# On a System of Generalized Mixed Equilibrium Problems Involving Variational-Like Inequalities in Banach Spaces: Existence and Algorithmic Aspects

Authors: H. Mahdioui; O. Chadli
Journal: Advances in Operations Research
 (2012)

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## Abstract

We study the existence and the algorithmic aspect of a System of Generalized Mixed Equilibrium Problems involving variational-like inequalities (SGMEPs) in the setting of Banach spaces. The approach adopted is based on the auxiliary principle technique and arguments from generalized convexity. A new existence theorem for the auxiliary problem is established; this leads us to generate an algorithm which converges strongly to a solution of (SGMEP) under weaker assumptions. When the study is reduced to the setting of reflexive Banach spaces, then it can be more relaxed by dropping the coercivity condition. The results obtained in this paper are new and improve some recent studies in this field.


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## Body

## 1. Introduction and the Problem Statement

LetK be a nonempty closed convex subset of a Banach space X, and let F:K×K→ℝ be a real-valued bifunction. By equilibrium problem, in short (EP), we mean the following problem: (1.1)(EP⁡)Findu¯∈Ksuch  thatF(u¯,v)≥0∀v∈K.
Equilibrium problems are suitable and common format for investigation of various applied problems arising in economics, mathematical physics, transportation, communication systems, engineering, and other fields. Moreover, equilibrium problems are closely related with other general problems in nonlinear analysis, such as fixed points, game theory, variational inequality, and optimization problems, see [1–4]. Various kinds of iterative algorithms to solve equilibrium problems and variational inequalities have been developed by many authors. There exists a vast literature on the approximation solvability of equilibrium problems and nonlinear variational inequalities using projection type methods, proximal-type methods, or resolvent operator type methods, see [5–11]. We observe that the projection method and its variant forms cannot be applied for constructing iterative algorithms of mixed variational-like inequalities or mixed equilibrium problems involving variational-like inequalities. This fact motivated many authors to develop the auxiliary principle technique to study the existence and algorithm of solutions for variational-like inequalities and its extensions to mixed equilibrium problems, see [12–15]. Kazmi and Khan [16] studied a system of generalized variational-like inequality problems in Hilbert spaces by using the auxiliary principle technique. Recently, Ding and Wang [17] and Ding [18] introduced new iterative algorithms for solving some class of system of generalized mixed variational-like inequalities and system of mixed equilibrium problems involving variational-like inequalities.In this paper, we study the existence and the algorithmic aspect of a System of Generalized Mixed Equilibrium Problems involving variational-like inequalities (SGMEPs) in the setting of Banach spaces. A new existence theorem for the auxiliary problem is established, this leads us to generate an algorithm which converges strongly to a solution of (SGMEP) under weaker assumptions. When the study is reduced to the setting of reflexive Banach spaces, then it can be more relaxed by dropping the coercivity condition. The results obtained in this paper are new and improve some recent studies in this field.Throughout this paper, letI={1,2} be an index set. For each i∈I, let Xi be a Banach space, Xi* its dual, 〈·,·〉i is the duality pairing between Xi* and Xi, ∥·∥i denotes the norm in Xi and Xi*, and let Ki be a nonempty closed convex subset of Xi. We will denote by 𝒞ℬ(Xi*) the family of all nonempty closed and bounded subsets of Xi*.For eachi∈I, let Ri:K1⇉𝒞ℬ(X1*) and Ai:K2⇉𝒞ℬ(X2*) be multivalued mappings, Ni:X1*×X2*→Xi* and ηi:Xi×Xi→Xi be single-valued mappings, Fi,ψi:Ki×Ki→ℝ be two real-valued functions, and let wi∈Xi*. We consider the following System of Generalized Mixed Equilibrium Problem involving variational-like inequalities (SGMEP).Find(x¯1,x¯2)∈K1×K2, (u¯1,v¯1)∈R1(x¯1)×A1(x¯2), and (u¯2,v¯2)∈R2(x¯1)×A2(x¯2) such that
(1.2)F1(x¯1,y1)+〈N1(u¯1,v¯1)-w1,η1(y1,x¯1)〉1+ψ1(x¯1,y1)-ψ1(x¯1,x¯1)≥0∀y1∈K1;F2(x¯2,y2)+〈N2(u¯2,v¯2)-w2,η2(y2,x¯2)〉2+ψ2(x¯2,y2)-ψ2(x¯2,x¯2)≥0∀y2∈K2.


