Generalized multivalued equilibrium like problems: auxiliary principle technique and predictor corrector methods

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R E S E A R C H

Open Access

Generalized multivalued equilibrium-like

problems: auxiliary principle technique and

predictor-corrector methods

Vahid Dadashi

1*

_{and Abdul Latif}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics, Sari}

Branch, Islamic Azad University, Sari, Iran

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

Abstract

This paper is dedicated to the introduction a new class of equilibrium problems named generalized multivalued equilibrium-like problems which includes the classes of hemiequilibrium problems, equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational inequalities as special cases. By utilizing the auxiliary principle technique, some new predictor-corrector iterative algorithms for solving them are suggested and analyzed. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved in generalized multivalued equilibrium-like problem. Results obtained in this paper include several new and known results as special cases.

MSC: Primary 47H05; secondary 47J20; 47J25; 49J40; 65K10; 90C33

Keywords: generalized multivalued equilibrium-like problems; multivalued hemiequilibrium problems; auxiliary principle technique; predictor-corrector methods; convergence analysis

1 Introduction

During the last decades, the theory of variational analysis including variational inequalities (VI) have attracted a lot of attention because of its applications in optimization, nonlinear analysis, game theory, economics, and so forth; see, for example, [] and the references therein. Because of the importance and active impact ofVIin branches of sciences, en-gineering, and social sciences, it has been extended and generalized in many diﬀerent directions. It has been used as a tool to study diﬀerent aspects of optimization problems; see, for example, [–] and the references therein. By replacing the linear term appear-ing in the formulation of variational inequalities by a vector-valued term, Paridaet al. [] and Yang and Chen [] independently introduced and studied a class of variational in-equalities known as variational-like inin-equalities or pre-variational inin-equalities which is an important extension of the variational inequalities. Another useful and important gener-alization of variational inequalities is a class of variational inequalities known as hemivari-ational inequalities involving the nonlinear Lipschitz continuous functions. It should be pointed out that the hemivariational inequalities are connected with nonconvex and

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sibly nonsmooth energy functions and have important applications in structural analysis and nonconvex optimization. For more details, we refer the reader to [–].

On the other hand, the concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics), economics, finance, optimization, im-age reconstruction, network, ecology, sociology, chemistry, biology, engineering sciences, transportation, and other fields. The study of equilibrium problems (EP), which was intro-duced by Blum and Oettli [] in , in terms of their formulation, qualitative analysis, and computation has been the focus of much research in the past several decades and has given rise to the development of a variety of mathematical methodologies. Examples of mathematical formulations that have been used for equilibrium problems are: nonlinear equations, optimization problems, complementarity problems, fixed point problems, and, most recently, variational inequality problems; see, for example, [–] and the references therein.

In recent years,EPhas received much attention by many authors due to the fact that it provides a uniﬁed model including the above mentioned problems, and various important generalizations of it have been proposed and analyzed; see, for example, [–] and the references therein. An important and useful generalization ofEPis the multivalued equi-librium problems involving a nonlinear bifunction. It has been shown that a wide class of unrelated odd order and nonsymmetric free, moving, obstacle and equilibrium problems can be studied via the multivalued equilibrium problems.

Inspired and motivated by the research going in this interesting and fascinating area, Noor [, ] introduced and investigated the class of equilibrium-like (or invex equilib-rium) problems as a useful and important extension of the class of equilibrium problems. It has been shown that equilibrium-like problems include equilibrium problems, variational-like inequalities, variational inequalities and their invariant forms as special cases. In the meanwhile, related to the hemivariational inequalities, Noor [] introduced and stud-ied the class of hemiequilibrium problems as a generalization of the class of equilibrium problems. It is shown that the hemiequilibrium problems include equilibrium problems, hemivariational inequalities and variational inequalities as special cases. It is worth men-tioning that the hemiequilibrium problems and equilibrium-like problems are two quite diﬀerent extensions of the classical equilibrium problems.

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and later was developed by many authors for solving various classes of variational inequal-ities and equilibrium problems; see, for example, [, –] and the references therein.

Motivated and inspired by the work mentioned above, the purpose of this paper is to in-troduce a new class of equilibrium problems named generalized multivalued equilibrium-like problems (GMELP), which includes hemiequilibrium problems, equilibrium-equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational-like in-equalities as special cases. By using the auxiliary principle technique, we suggest and ana-lyze some predictor-corrector methods for solvingGMELP. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved inGMELP. As special cases, one can ob-tain several new and known methods for solving variational inequalities and equilibrium problems. The results presented in this paper generalize and improve some recent results in this ﬁeld.

