Strong convergence of gradient projection method for generalized equilibrium problem in a Banach space

(1)

R E S E A R C H

Open Access

Strong convergence of gradient

projection method for generalized

equilibrium problem in a Banach space

Mohammad Farid

1*

_{, Syed Shakaib Irfan}

2

_{, Mohammad Firdosh Khan}

2

_{and Suhel Ahmad Khan}

3

*_{Correspondence:} [email protected] 1_{Unaizah College of Engineering,} Qassim University, Unaizah, 51911, Saudi Arabia

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

Abstract

In this paper, we propose and analyze a hybrid iterative method for ﬁnding a

common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a variational inequality problem, and the set of ﬁxed points of a relatively nonexpansive mapping in a real Banach space. Further, we prove the strong convergence of the sequences generated by the iterative scheme. Finally, we derive some consequences from our main result. Our work is an improvement and extension of some previously known results recently obtained by many authors.

Keywords: generalized equilibrium problem; variational inequality problem; ﬁxed point problem; relatively nonexpansive mappings; iterative scheme

1 Introduction

LetXbe a real Banach space with its dual spaceX∗, let·,·be the duality pairing between XandX∗, and let · denote the norm ofXandX∗. LetKbe a nonempty closed convex subset ofX, and let X=∅be the set of all subsets ofX.

LetG,ξ :K×K →Rbe bifunctions. The generalized equilibrium problem (GEP) is ﬁndingx∈Ksuch that

G(x,y) +ξ(y,x) –ξ(x,x)≥, ∀y∈K. (.)

We denote the solution set of GEP (.) by Sol(GEP(.)). Problem (.) includes ﬁxed point problems, optimization problems, variational inequality problems, Nash equilib-rium problems,etc. as particular cases. In recent past, many iterative methods have been proposed to solve GEP (.); see, for example, [–].

Forξ= , GEP (.) reduces to the following equilibrium problem (EP): Findx∈Ksuch that

G(x,y)≥ for ally∈K. (.)

Problem (.) was introduced and studied by Blum and Oettli []. The variational inequality problem (VIP) is to ﬁndx∈Ksuch that

Sx,y–x ≥ for ally∈K, (.)

(2)

where S:K →X∗ is a nonlinear mapping. We denote the solution set of VIP (.) by Sol(VIP(.)).

A mappingS:K→X∗is said to be

(i) monotone ifx–y,Sx–Sy ≥for allx,y∈K;

(ii) γ-inverse strongly monotone if there exists a positive real numberγ such that x–y,Sx–Sy ≥γSx–Sy_{for all}_x,_y_∈_K_;

(iii) Lipschitz continuous if there exists a constantL> such thatSx–Sy ≤Lx–y.

IfSisγ-inverse strongly monotone, then it is Lipschitz continuous with constant_γ, that is,Sx–Sy ≤ _γx–yfor allx,y∈K.

Theﬁxed point problem(FPP):

Findx∈K such thatx∈Fix(T), (.)

whereT:K→Kis a nonlinear mapping, andFix(T) is the ﬁxed point set.

In , Takahashi and Zembayashi [] studied weak and strong convergence theorems for ﬁnding a common solution of EP (.) and FPP (.) of a relatively nonexpansive map-ping in a real Banach space. Later on, Petrotet al. [] extended the work [] by using the hybrid projection method, which plays an important role for establishing strong conver-gence results.

Nadezhkinaet al. [] proposed a convex combination of a nonexpansive mapping and the extragradient method and considered the iterative scheme by the hybrid method. They proved the strong convergence theorem in a Hilbert space.

Very recently, in , Nakajoet al. [] proposed a composition and convex combina-tion of a relatively nonexpansive mapping and the gradient method. Further, they proved the strong convergence to a common element of solutions of the variational inequality problem and ﬁxed point problem by using the hybrid method.

Motivated and inspired by the recent work of Takahashi and Zembayashi [], Petrotet al. [], Nadezhkinaet al. [], and Nakajoet al. [], we propose an iterative scheme to ﬁnd the common solution of GEP (.), VIP (.), and FPP (.) for a relatively nonexpansive map-ping in a real Banach space. Further, by using the hybrid projection we prove the strong convergence of the sequences generated by the iterative algorithm, which improves and extends the corresponding results of [, , –].

2 Preliminaries

Now, we use the following results and deﬁnitions to prove our main result. The normalized duality mapping is deﬁned as

J(u) =v∈X∗:u,v=u=v

for everyu∈X, whereJ:X→X∗_.

The mappingρX: [,∞)→[,∞) deﬁned by

ρX(s) =sup

u+v+u–v

 –  :u= ,v=s

is called the modulus of smoothness ofX. The spaceXis said to be smooth ifρX(s) >  for

alls> , andXis called uniformly smooth ifρX(s)

s → ass→. A Banach spaceXis said

(3)

well known that ifXisq-uniformly smooth, thenq≤, andXis uniformly smooth. Note that ifXis uniformly smooth, thenJis uniformly continuous on bounded subsets ofX.

