Theorem , we have the following corollary.

Corollary  Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.For each i= , , . . . ,m,letθi be a bifunction from C×C toRsatisfying the conditions(A)-(A).Let Aj⊂E×E*be a maximal monotone operator satisfying D(A)⊂C and JλAjj,n= (J+λj,nAj)

j,n>

and j= , , . . . ,l.Let S:C→C be a closed and quasi-φ-asymptotically nonexpansive mapping.Assume that S uniformly L-Lipschitz continuous andF=F(S)∩(m_i_=EP(θi))∩

(l_j_=A–_j )=∅.For an initial point x∈E,deﬁne C=C and the sequence{xn}in C by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

λl–,n◦ · · · ◦J A

θm–

rm–,n◦ · · · ◦T θ

Cn+={v∈Cn:φ(v,yn)≤αnφ(q,x) + ( –αn)φ(q,xn) +ζn}, xn+=ΠCn+x

()

for all n≥,where{αn},{βn} and{γn}are the sequences in(, )withαn+βn+γn= , ζn=supq∈F(kn– )φ(q,xn)and,for each i= , , , . . . ,m,{ri,n} ⊂[d,∞)for some d> .

Iflimn→∞αn= ,lim infn→∞βn< andlim infn→∞λj,n> for each j= , , . . . ,l,then the sequence{xn}converges strongly to a pointΠ_Fx.

Competing interests

The author 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 and Statistics, Faculty of Science, Thaksin University, Phatthalung, 93110, Thailand.} 2_{Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangmod,}

Thrungkru, Bangkok, 10140, Thailand.3_{Department of Mathematics Education and RINS, Gyeongsang National}

University, Chinju, 660-701, Korea.

Acknowledgements

This work was supported by Thaksin University Research Fund and YJ Cho was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170).

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doi:10.1186/1687-1812-2013-199