2 Formulations, algorithms, and convergence results

LetHbe a real Hilbert space whose norm and inner product are denoted by · and ·,·, respectively. LetKbe a nonempty closed set inHand letCB(H) be the family of all nonempty, closed, and bounded subsets ofH. Suppose further thatS,T:K→CB(H) are two multivalued operators and letg:K→Kandη:K×K→Hbe two operators. For given bifunctionsF,G:H×H→R, we consider the problem of ﬁndingu∈K,ν∈T(u), andϑ∈S(u) such that

Fν,g(v)+Gϑ,ηg(v),g(u)≥, ∀v∈K, (.)

which is called thegeneralized multivalued equilibrium-like problem(GMELP).

Ifη(x,y) =x–y, for allx,y∈K, then the problem (.) reduces to the problem of ﬁnding u∈K,ν∈T(u), andϑ∈S(u) such that

Fν,g(v)+Gϑ,g(v) –g(u)≥, ∀v∈K, (.)

which is called thegeneralized multivalued hemiequilibrium problem(GMHEP).

It should be remarked that by taking diﬀerent choices of the operatorsS,T,η,g, and the bifunctionsFandGin the above problems, one can easily obtain the problems studied in [, ] and the references therein.

In the sequel, we denote byGMELP(F,G,S,T,η,g,K) andGMHEP(F,G,S,T,g,K) the set of solutions of the problems (.) and (.), respectively.

Lemma .[] Let X be a complete metric space,T:X→CB(X)be a multivalued map-ping. Then for anyε> and for any given x,y∈X,u∈T(x),there exists v∈T(y)such that

d(u,v)≤( +ε)MT(x),T(y),

where M(·,·)is the Hausdorﬀ metric on CB(X)deﬁned by

M(A,B) =max

sup

x∈A

inf

y∈Bx–y,sup_y_∈_Bxinf∈Ax–y

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LetF,G,S,T,η, andgbe the same as inGMELP(.). For givenu∈K,ν∈T(u), and ϑ∈S(u), we consider the auxiliary generalized multivalued hemiequilibrium-like problem of ﬁndingw∈Ksuch that

ρFν,g(v)+ρGϑ,ηg(v),g(u)+g(w) –g(u),g(v) –g(w)≥, ∀v∈K, (.)

whereρ>  is a constant. Obviously, ifw=u, then (w,ν,ϑ) is a solution ofGMELP(.). This observation and Nadler’s technique [] enables us to suggest the following ﬁnite step predictor-corrector method for solvingGMELP(.).

Algorithm . LetF,G,S,T,η, andgbe the same as inGMELP(.). For givenu∈K,

ν∈T(u), andϑ∈S(u), compute the iterative sequences{un},{νn}, and{ϑn}by the

iterative schemes

ρFν,n,g(v)

+ρGϑ,n,η

g(v),g(u,n)

+g(un+) –g(y,n),g(v) –g(un+)

≥, ∀v∈K, (.)

ρi+F

νi+,n,g(v)

+ρi+G

ϑi+,n,η

g(v),g(ui+,n)

+g(yi,n) –g(yi+,n),g(v) –g(yi,n)

≥, i= , , . . . ,q– ,∀v∈K, (.)

ρqF

νn,g(v)

+ρqG

ϑn,η

g(v),g(un)

+g(yq–,n) –g(un),g(v) –g(yq–,n)

≥, ∀v∈K, (.)

νi,n∈T(yi,n) :νi,n+–νi,n ≤

 + ( +n)–MT(yi,n+),T(yi,n)

,

i= , , . . . ,q– , (.)

ϑi,n∈S(yi,n) :ϑi,n+–ϑi,n ≤

 + ( +n)–MS(yi,n+),S(yi,n)

,

i= , , . . . ,q– , (.)

νn∈T(un) :νn+–νn ≤

 + ( +n)–MT(un+),T(un)

, (.)

ϑn∈S(un) :ϑn+–ϑn ≤

 + ( +n)–MS(un+),S(un)

, (.)

whereρi>  (i= , , . . . ,q) are constants andn= , , , . . . .

Ifη(x,y) =x–y, for allx,y∈K, then Algorithm . reduces to the following predictor-corrector method.

Algorithm . For givenu∈K,ν∈T(u) and ϑ∈S(u), compute the iterative

se-quences{un},{νn}, and{ϑn}by the iterative schemes

ρFν,n,g(v)

+ρGϑ,n,g(v) –g(u,n)

+g(un+) –g(y,n),g(v) –g(un+)

≥, ∀v∈K,

ρi+F

νi+,n,g(v)

+ρi+G

ϑi+,n,g(v) –g(ui+,n)

+g(yi,n) –g(yi+,n),g(v) –g(yi,n)

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ρqF

νn,g(v)

+ρqG

ϑn,g(v) –g(un)

+g(yq–,n) –g(un),g(v) –g(yq–,n)

≥, ∀v∈K,

νi,n∈T(yi,n) :νi,n+–νi,n ≤

 + ( +n)–MT(yi,n+),T(yi,n)

, i= , , . . . ,q– ,

ϑi,n∈S(yi,n) :ϑi,n+–ϑi,n ≤

 + ( +n)–MS(yi,n+),S(yi,n)

, i= , , . . . ,q– ,

νn∈T(un) :νn+–νn ≤

 + ( +n)–MT(un+),T(un)

,

ϑn∈S(un) :ϑn+–ϑn ≤

 + ( +n)–MS(un+),S(un)

,

In order to study the convergence analysis of the iterative sequences generated by Algo-rithm ., we need the following deﬁnitions.