The modulus of convexity ofXis the functionδX: (, ]→[, ] deﬁned by

δX(t) =inf

 –

x+y:x=y= ,x–y=t

fort∈(, ]. A Banach spaceXis said to be uniformly convex ifδX(t) >  for allt∈(, ].

Letp> . The spaceXis said to bep-uniformly convex if there exists a constantc>  such thatδX(t)≥ctp for allt∈(, ]. Note that everyp-uniformly convex space is uniformly

convex (for more details, see []).

LetXbe a smooth, strictly convex, and reﬂexive Banach space.

Following Takahashi and Zembayashi [], a pointx∈K is said to be an asymptotic

ﬁxed point of T if K contains a sequence {xn} that converges weakly tox such that lim_n→∞xn–Txn= . The set of asymptotic ﬁxed points ofT is denoted byFix(T).

A mappingTfromKinto itself is said to be relatively nonexpansive ifFix(T)=∅,Fix(T) =

Fix(T), andφ(x,Tx)≤φ(x,x) for allx∈Kandx∈Fix(T), whereφ:X×X→R+is the

Lyapunov functional deﬁned by

φ(u,v) =u– u,Jv+v foru,v∈X. (.)

The generalized projectionK:X→Kis deﬁned as

K(u) =inf

x∈Kφ(u,x) foru∈X,

whereφ(u,x) is deﬁned by (.) (for more details, see []).

Lemma .([, ]) Let X be a smooth,strictly convex,and reﬂexive Banach space,and let K=∅be a closed convex subset of X.Then,the following hold:

(i) φ(x,Ku) +φ(Ku,u)≤φ(x,u)for allx∈K,u∈X. (ii) Foru∈Xandx∈K,we have

x=K(u) ⇔ x–y,Ju–Jx ≥ for ally∈K.

Remark .([])

(i) Using (.), we get

u–v≤φ(u,v)≤u+v for allu,v∈X.

(ii) IfX=His a real Hilbert space, thenφ(u,v) = (u–v)_{, and}

K=PK, the metric projection ofHontoK.

(iii) IfXis a smooth, strictly convex, and reﬂexive Banach space, thenφ(u,v) = for

u,v∈Xif and only ifu=v.

Lemma .([]) Let X be a smooth Banach space.Then,the following are equivalent:

(i) Xis-uniformly convex.

(ii) There exists a constantc> such thatu+v≥ u+ v,Ju+cvfor all

(4)

Lemma .([]) Let X be a-uniformly convex and smooth Banach space.Thenφ(u,v)≥ cu–vfor all u,v∈X,where cis the constant in Lemma..

Lemma .([]) Let X be a-uniformly convex Banach space.Then,for all u,v∈X,we have

u–v ≤  c

Ju–Jv,

where J is the normalized duality mapping of X,and <c≤.

Lemma .([]) Let K=∅be a closed convex subset of a smooth,strictly convex,and reﬂexive Banach space X,and let T:K→K be a relatively nonexpansive mapping.Then,

Fix(T)is closed and convex.

Lemma .([]) Let X be a smooth and uniformly convex Banach space,and let{un} and{vn}be sequences in X such that either{un}or{vn}is bounded.Iflim_n→∞φ(un,vn) = ,

thenlimn→∞un–vn= .

Lemma .([]) Let K be a nonempty closed convex subset of a Banach space X,and let S be a monotone and hemicontinuous operator of K into X∗.Deﬁne the mapping M⊂X×X∗ as

M(z) = S(z) +NK(z) if z∈K,

∅ if z∈/K,

where NK(z) :={u∈X∗:z–x,u ≥,∀x∈K}is the normal cone to K at z∈K.Then,M

is maximal monotone,and M–_{() =}_{Sol(VIP(.)).}

Lemma . ([, ]) Let X be a uniformly convex Banach space, and let r> .Then there exists a strictly increasing,continuous,and convex function g: [, r]→Rsuch that g() = and

αx+ ( –α)y≤αx+ ( –α)y–α( –α)gx–y

for all x,y∈Brandα∈[, ],where Br={u∈X:u ≤r}.

Lemma .([]) Let X be a smooth and uniformly convex Banach space,and let r> . Then there exists a strictly increasing,continuous,and convex function g: [, r]→Rsuch that g() = and

gx–y≤φ(x,y) for all x,y∈Br.

The functionF:X×X∗→Rdeﬁned by

Fu,u∗=u– u,u∗+u∗ foru∈Xandu∗∈X∗

(5)

Lemma .([]) Let X be a reﬂexive strictly convex and smooth Banach space with its dual X∗.Then

Gu,u∗+ J–u∗–u,v∗≤Gu,u∗+v∗ for all u∈X and u∗,v∗∈X∗.

Assumption . LetGandξsatisfy the following conditions:

(i) G(x,x) = forx∈K.

(ii) Gis monotone, that is,G(x,y) +G(y,x)≤forx,y∈K. (iii) For eachy∈K,x→G(x,y)is weakly upper semicontinuous. (iv) For eachx∈K,y→G(x,y)is convex and lower semicontinuous.

(v) ξ(·,·)is weakly continuous, andξ(·,y)is convex. (vi) ξ is skew-symmetric, that is,

ξ(x,x) –ξ(x,y) +ξ(y,y) –ξ(y,x)≥ for allx,y∈K.

Theorem . Let K be a nonempty closed and convex subset of a smooth,strictly convex, and reﬂexive Banach space X.Let G,ξ:K×K→Rbe nonlinear mappings satisfying As-sumption..For t> and u∈X,deﬁne the mappingϒt:X→K as follows:

ϒt(u) =

z∈K:G(z,y) +ξ(y,z) –ξ(z,z) +

ty–z,Jz–Ju ≥,∀y∈K

.

Then,the following conclusions hold:

(i) ϒtis single-valued;

(ii) ϒtis ﬁrmly nonexpansive mapping,that is,for allu,u∈X, ϒtu–ϒtu,Jϒtu–Jϒtu ≤ ϒtu–ϒtu,Ju–Ju;

(iii) Fix(ϒt) =Sol(GEP(.));

(iv) Sol(GEP(.))is closed and convex.

Proof (i) We claim thatϒtis single-valued. Indeed, forx∈Kandt> , letz,z∈ϒt(x).

Then

G(z,y) +ξ(y,z) –ξ(z,z) +



ty–z,Jz–Jx ≥ for ally∈K (.)

and

G(z,y) +ξ(y,z) –ξ(z,z) +



ty–z,Jz–Jx ≥ for ally∈K. (.)

Lettingy=zin (.) andy=zin (.) and then adding, we have

G(z,z) +G(z,z) +ξ(z,z) –ξ(z,z) +ξ(z,z) –ξ(z,z) +



tz–z,Jz–Jz ≥.

SinceGis monotone,ξ is skew symmetric, and sincet> , we have

z–z,Jz–Jz ≥.

(6)

(ii) For anyu,u∈X, letx=ϒtuandx=ϒtu. Then

G(x,y) +ξ(y,x) –ξ(x,x) +



ty–x,Jx–Ju ≥ for ally∈K, (.)

and

G(x,y) +ξ(y,x) –ξ(x,x) +



ty–x,Jx–Ju ≥ for ally∈K. (.)

By puttingy=xin (.) andy=xin (.) and taking their sum, we have

G(x,x) +G(x,x) +ξ(x,x) –ξ(x,x) +ξ(x,x) –ξ(x,x)

+

tx–x,Jx–Jx–Ju+Ju ≥.

Using the monotonicity ofGand properties ofξ, we have 

tx–x,Jx–Jx–Ju–Ju ≥.

Hence, we have

x–x,Jx–Jx+x–x,Ju–Ju ≥

or

x–x,Jx–Jx ≤ x–x,Ju–Ju,

that is,

ϒtu–ϒtu,Jϒtu–Jϒtu ≤ ϒtu–ϒtu,Ju–Ju. (.)

Thus,ϒtis a ﬁrmly nonexpansive mapping.

(iii) Letx∈Fix(ϒt). Then

G(x,y) +ξ(y,x) –ξ(x,x) +

ty–x,Jx–Jx ≥ for ally∈K,

and so

G(x,y) +ξ(y,x) –ξ(x,x)≥ for ally∈K. Thus,x∈Sol(GEP(.)).

Letx∈Sol(GEP(.)). Then

G(x,y) +ξ(y,x) –ξ(x,x)≥ for ally∈K, and so

G(x,y) +ξ(y,x) –ξ(x,x) +

ty–x,Jx–Jx ≥ for ally∈K.

(7)

(iv) First, we show thatϒtis a relatively nonexpansive mapping.

Using the deﬁnition ofξ, for anyu,u∈X, we have

φ(ϒtu,ϒtu) +φ(ϒtu,ϒtu)

= ϒtu– ϒtu,Jϒtu– ϒtu,Jϒtu+ ϒtu

= ϒtu,Jϒtu–Jϒtu+ ϒtu,Jϒtu–Jϒtu

= ϒtu–ϒtu,Jϒtu–Jϒtu

and

φ(ϒtu,u) +φ(ϒtu,u) –φ(ϒtu,u) –φ(ϒtu,u)

=ϒtu– ϒtu,Ju+u+ϒtu+u

– ϒtu,Ju–ϒtu+ ϒtu,Ju–u

–ϒtu+ ϒtu,Ju–u

= ϒtu,Ju–Ju– ϒtu,Ju–Ju

= ϒtu–ϒtu,Ju–Ju.