Deﬁnition . LetT:K→CB(H) be a multivalued operator andg:K→Kbe an oper-ator. The bifunctionF:H×H→Ris said to be

(a) g-monotonewith respect toT, if

Fw,g(u)+Fw,g(u)≤, ∀u,u∈K,w∈T(u),w∈T(u);

(b) partiallyα-relaxedg-monotonewith respect toT, if there exists a constantα> 

such that

Fw,g(u)

+Fw,g(z)

≤αg(z) –g(u) 

,

∀u,u,z∈K,w∈T(u),w∈T(u).

It should be pointed out that ifz=u, then the partiallyα-relaxedg-monotonicity of the bifunctionFwith respect toTreduces tog-monotonicity with respect toT. Meanwhile, ifg≡I, then parts (a) and (b) of Deﬁnition . reduce to the deﬁnition of monotonicity and partiallyα-relaxed monotonicity of the bifunctionFwith respect toT, respectively.

Deﬁnition . LetS:K→CB(H) be a multivalued operator and letg:K→Kandη: K×K→Hbe two operators. The bifunctionG:H×H→Ris said to be

(a) g-η-monotonewith respect toS, if

Gw,η

g(u),g(u)

+Gw,η

g(u),g(u)

≤, ∀u,u∈K,w∈S(u),w∈S(u);

(b) partiallyβ-relaxedg-η-monotonewith respect toS, if there exists a constantβ> 

such that

Gw,ηg(u),g(u)+Gw,ηg(z),g(u)≤βg(z) –g(u),

∀u,u,z∈K,w∈S(u),w∈S(u).

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respect toS. Furthermore, for the case whenη(x,y) =x–y, for allx,y∈K, then parts (a) and (b) of Deﬁnition . reduce to the deﬁnition ofg-monotonicity and partiallyβ-relaxed g-monotonicity of the bifunctionGwith respect toS, respectively.

Deﬁnition . A multivalued operatorT:H→CB(H) is said to beM-Lipschitz contin-uous with constantδ, if there exists a constantδ>  such that

MT(u),T(v)≤δu–v, ∀u,v∈H,

whereM(·,·) is the Hausdorﬀ metric onCB(H).

The next proposition plays a crucial role in the study of convergence analysis of the iterative sequences generated by Algorithm ..

Proposition . Let F,G,S,T,η,and g be the same as inGMELP(.)and letuˆ∈K, ˆ

ν∈T(u),andϑˆ ∈G(ˆu)be the solution ofGMELP(.).Suppose further that{un}and{yi,n}

(i= , , . . . ,q– )are the sequences generated by Algorithm..If F is partiallyα-relaxed g-η-monotone with respect to T,and G is partiallyβ-relaxed g-η-monotone with respect to S,then

g(u) –ˆ g(un+)



≤g(ˆu) –g(y,n)



– – (α+β)ρg(un+) –g(y,n)



, (.)

g(u) –ˆ g(yi,n)



≤g(u) –ˆ g(yi+,n)



– – (α+β)ρi+g(yi,n) –g(yi+,n)



, (.)

g(u) –ˆ g(yq–,n)



≤g(u) –ˆ g(un)



– – (α+β)ρqg(yq–,n) –g(un)



, (.)

for all n≥,where i= , , . . . ,q– .

Proof Sinceuˆ∈K,νˆ∈T(u), andϑˆ ∈S(u) are the solution ofGMELP(.), we have

Fνˆ,g(v)+Gϑˆ,ηg(v),g(ˆu)≥, ∀v∈K. (.)

Takingv=un+in (.) andv=uin (.), we get

Fνˆ,g(un+)

+Gϑˆ,ηg(un+),g(ˆu)

≥ (.)

and

ρFν,n,g(u)ˆ

+ρG

ϑ,n,η

g(ˆu),g(u,n)

+g(un+) –g(y,n),g(ˆu) –g(un+)

≥. (.)

By combining (.) and (.) and taking into account of the facts that the bifunction F is partiallyα-relaxedg-monotone with respect toT, and the bifunctionGis partially β-relaxed stronglyg-η-monotone with respect toG, it follows that

g(un+) –g(y,n),g(ˆu) –g(un+)

≥–ρFν,n,g(u)ˆ

–ρGϑ,n,η

g(ˆu),g(u,n)

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≥–ρ

Fν,n,g(ˆu)

+Fνˆ,g(un+)

+Gϑ,n,η

g(u),ˆ g(u,n)

+Gϑˆ,ηg(un+),g(u)ˆ

≥–ραg(un+) –g(y,n)–ρβg(un+) –g(y,n)

= –ρ(α+β)g(un+) –g(y,n)



. (.)