Sinceϒtis ﬁrmly nonexpansive, from the above two equalities we have

φ(ϒtu,ϒtu) +φ(ϒtu,ϒtu)≤φ(ϒtu,u) +φ(ϒtu,u) –φ(ϒtu,u) –φ(ϒtu,u).

Thus,

φ(ϒtu,ϒtu) +φ(ϒtu,ϒtu)≤φ(ϒtu,u) +φ(ϒtu,u).

Takingu=u∈Fix(ϒt), we have

φ(u,ϒtu)≤φ(u,u).

Further, we prove thatFix(ϒt) = Sol(GEP(.)).

Let x ∈ Fix(ϒt). Then there exists a sequence {un} ⊂ X such that un x and lim_n→∞un–ϒtun= . Thus,ϒtunx. Hence, we getx∈K.

SinceJis uniformly continuous on bounded sets, we have

lim n→∞

Jun–Jϒtun

t = , t> .

From the deﬁnition ofϒt, for anyy∈K, we have

G(ϒtun,y) +ξ(y,ϒtun) –ξ(ϒtun,ϒtun) +



ty–ϒtun,Jϒtun–Jun ≥. Letyp= ( –p)x+pyforp∈(, ]. Sincey∈Kandx∈K, we haveyp∈K, and thus

G(ϒtun,yp) +ξ(yp,ϒtun) –ξ(ϒtun,ϒtun) +



(8)

Sinceξ is weakly continuous andGis weakly lower semicontinuous in the second argu-ment, lettingn→ ∞, we get

G(x,yp) +ξ(yp,x) –ξ(x,x)≥,

ξ(yp,x) –ξ(x,x)≥G(yp,x).

Forp> , we have

 =G(yp,yp)

≤pG(yp,y) + ( –p)G(yp,x) ≤pG(yp,y) + ( –p)

ξ(yp,x) –ξ(x,x)

≤pG(yp,y) + ( –p)p

ξ(y,x) –ξ(x,x)

≤pG(yp,y) + ( –p)

ξ(y,x) –ξ(x,x). Dividing byp>  and lettingp→+, we have

G(x,y) +ξ(y,x) –ξ(x,x)≥.

This implies thatx∈Sol(GEP(.)), and henceFix(ϒt) = Sol(GEP(.)) =Fix(ϒt). Thus,

ϒtbe a relatively nonexpansive mapping. By Lemma ., Sol(GEP(.)) =Fix(ϒt) is closed

and convex.

Next, we have the following lemma whose proof is on the similar lines of the proof of Lemma . [] and hence omitted.

Lemma . Let X,K,G,ξ,ϒt be same as in Theorem.,and let t> .Then,for x∈X

and u∈Fix(ϒt),we have

φ(u,ϒtx) +φ(ϒtx,x)≤φ(u,x).

3 Main result

Now, we prove the following convergence theorem.

Theorem . Let X be a-uniformly convex and uniformly smooth Banach space,and let K be a nonempty closed and convex subset of X.Let S:K→X∗be aγ-inverse strongly monotone mapping with constantγ ∈(, ),let G,ξ:K×K→Rbe nonlinear mappings satisfying Assumption.,and let T:K→K be a relatively nonexpansive mapping such that:=Sol(GEP(.))∩Sol(VIP(.))∩Fix(T)=∅.Let the iterative sequence{xn}be gen-erated as follows:

x=x∈K,

zn=

K

J–(Jxn–λnSxn),

yn=J–

θnJxn+ ( –θn)JTzn

,

(9)

such that

G(un,y) +ξ(y,un) –ξ(un,un) +

 tn

y–un,Jun–Jyn ≥, ∀y∈K,

Pn=

z∈K:φ(z,un)≤φ(z,xn) – ( –θn)φ(zn,xn) – ( –θn)λnzn–z,Sxn–Szn

,

Qn=

z∈K:xn–z,Jx–Jxn ≥

,

xn+=

Pn∩Qn

x, ∀n∈N∪ {},

where J is the normalized duality mapping on X,tn∈(,∞),and{λn}and{θn}are the

sequences in(,∞)and(,)satisfying the following:

(i)  <lim inf_n→∞λn≤lim supn→∞λn<c

 γ

 ,wherecis the constant in Lemma.; (ii)  <lim inf_n→∞θn≤lim supn→∞θn< .

Then,{xn}converges strongly tox,wherex is the generalized projection of X onto. Proof SinceT is a relatively nonexpansive mapping fromK into itself, it follows from Lemma . and Theorem .(iv) thatis closed and convex. First, we show thatPn∩Qn

is closed and convex for alln∈N∪ {}. By the deﬁnition ofQnit is closed and convex.

Further, by the deﬁnition ofφwe observe thatPnis closed and

φ(z,un)≤φ(z,xn) – ( –θn)φ(yn,xn) – ( –θn)λnyn–z,Sxn–Syn, z– z,Jxn+xn– ( –θn)φ(yn,xn) – ( –θn)λnyn–z,Sxn–Syn

–z+ z,Jun–un≥,

z,Jun–Jxn– ( –θn)λnyn–z,Sxn–Syn+xn–un– ( –θn)φ(yn,xn)≥,

and hencePnis closed and convex for alln∈N∪ {}. Thus,Pn∩Qnis closed and convex

for alln∈N∪ {}.

Next, we show that⊂Pn∩Qnand{xn}is well deﬁned.

Letx∗∈. Thenx∗∈Sol(VIP(.)), that is,zn–x∗,Szn ≥ zn–x∗,Sx∗ ≥.

Sincex∗∈, using Lemma ., we have

zn–x∗,Jxn–Jzn–λnSxn

≥.

Thus,

zn–x∗,Jxn–Jzn

≥λn

zn–x∗,Sxn

=λn

zn–x∗,Sxn–Szn

+λn

zn–x∗,Szn

≥λn

for eachn∈N∪ {}, which implies

–zn–x∗,Jxn–Jzn

≤–λn

,

x∗,Jxn–Jzn

– zn,Jxn–Jzn ≤–λn

,

+ zn,Jzn– zn,Jxn ≤–λn

(10)

+ zn– zn,Jxn ≤–λn

, x∗– x∗,Jzn

+zn≤x∗– x∗,Jxn

+xn–zn+ zn,Jxn

–xn– λn

,

φx∗,zn

≤φx∗,xn

–φ(zn,xn) – λn

. (.)

Sinceun=ϒtnynfor alln∈N∪ {}andϒtnis relatively nonexpansive, we have

φx∗,un

=φx∗,ϒtnyn

≤φx∗,yn

. (.)

Now, we estimate

φx∗,yn

=φx∗,J–θnJxn+ ( –θn)JTzn

=x∗– x∗,θnJxn+ ( –θn)JTzn

+θnJxn+ ( –θn)JTzn 

≤x∗– θn

x∗,Jxn

– ( –θn)

x∗,JTzn

+θnxn+ ( –θn)Tzn ≤θnφ

x∗,xn

+ ( –θn)φ

x∗,Tzn

≤θnφ

x∗,xn

+ ( –θn)φ

x∗,Tzn

≤θnφ

x∗,xn

+ ( –θn)φ

x∗,zn

. (.)

By (.), (.), and (.) we observe that

φx∗,un

≤φx∗,xn

– ( –θn)φ(zn,xn) – ( –θn)λn

.

This implies thatx∗∈Pn. Therefore,⊂Pnfor alln∈N∪ {}.

Next, we show by induction that⊂Pn∩Qn for alln∈N∪ {}. SinceQ=K, we

have ⊂P∩Q. Suppose that⊂Pk∩Qk for somek∈N∪ {}. Then there exists

xk+∈Pk∩Qksuch thatxk+=

Pk∩Qkx. From the deﬁnition ofxk+we have, for allz∈

Pk∩Qk,

xk+–z,Jx–Jxk+ ≥.

Since⊂Pk∩Qk, we have

xk+–z,Jx–Jxk+ ≥ for allz∈,

and hencez∈Qk+. So, we have⊂Qk+. Therefore, we have⊂Pk+∩Qk+.

Thus, we have that⊂Pn∩Qnfor alln∈N∪ {}. This means that{xn}is well-deﬁned.

(11)

By the deﬁnition ofQnwe getxn=

Qnx. Usingxn=

Qnxand Lemma ., we have,

for allx∗∈⊂Qn,

φ(xn,x) =φ

Qn

x,x

≤φx∗,x–φ

x∗,

Qn

x

≤φx∗,x.

Thus{φ(xn,x)}is bounded. Therefore{xn}is bounded.

Lettingx∗∈, we have

Sxn ≤Sxn–Sx∗+Sx∗

≤ 

γxn–x

∗₊_Sx∗_.

So,{Sxn}is bounded.

Fromφ(zn,J–(Jxn–λnSxn))≤φ(x∗,J–(Jxn–λnSxn)) we have

≥ zn– zn,Jxn–λnSxn–x∗ 

+ x∗,Jxn–λnSxn

≥ zn– znxn+λnSxn

–x∗– x∗xn+λnSxn. DenoteM=sup{xn,Sxn}. Now, we have

zn–Mzn–x∗–Mx∗≤ for alln∈N∪ {}.

Thus{zn}is bounded. Sincexn+=

Pn∩Qnx∈Pn∩Qn⊂Qnandxn=

Qnx, from the deﬁnition of

Qn we

have

φ(xn,x)≤φ(xn+,x) for alln∈N∪ {}.