On the other hand, lettingx=g(u) –ˆ g(un+) andy=g(un+) –g(y,n) and by utilizing the

well-known property of the inner product, we have

g(un+) –g(y,n),g(u) –ˆ g(un+)

=g(u) –ˆ g(y,n)



–g(u) –ˆ g(un+) 

–g(un+) –g(y,n)



. (.)

Applying (.) and (.), it follows that

_g(ˆ_{u) –}_g(u_n_+)

≤g(ˆu) –g(y,n)



– – (α+β)ρg(un+) –g(y,n)

 ,

which is the required result (.).

Takingv=yi,n(i= , , . . . ,q– ) in (.) andv=uˆin (.), for eachi= , , . . . ,q– , we

have

Fνˆ,g(yi,n)

+Gϑˆ,ηg(yi,n),g(u)ˆ

≥ (.)

and

ρi+F

νi+,n,g(u)ˆ

+ρi+G

ϑi+,n,η

g(u),ˆ g(ui+,n)

+g(yi,n) –g(yi+,n),g(ˆu) –g(yi,n)

≥. (.)

Lettingx=g(u) –ˆ g(yi,n) andy=g(yi,n) –g(yi+,n) for eachi= , , . . . ,q– , and by using the

well-known property of the inner product, one has

g(yi,n) –g(yi+,n),g(u) –ˆ g(yi,n)

=g(u) –ˆ g(yi+,n)



–g(u) –ˆ g(yi,n)



–g(yi,n) –g(yi+,n)



. (.)

In a similar fashion to the preceding analysis, employing (.)-(.) and considering the facts that the bifunctionFis partiallyα-relaxedg-η-monotone with respect toT, and the bifunctionGis partiallyβ-relaxedg-η-monotone with respect toS, for eachi= , , . . . , q– , one can deduce that

g(u) –ˆ g(yi,n)



≤g(u) –ˆ g(yi+,n)



– – (α+β)ρi+g(yi,n) –g(yi+,n)

 ,

which is the required result (.).

Takingv=yq–,nin (.) andv=uˆin (.), we have

Fνˆ,g(yq–,n)

+Gϑˆ,ηg(yq–,n),g(ˆu)

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and

ρqF

νn,g(ˆu)

+ρqG

ϑn,η

g(u),ˆ g(un)

+g(yq–,n) –g(un),g(u) –ˆ g(yq–,n)

≥. (.)

By assumingx=g(ˆu) –g(yq–,n) andy=g(yq–,n) –g(un), and by utilizing the well-known

property of the inner product, we obtain

g(yq–,n) –g(un),g(u) –ˆ g(yq–,n)

=g(u) –ˆ g(un)



–g(u) –ˆ g(yq–,n)



–g(yq–,n) –g(un)



. (.)

By a similar way to that of proof of (.), by using (.)-(.) and in light of the facts that the bifunctionFis partiallyα-relaxedg-η-monotone with respect toT, and the bifunction Gis partiallyβ-relaxedg-η-monotone with respect toS, we can show that

g(u) –ˆ g(yq–,n)≤g(u) –ˆ g(un)



– – (α+β)ρqg(yq–,n) –g(un)

 ,

which is the required result (.). This completes the proof.

In the next theorem, the strong convergence of the iterative sequences generated by Algorithm . to a solution ofGMELP(.) is established.

Theorem . LetHbe a ﬁnite dimensional real Hilbert space and let g:K→K be a

continuous and invertible operator.Suppose that the bifunction F:H×H→Ris con-tinuous in the ﬁrst argument,the bifunction G:H×H→Ris continuous in both argu-ments and the operatorη:K×K →His continuous in the second argument.Assume that the multivalued operators S,T:K→CB(H)are M-Lipschitz continuous with con-stantsσ andδ,respectively.Moreover, let all the conditions of Proposition.hold and GMELP(F,G,S,T,η,g,K)=∅.Ifρi∈(,_(α+β)), for each i= , , . . . ,q, then the iterative sequences {un}, {νn}, and {ϑn} generated by Algorithm. converge strongly to uˆ ∈K,

ˆ

ν∈T(u),ˆ andϑˆ∈S(u),ˆ respectively,and(u,ˆ νˆ,ϑˆ)is a solution ofGMELP(.).