Thus{φ(xn,x)}is nondecreasing. So, the limit of{φ(xn,x)}exists. By the construction of

Qnwe haveQm⊂Qnandxm=Qmx∈Qnform≥n. It follows that

φ(xm,xn) =φ

xm,

Qn

x

≤φ(xm,x) –φ

Qn

x,x

=φ(xm,x) –φ(xn,x). (.)

(12)

andK is closed and convex, we can assume thatxn→x∗∈K asn→ ∞. From (.) we

get

φ(xn+,xn)≤φ(xn+,x) –φ(xn,x) for alln∈N∪ {},

which implies

lim

n→∞φ(xn+,xn) = . (.)

Using Lemma ., we get

lim

n→∞xn+–xn= . (.)

By Lemma . andxn+∈Pnwe estimate

φ(xn+,un)≤φ(xn+,xn) –φ(zn,xn) – λnzn–xn+,Sxn–Szn

=φ(xn+,xn) –φ(zn,xn) – λnzn–xn,Sxn–Szn

– λnxn–xn+,Sxn–Szn

≤φ(xn+,xn) –φ(zn,xn) + λnzn–xnSxn–Szn

+ λnxn–xn+Sxn–Szn ≤φ(xn+,xn) +

λn

γ –c

xn–zn+

λn

γ xn–xn+xn–zn.

Using (.), (.), and the inequalitysup_n∈Nλn< c

γ

 , we have

lim n→∞

c–

λn

γ

xn–zn=_n→∞lim φ(xn+,un) = ,

which implies

lim

n→∞xn–zn=n→∞lim xn+–un= . (.)

Using (.) and (.), we have

lim

n→∞xn–un= . (.)

The uniform continuity ofJimplies that

lim

(13)

Using the property ofφand Lemma ., we have, for allx∗∈,

φx∗,yn

=φx∗,J–θnJxn+ ( –θn)JTzn

=x∗– x∗,θnJxn+ ( –θn)JTzn

+θnJxn+ ( –θn)JTzn 

≤x∗– θn

x∗,Jxn

– ( –θn)

x∗,JTzn

+θnJxn + ( –θn)JTzn–θn( –θn)g

Jxn–JTzn

=θnφ

x∗,xn

+ ( –θn)φ

x∗,zn

–θn( –θn)g

Jxn–JTzn

. (.)

Next, we estimate

φx∗,zn

=φ

x∗,J–_(Jx

n–λnSxn)

≤φx∗,J–(Jxn–λnSxn)

=Fx∗,Jxn–λnSxn

≤Fx∗, (Jxn–λnSxn) +λnSxn

– J–(Jxn–λnSxn) –p,λnSxn

=Fx∗,Jxn

– λn

J–(Jxn–λnSxn) –x∗,Sxn

=φx∗,xn

– λn

xn–x∗,Sxn

– λn

J–(Jxn–λnSxn) –xn,Sxn

≤φx∗,xn

– λn

xn–x∗,Sxn–Sx∗

– λn

xn–x∗,Sx∗

+ J–(Jxn–λnSxn) –xn,Sxn

≤φx∗,xn

– λnγSxn–Sx∗ 

+ J–(Jxn–λnSxn) –xn,Sxn

.

From this, using Lemma . and the inequalitySx ≤ Sx–Sx∗forx∈Kandx∗∈, we have

φx∗,zn

≤φx∗,xn

– λnγSxn–Sx∗ 

+ 

c 

λ_nSxn–Sx∗ 

=φx∗,xn

+ λn

 c_λn–γ

Sxn–Sx∗. (.)

From (.) and (.) we have

φx∗,yn

≤φx∗,xn

+ ( –θn)λn

Sxn–Sx∗ 

–θn( –θn)g

Jxn–JTzn

. (.)

Using (.) in (.), we have

φx∗,un

≤φx∗,xn

+ ( –θn)λn

Sxn–Sx∗ 

–θn( –θn)g

Jxn–JTzn

(14)

Sinceλn≤c

 γ

 , we get

θn( –θn)g

Jxn–JTzn

≤φx∗,xn

–φx∗,un

. (.)

Now,

φx∗,xn

–φx∗,un

=xn–un– x∗,Jxn–Jun

≤ xn–un

xn+un+ x∗Jxn–Jun

.

It follows from (.) and (.) that

φx∗,xn

–φx∗,un

→ asn→ ∞. (.)

Thus, from (.) and (.) we have

gJxn–JTzn

→ asn→ ∞.

Using Lemma ., we obtain

Jxn–JTzn → asn→ ∞.

SinceJ–_{is uniformly norm-to-norm continuous, we have}

xn–Tzn → asn→ ∞. (.)

From (.) we have

( –θn)λn

γ –  c 

λn

Sxn–Sx∗ 

≤φx∗,xn

–φx∗,un

,

which implies that

lim

n→∞Sxn–Sx

∗_{= .} _(.)

Using Lemmas . and . and (.), we estimate

φ(xn,zn) =φ

xn,

K

J–(Jxn–λnSxn)

≤φxn,J–(Jxn–λnSxn)

=F(xn,Jxn–λnSxn)

≤Fxn, (Jxn–λnSxn) +λnSxn

– J–(Jxn–λnSxn) –xn,λnSxn

=φ(xn,xn) + 

J–(Jxn–λnSxn) –xn, –λnSxn

= J–(Jxn–λnSxn) –xn, –λnSxn

(15)

By Lemma ., using the inequalitySx ≤ Sx–Sx∗forx∈K,x∗∈, we have

φ(xn,zn)≤

 c

γ

Sxn–Sx∗ 

.

It follows from (.) and Lemma . that

lim

n→∞xn–zn= . (.)

Thuszn→x∗asn→ ∞.

Sinceun=ϒtnyn, using Lemma . and (.), we get

φ(un,yn) =φ(ϒtnyn,yn)

=φx∗,yn

–φx∗,un

≤θnφ

x∗,xn

+ ( –θn)φ

x∗,zn

–φx∗,un

≤θnφ

x∗,xn

+ ( –θn)

φx∗,xn

+ λn

 c 

λn–γ

Sxn–Sx∗ 

–φx∗,un

≤φx∗,xn

+ ( –θn)λn

Sxn–Sx∗–φ

x∗,un

.

From this, using (.) and the restrictions on the sequences{θn}and{λn}, we get

lim

n→∞φ(un,yn) = .

By Lemma .,

lim

n→∞un–yn= .

Using the uniform continuity ofJ, we have

lim

n→∞Jun–Jyn= . (.)

From (.) and (.) we get

Tzn–zn ≤ Tzn–xn+zn–xn → asn→ ∞,

which implies thatx∗∈Fix(T).

Further, we show thatx∗∈Sol(VIP(.)). Since{xn}is bounded, there exists a subse-quence{xnk} of {xn} that converges weakly to x

∗_{. Deﬁne the mapping} _M_⊂_X_×_X∗ _as

follows:

M(z) = S(z) +NK(z) ifz∈K,

(16)

where NK(z) :={w∈X:z–x,w ≥,∀x∈K} is the normal cone to K at z∈K. By

Lemma ., M is a maximal monotone operator, and M–_{() =}_VI_(K,_{S). Let (z,}_w)_∈ graph(M). Sincew∈M(z) =S(z) +NK(z), we getw–Sz∈NK(z). Sincezn∈K, we obtain

z–znk,w–Sz ≥. (.)

On the other hand,znk=

KJ–(Jxnk–λnkSxnk), and using Lemma ., we obtain

z–znk,Jznk– (Jxnk–λnkSxnk)

≤,

and thus

z–znk,

Jxnk–Jznk

λnk

–Sxnk

≤. (.)

Therefore, it follows from the monotonicity ofS, (.), and (.) that

z–znk,w ≥ z–znk,Sz

≥ z–znk,Sz+

z–znk,

Jxnk–Jznk

λnk

–Sxnk

=

z–znk,Sz–Sxnk+

Jxnk–Jznk

λnk

=z–znk,Sz–Sznk+z–znk,Sznk–Sxnk+

z–znk,

Jxnk–Jznk

λnk

≥–z–znkSznk–Sxnk–z–znk

Jxnk–Jznk

a

≥–

γz–znkznk–xnk–z–znk

Jxnk–Jznk

a

≥–ρ



γznk–xnk+

Jxnk–Jznk

a

,

whereρ=sup_k∈N{z–znk}anda<lim supλn. Taking the limit ask→ ∞and using the

fact that{z–znk}k∈N is bounded, we see thatz–x∗,w ≥. Thusx∗∈Sol(VIP(.)).

Next, we prove thatx∗∈Sol(GEP(.)). The relationun=ϒtnynimplies that

G(un,y) +ξ(y,un) –ξ(un,un) +

 tn

y–un,Jun–Jyn ≥ for ally∈K.

Letyp= ( –p)x∗+pyforp∈(, ]. Sincey∈Kandx∗∈K, we getyp∈K, and hence

G(un,yp) +ξ(yp,un) –ξ(un,un) +

 tn

yp–un,Jun–Jyn ≥.

Using (.) andlim infn→∞tn> , we have

lim n→∞

Jyn–Jϒtnyn

tn

(17)

Further, sinceξis weakly continuous andGis weakly lower semicontinuous in the second argument, lettingn→ ∞, we get

Gx∗,yp

+ξyp,x∗

–ξx∗,x∗≥,

ξyp,x∗

–ξx∗,x∗≥Gyp,x∗

.

Now, forp> ,

 =G(yp,yp)

≤pG(yp,y) + ( –p)G

yp,x∗

≤pG(yp,y) + ( –p)

ξyp,x∗

–ξx∗,x∗

≤pG(yp,y) + ( –p)p

ξy,x∗–ξx∗,x∗

≤pG(yp,y) + ( –p)

ξy,x∗–ξx∗,x∗. Dividing byp>  and lettingp→+, we have

Gx∗,y+ξy,x∗–ξx∗,x∗≥.