Proof Letu∈K,ν∈T(u), andϑ∈S(u) be the solution ofGMELP(.). In view of the fact that all the conditions of Proposition . hold, then Proposition . implies that for alln≥

g(u) –g(un+)≤g(u) –g(y,n)–

 – (α+β)ρg(un+) –g(y,n), (.)

g(u) –g(yi,n)



≤g(u) –g(yi+,n)



– – (α+β)ρi+g(yi,n) –g(yi+,n)



, (.)

g(u) –g(yq–,n)≤g(u) –g(un)



– – (α+β)ρqg(yq–,n) –g(un)



, (.)

wherei= , , . . . ,q– . From the inequalities (.)-(.), it follows that the sequence {g(un) –g(u)}is nonincreasing and hence the sequence{g(un)}is bounded. Since the

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by (.)-(.), we have

 – (α+β)

q

i=

ρi g(un+) –g(y,n)+ q–

i=

g(yi,n) –g(yi+,n)

+g(yq–,n) –g(un)



≤g(u) –g(un)



–g(u) –g(un+) 

,

which implies that

∞

n=

 – (α+β)

q

i=

ρi g(un+) –g(y,n)

 +

q–

i=

g(yi,n) –g(yi+,n)



+g(yq–,n) –g(un)



≤g(u) –g(u). (.)

The inequality (.) guarantees that

g(un+) –g(y,n)→, g(yi,n) –g(yi+,n)→, g(yq–,n) –g(un)→,

for eachi= , , . . . ,q– , asn→ ∞. Letuˆ be a cluster point of the sequence{un}. Since

{un} is bounded, there exists a subsequence{unj}of{un}such thatunj → ˆu, asj→ ∞.

Taking into consideration the fact that the multivalued operatorsSandTareM-Lipschitz continuous with constants σ andδ, respectively, in virtue of the inequalities (.) and (.), we have

νnj+–νnj ≤

 + ( +nj)–

MT(unj+),T(unj)

≤ + ( +nj)–

δunj+–unj (.)

and

ϑnj+–ϑnj ≤

 + ( +nj)–

MS(unj+),S(unj)

≤ + ( +nj)–

σunj+–unj. (.)

The inequalities (.) and (.) imply thatνnj+–νnj → andϑnj+–ϑnj →, as

j→ ∞, that is,{νnj}and{ϑnj}are Cauchy sequences inH. Thus,νnj→ ˆνandϑnj→ ˆϑfor

someνˆ,ϑˆ ∈H, asj→ ∞. By using (.), we have

ρqF

νnj,g(v)

+ρqG

ϑnj,η

g(v),g(unj)

+g(yq–,nj) –g(unj),g(v) –g(yq–,nj)

≥. (.)

In view of the fact thatFis continuous in the ﬁrst argument,Gis continuous in both argu-ments,ηis continuous in the second argument,gis continuous andg(yq–,n)–g(un) →,

asn→ ∞, lettingi→ ∞and by using (.), we deduce that

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In the meantime, from theM-Lipschitz continuity ofTwith constantδ, it follows that

dνˆ,T(u)ˆ =infˆν–z:z∈T(u)ˆ

≤ ˆν–νnj+d

νnj,T(u)ˆ

≤ ˆν–νnj+M

T(unj),T(u)ˆ

≤ ˆν–νnj+δunj–u.ˆ (.)

Notice that the right side of the above inequality tends to zero asj→ ∞. SinceT(ˆu)∈ CB(H) it follows thatνˆ∈T(ˆu). Taking into account of the fact that the multivalued opera-torSisM-Lipschitz continuous with constantσ, in a similar way to that of proof of (.), one can deduce that

dϑˆ,S(u)ˆ ≤ ˆϑ–ϑnj+σunj–u,ˆ

which relying on the fact thatS(u)ˆ ∈CB(H) implies thatϑˆ∈S(u). Hence,ˆ uˆ∈K,νˆ∈T(ˆu), andϑˆ ∈S(u) are the solution ofˆ GMELP(.). Now, the inequalities (.)-(.) imply that

g(un+–g(u)ˆ ≤g(un) –g(ˆu), ∀n≥. (.)

The inequality (.) guarantees that g(un)→g(u), asˆ n→ ∞ and henceun→ ˆu, as

n→ ∞, sinceg is continuous and invertible. Consequently, the sequence{un}has

ex-actly one cluster pointu. Considering the facts thatˆ SandT areM-Lipschitz continuous with constantsσandδ, respectively, by using (.) and (.), we have

νn+–νn ≤

 + ( +n)–MT(un+),T(un)

≤ + ( +n)–δun+–un (.)

and

ϑn+–ϑn ≤

 + ( +n)–MS(un+),S(un)

≤ + ( +n)–σun+–un. (.)

The inequalities (.) and (.) imply that{νn} and{ϑn}are Cauchy sequences in H.

Sinceνˆ andϑˆ are cluster points of the sequences{νn}and{ϑn}, respectively, it follows

thatνn→ ˆνandϑn→ ˆϑ, asn→ ∞, that is, the sequences{νn}and{ϑn}have exactly one

cluster pointνˆandϑˆ, respectively. The proof is completed.

The next proposition is a main tool for studying the convergence analysis of the iterative sequences generated by Algorithm ..