Thus,x∗∈Sol(GEP(.)), and hencex∗∈.

Finally, we have the following consequences of Theorem ..

Corollary . Let X be a-uniformly convex and uniformly smooth Banach space,and let K be a nonempty closed and convex subset of X.Let S:K→X∗be aγ-inverse strongly monotone mapping with constantγ ∈(, ),let G:K×K→Rbe a nonlinear mapping satisfying Assumption.(i)-(iv),and let T:K→K be a relatively nonexpansive mapping such that:=Sol(EP(.))∩Sol(VIP(.))∩Fix(T)=∅.Let the iterative sequence{xn}be generated as follows:

x=x∈K,

zn=

K

yn=J–

,

un∈K

such that

G(un,y) +

 tn

y–un,Jun–Jyn ≥, ∀y∈K,

Pn=

,

Qn=

,

xn+=

Pn∩Qn

(18)

where J is the normalized duality mapping on X,tn∈(,∞),and{λn}and{θn}are the

sequences in(,∞)and(,)satisfying the following:

(i)  <lim inf_n→∞λn≤lim supn→∞λn< c_γ

Then,{xn}converges strongly tox.

Proof The proof follows by takingξ=  in Theorem .. Corollary . Let X be a-uniformly convex and uniformly smooth Banach space,and

let K be a nonempty closed and convex subset of X.Let S:K→X∗be aγ-inverse strongly monotone mapping with constantγ ∈(, ),and let T:K→K be a relatively nonexpan-sive mapping such that:=Sol(VIP(.))∩Fix(T)=∅.Let the iterative sequence{xn}be generated as follows:

x=x∈K,

zn=

K

yn=J–

,

Pn=

,

Qn=

,

xn+=

Pn∩Qn

x, ∀n∈N∪ {},

where J is the normalized duality mapping on X,and{λn}and{θn}are sequences in(,∞) and(, )satisfying the following:

(i)  <lim inf_n→∞λn≤lim supn→∞λn< c_γ

Then,{xn}converges strongly tox.

Proof The proof follows by takingξ=  andG=  in Theorem ..

4 Conclusion

In this paper, we propose an iterative algorithm to ﬁnd the common solution of the gener-alized equilibrium problem, variational inequality problem, and ﬁxed point problem for a relatively nonexpansive mapping in a real Banach space. Further, using the hybrid projec-tion method, we proved the strong convergence of the sequences generated by the iterative algorithm. Finally, we derived some consequences from our main result. The result pre-sented in this paper is an improvement and extension of the corresponding results of [, , –].

Acknowledgements

(19)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Unaizah College of Engineering, Qassim University, Unaizah, 51911, Saudi Arabia.}2_{College of Engineering, Qassim} University, Buraidah, 51452, Saudi Arabia. 3_{Department of Mathematics, BITS-Pilani, Dubai Campus, Dubai, 345055,} United Arab Emirates.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 11 August 2017 Accepted: 20 November 2017 References

1. Takahashi, W, Zembayashi, K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal.70, 45-57 (2009)

2. Petrot, N, Wattanawitoon, K, Kumam, P: A hybrid projection method for generalized mixed equilibrium problems and ﬁxed point problems in Banach spaces. Nonlinear Anal.4, 631-643 (2010)

3. Takahashi, S, Takahashi, W: Viscosity approximation method for equilibrium problems and ﬁxed point problems in Hilbert space. J. Math. Anal. Appl.331, 506-515 (2007)

4. Djafari Rouhani, B, Farid, M, Kazmi, KR: Common solution to generalized mixed quilibrium problem and ﬁxed point problem for a nonexpansive semigroup in Hilbert space. J. Korean Math. Soc.53(1), 89-114 (2016)

5. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud.63, 123-145 (1994)

6. Nadezhkina, N, Takahashi, W: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim.16(4), 1230-1241 (2006)

7. Nakajo, K: Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces. Appl. Math. Comput.271, 251-258 (2015)

8. Kazmi, KR, Rizvi, SH: A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a ﬁxed point problem. Appl. Math. Comput.218, 5439-5452 (2012)

9. Tada, A, Takahashi, W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In: Takahashi, W, Tanaka, T (eds.) Nonlinear Analysis and Convex Analysis, pp. 609-617. Yokohama, Japan (2006) 10. Matsushita, S, Takahashi, W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach

spaces. Fixed Point Theory Appl.2004, 37-47 (2004)

11. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal.16, 1127-1138 (1991)

12. Alber, YI: Metric and generalized projection operators in Banach spaces. In: Kartosatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15-50. Marcel Dekker, New York (1996) 13. Kamimura, S, Takahashi, W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim.13,

938-945 (2002)