Proposition . Let F, G,S,T, and g be the same as inGMHEP (.)and letuˆ ∈K, ˆ

ν∈T(u),andϑˆ ∈G(u)ˆ be the solution ofGMHEP(.).Furthermore,let{un}and{yi,n}

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Proof It follows from Proposition . by deﬁning the operatorη:K×K→Hasη(x,y) =

x–yfor allx,y∈K.

The next assertion provides us the required conditions under which the iterative se-quences generated by Algorithm . converge strongly to a solution ofGMHEP(.).

Corollary . Suppose thatHis a ﬁnite dimensional real Hilbert space and let g:K→K be a continuous and invertible operator. Assume that the bifunction F:H×H→Ris continuous in the ﬁrst argument and the bifunction G:H×H→Ris continuous in both arguments.Let the multivalued operators S,T:K→CB(H)be M-Lipschitz continuous with constantsσandδ,respectively.Furthermore,let all the conditions of Proposition. hold andGMHEP(F,G,S,T,g,K)=∅.Ifρi∈(,_(α+β)),for each i= , , . . . ,q,then the iter-ative sequences{un},{νn},and{ϑn}generated by Algorithm.converge strongly touˆ∈K,

ˆ

ν∈T(u)ˆ andϑˆ ∈S(u),ˆ respectively,and(u,ˆ νˆ,ϑˆ)is a solution ofGMHEP(.).

Proof By deﬁning the operatorη:K×K→Hasη(x,y) =x–yfor allx,y∈K, the desired

result follows from Theorem ..

It is well known that to implement the proximal point methods, one has to calculate the approximate solution implicitly, which is itself a diﬃcult problem. In order to overcome this drawback, we consider another auxiliary problem and with the help of it, we construct an iterative algorithm for solving the problem (.).

LetS,T,F,G,η, andgbe the same as inGMELP(.). For givenu∈K,ν∈T(u), and ϑ∈S(u), we consider the auxiliary generalized multivalued equilibrium-like problem of ﬁndingw∈K,ξ∈T(w), andγ ∈S(w) such that

ρFξ,g(v)+ρGγ,ηg(v),g(u)+g(w) –g(u),g(v) –g(w)≥, ∀v∈K, (.)

whereρ>  is a constant. It should be pointed out that the two problems (.) and (.) are quite diﬀerent. Ifw=u, then clearly (w,ξ,γ) is a solution ofGMELP(.). By using this observation and Nadler’s technique [], we are able to suggest the following predictor-corrector method for solvingGMELP(.).

Algorithm . LetF,G,S,T,η, andgbe the same as inGMELP(.). For givenu∈K, ξ∈T(u), andγ∈S(u), compute the iterative sequences{un}, {ξn}, and{γn} by the

iterative schemes

ρFξ,n,g(v)

+ρGγ,n,η

g(v),g(y,n)

+g(un+) –g(y,n),g(v) –g(un+)

≥, ∀v∈K, (.)

ρi+F

ξi+,n,g(v)

+ρi+G

γi+,n,η

g(v),g(yi+,n)

+g(yi,n) –g(yi+,n),g(v) –g(yi,n)

≥, i= , , . . . ,q– ,∀v∈K, (.)

ρqF

ξn,g(v)

+ρqG

γn,η

g(v),g(un)

+g(yq–,n) –g(un),g(v) –g(yq–,n)

≥, ∀v∈K, (.)

ξi,n∈T(yi,n) :ξi,n+–ξi,n ≤

 + ( +n)–MT(yi,n+),T(yi,n)

,

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γi,n∈S(yi,n) :γi,n+–γi,n ≤

 + ( +n)–MS(yi,n+),S(yi,n)

,

i= , , . . . ,q– , (.)

ξn∈T(un) :ξn+–ξn ≤

 + ( +n)–MT(un+),T(un)

, (.)

γn∈S(un) :γn+–γn ≤

 + ( +n)–MS(un+),S(un)

, (.)

Ifη(x,y) =x–y, for allx,y∈K, then Algorithm . reduces to the following predictor-corrector method.

Algorithm . For givenu∈K,ξ∈T(u), andγ∈S(u), compute the iterative se-quences{un},{ξn}, and{γn}by the iterative schemes

ρFξ,n,g(v)

+ρGγ,n,g(v) –g(y,n)

+g(un+) –g(y,n),g(v) –g(un+)

≥, ∀v∈K,

ρi+F

ξi+,n,g(v)

+ρi+G

γi+,n,g(v) –g(yi+,n)

+g(yi,n) –g(yi+,n),g(v) –g(yi,n)

≥, i= , , . . . ,q– ,∀v∈K,

ρqF

ξn,g(v)

+ρqG

γn,g(v) –g(un)

+g(yq–,n) –g(un),g(v) –g(yq–,n)

≥, ∀v∈K,

ξi,n∈T(yi,n) :ξi,n+–ξi,n ≤

 + ( +n)–MT(yi,n+),T(yi,n)

, i= , , . . . ,q– ,

γi,n∈S(yi,n) :γi,n+–γi,n ≤

 + ( +n)–MS(yi,n+),S(yi,n)

, i= , , . . . ,q– ,

ξn∈T(un) :ξn+–ξn ≤

 + ( +n)–MT(un+),T(un)

,

γn∈S(un) :γn+–γn ≤

 + ( +n)–MS(un+),S(un)

,

To prove the strong convergence of the sequences generated by Algorithm . to a so-lution ofGMELP(.), we need the following deﬁnition.

Deﬁnition . LetS,T:K→CB(H),η:K×K→H, andg:K→Kbe operators. The bifunctionsF,G:H×H→Rare said to bejointly g-η-pseudomonotonewith respect to SandT, if

Fw,g(u)+Gϑ,ηg(u),g(u)≥, implies that

Fw,g(u)

+Gϑ,η

g(u),g(u)

≤,

∀u,u∈K,w∈T(u),w∈T(u),ϑ∈S(u),ϑ∈S(u).

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Before turning to the study of convergence analysis of the iterative sequences gener-ated by Algorithm ., we would like to present the following proposition which plays an important and key role in it.

Proposition . Let F,G,S,T,η,and g be the same as inGMELP(.)and letuˆ∈K, ˆ

ν∈T(u),ˆ andϑˆ ∈S(ˆu)be the solution ofGMELP(.).Assume further that{un}and{yi,n}

(i= , , . . . ,q– )are the sequences generated by Algorithm..If F and G are jointly g-η -pseudomonotone with respect to S and T,then for all n≥

g(u) –ˆ g(un+)



≤g(u) –ˆ g(y,n)



–g(un+) –g(y,n)



, (.)

g(u) –ˆ g(yi,n)



≤g(u) –ˆ g(yi+,n)



–g(yi,n) –g(yi+,n)

 ,

i= , , . . . ,q– , (.)

g(u) –ˆ g(yq–,n)



≤g(u) –ˆ g(un)



–g(yq–,n) –g(un)



. (.)

Proof Sinceuˆ∈K,νˆ∈T(ˆu), andϑˆ∈S(u) are the solution ofˆ GMELP(.), it follows that (u,ˆ v,ˆ ϑˆ) satisﬁes (.). Takingv=y,nin (.), we have

Fνˆ,g(y,n)

+Gϑˆ,ηg(y,n),g(ˆu)

≥. (.)

Taking into consideration the fact that the bifunctionsF andGare jointlyg-η -pseudo-monotone with respect toSandT, from (.), we conclude that

Fξ,n,g(ˆu)

+Gγ,n,η

g(ˆu),g(y,n)

≤. (.)

Takingv=uˆin (.), we obtain

ρFξ,n,g(ˆu)

+ρGγ,n,η

g(u),ˆ g(y,n)

+g(un+) –g(y,n),g(ˆu) –g(un+)

≥. (.)

By combining (.) and (.), we get

g(un+) –g(y,n),g(ˆu) –g(un+)

≥–ρFξ,n,g(ˆu)

–ρGγ,n,η

g(u),ˆ g(y,n)

≥. (.)

Relying on (.) and (.), we get

_g(ˆu) –g(un+)



≤g(ˆu) –g(y,n)



–g(un+) –g(y,n)

 ,

which is the required result (.).

Takingv=yi+,n(i= , , . . . ,q– ) in (.), we have

Fνˆ,g(yi+,n)

+Gϑˆ,ηg(yi+,n),g(u)ˆ

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Considering the fact thatFandGare jointlyg-η-pseudomonotone with respect toSandT, the inequality (.) implies that, for eachi= , , . . . ,q– ,

Fξi+,n,g(u)ˆ

+Gγi+,n,η

g(u),ˆ g(yi+,n)

≤. (.)

Lettingv=uˆin (.), for eachi= , , . . . ,q– , we obtain

ρi+F

ξi+,n,g(ˆu)

+ρi+G

γi+,n,η

g(u),ˆ g(yi+,n)

+g(yi,n) –g(yi+,n),g(ˆu) –g(yi,n)

≥. (.)

It follows from (.), (.), and (.) that, for eachi= , , . . . ,q– ,

_g(ˆu) –g(yi,n)



≤g(ˆu) –g(yi+,n)



–g(yi,n) –g(yi+,n)

 ,

which is the required result (.). Takingv=yq,nin (.), we have

Fνˆ,g(yq,n)

+Gϑˆ,ηg(yq,n),g(u)ˆ

≥. (.)

In light of the fact thatFandGare jointlyg-η-pseudomotone with respect toSandT, we deduce that

Fξn,g(u)ˆ

+Gγn,η

g(u),ˆ g(un)

≤. (.)

Lettingv=uˆin (.), we get

ρqF

ξn,g(ˆu)

+ρqG

γn,η

g(u),ˆ g(un)

+g(yq–,n) –g(un),g(u) –ˆ g(yq–,n)

≥. (.)

Applying (.), (.) and (.), one can deduce that

g(u) –ˆ g(yq–,n)≤g(u) –ˆ g(un)



–g(yq–,n) –g(un)

 ,

which is the required result (.). This completes the proof.

Now we establish the strong convergence of the iterative sequences generated by Algo-rithm . to a solution ofGMELP(.).

Theorem . LetHbe a ﬁnite dimensional real Hilbert space and let g:K→K be a continuous and invertible operator.Suppose that the bifunction F:H×H→Ris con-tinuous in the ﬁrst argument,the bifunction G:H×H→Ris continuous in both argu-ments and the operatorη:K×K →His continuous in the second argument.Assume that the multivalued operators S,T:K→CB(H)are M-Lipschitz continuous with con-stants σ andδ,respectively.Further,let all the conditions of Proposition.hold and GMELP(F,G,S,T,η,g,K)=∅.Then the iterative sequences{un},{ξn},and{γn}generated

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Proof Letu∈K,ν∈T(u), andϑ∈S(u) be the solution ofGMELP(.). Since all the conditions of Proposition . hold, according to Proposition ., for alln≥ we have

g(u) –g(un+)≤g(u) –g(y,n)–g(un+) –g(y,n), (.)

g(u) –g(yi,n)≤g(u) –g(yi+,n)–g(yi,n) –g(yi+,n),

i= , , . . . ,q– , (.)

g(u) –g(yq–,n)≤g(u) –g(un)–g(yq–,n) –g(un). (.)

It is easy to see that the inequalities (.)-(.) imply that the sequence{g(un) –g(u)}

is nonincreasing and hence the sequence{g(un)}is bounded. Considering the fact that

the operatorgis invertible, it follows that the sequence{un}is also bounded. Meanwhile,

relying on (.)-(.), we have

g(un+) –g(y,n)

 +

q–

i=

g(yi,n) –g(yi+,n)



+g(yq–,n) –g(un)



≤g(u) –g(un)–g(u) –g(un+),

whence we deduce that

∞

n=

g(un+) –g(y,n)

 +

q–

i=

g(yi,n) –g(yi+,n)



+g(yq–,n) –g(un)



≤g(u) –g(u). (.)

From (.), it follows that

g(un+) –g(y,n)→, g(yi,n) –g(yi+,n)→, g(yq–,n) –g(un)→,

for eachi= , , . . . ,q– , asn→ ∞. Letuˆ be a cluster point of the sequence{un}. Since

{un}is bounded, there exists a subsequence{unj}of{un}such thatunj→ ˆu, asj→ ∞. In

a similar way to that of proof of Theorem ., one can establish that{ξnj}and{γnj}are

Cauchy sequences inHandξnj→ ˆν, andγnj→ ˆϑfor someνˆ,ϑˆ∈H, asj→ ∞.

Further-more,uˆ∈K,νˆ∈T(u), andˆ ϑˆ ∈S(u) are the solution ofˆ GMELP(.) and the sequences {un},{ξn}, and{γn}have exactly one cluster pointu,ˆ νˆ, andϑˆ, respectively. This gives us

the desired result.

The next proposition plays a crucial role in the study of convergence analysis of the iterative sequences generated by Algorithm ..

Proposition . Let F,G,S,T,and g be the same as inGMHEP(.)and letuˆ∈K, ˆ

ν∈T(u),ˆ andϑˆ ∈S(ˆu)be the solution ofGMHEP(.).Moreover,let{un}and{yi,n}(i=

, , . . . ,q– )be the sequences generated by Algorithm..If the bifunctions F and G are jointly g-pseudomonotone with respect to S and T,then the inequalities(.)-(.)hold.

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We now conclude this paper with the following result in which the strong convergence of the iterative sequence generated by Algorithm . is established.

Corollary . Assume thatHis a ﬁnite dimensional real Hilbert space and let g:K→K be a continuous and invertible operator.Let the bifunction F:H×H→Rbe continuous in the ﬁrst argument and the bifunction G:H×H→Rbe continuous in both arguments. Suppose that the multivalued operators S,T:K→CB(H)are M-Lipschitz continuous with constantsσandδ,respectively.Moreover,let all the conditions of Proposition.hold and GMHEP(F,G,S,T,g,K)=∅.Then the iterative sequences{un},{ξn},and{γn}generated by

Algorithm.converge strongly touˆ∈K,νˆ∈T(u),ˆ andϑˆ ∈S(u),ˆ respectively,and(u,ˆ νˆ,ϑˆ) is a solution ofGMHEP(.).

Proof We obtain the desired result from Theorem . by deﬁningη:K×K→Has

η(x,y) =x–yfor allx,y∈K.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran.}2_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

Acknowledgements

The first author is supported by the Sari Branch, Islamic Azad University, Iran. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support.

Received: 4 January 2016 Accepted: 1 February 2016 References

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