### 1.1. Some Special Cases

(1)
If for eachi∈I, Xi=Xi*=Hi is a Hilbert space, Ki=Hi, Fi≡0,  wi=0 and for each (x,y)∈H1×H2, Ri(x)=x and Ai(y)=y, then problem (1.2) reduces to the problem of finding (x¯1,x¯2)∈H1×H2 such that
(1.3)〈N1(x¯1,x¯2),η1(y1,x¯1)〉1+ψ1(x¯1,y1)-ψ1(x¯1,x¯1)≥0∀y1∈H1;〈N2(x¯1,x¯2),η2(y2,x¯2)〉2+ψ2(x¯2,y2)-ψ2(x¯2,x¯2)≥0∀y2∈H2.
Problem (1.3) has been introduced and studied by Kazmi and Khan in [16].
(2)
If for eachi∈I, Xi is a reflexive Banach space, Ki=Xi and Fi≡0, then problem (1.2) reduces to the problem of finding (x¯1,x¯2)∈X1×X2, (u¯1,v¯1)∈R1(x¯1)×A1(x¯2) and (u¯2,v¯2)∈R2(x¯1)×A2(x¯2) such that
(1.4)〈N1(u¯1,v¯1)-w1,η1(y1,x¯1)〉1+ψ1(x¯1,y1)-ψ1(x¯1,x¯1)≥0∀y1∈X1;〈N2(u¯2,v¯2)-w2,η2(y2,x¯2)〉2+ψ2(x¯2,y2)-ψ2(x¯2,x¯2)≥0∀y2∈X2.
Problem (1.4) has been considered recently by Ding and Wang in [17].
(3)
In finite dimensional spaces a particular problem of (1.2) has been considered by Mordukhovich et al. in [19] for finding a common solution of a variational inequality problem and an equilibrium problem by an approach based on an hybrid proximal point algorithm.The organization of the paper is as follows. In Section2, we give some definitions and preliminary results that we will need in the sequel. We introduce in Section 3 an auxiliary principle, for the problem studied and we show that under some suitable conditions this problem has a unique solution. Further, by using the auxiliary principle we consider an algorithm to approach the solution of the main problem studied in this paper and discuss its convergence. Finally, we end the paper by some commentaries on the approach used and we give some comparisons with some known results in this direction.


## 1.1. Some Special Cases

(1)
If for eachi∈I, Xi=Xi*=Hi is a Hilbert space, Ki=Hi, Fi≡0,  wi=0 and for each (x,y)∈H1×H2, Ri(x)=x and Ai(y)=y, then problem (1.2) reduces to the problem of finding (x¯1,x¯2)∈H1×H2 such that
(1.3)〈N1(x¯1,x¯2),η1(y1,x¯1)〉1+ψ1(x¯1,y1)-ψ1(x¯1,x¯1)≥0∀y1∈H1;〈N2(x¯1,x¯2),η2(y2,x¯2)〉2+ψ2(x¯2,y2)-ψ2(x¯2,x¯2)≥0∀y2∈H2.
Problem (1.3) has been introduced and studied by Kazmi and Khan in [16].
(2)
If for eachi∈I, Xi is a reflexive Banach space, Ki=Xi and Fi≡0, then problem (1.2) reduces to the problem of finding (x¯1,x¯2)∈X1×X2, (u¯1,v¯1)∈R1(x¯1)×A1(x¯2) and (u¯2,v¯2)∈R2(x¯1)×A2(x¯2) such that
(1.4)〈N1(u¯1,v¯1)-w1,η1(y1,x¯1)〉1+ψ1(x¯1,y1)-ψ1(x¯1,x¯1)≥0∀y1∈X1;〈N2(u¯2,v¯2)-w2,η2(y2,x¯2)〉2+ψ2(x¯2,y2)-ψ2(x¯2,x¯2)≥0∀y2∈X2.
Problem (1.4) has been considered recently by Ding and Wang in [17].
(3)
In finite dimensional spaces a particular problem of (1.2) has been considered by Mordukhovich et al. in [19] for finding a common solution of a variational inequality problem and an equilibrium problem by an approach based on an hybrid proximal point algorithm.The organization of the paper is as follows. In Section2, we give some definitions and preliminary results that we will need in the sequel. We introduce in Section 3 an auxiliary principle, for the problem studied and we show that under some suitable conditions this problem has a unique solution. Further, by using the auxiliary principle we consider an algorithm to approach the solution of the main problem studied in this paper and discuss its convergence. Finally, we end the paper by some commentaries on the approach used and we give some comparisons with some known results in this direction.


## 2. Preliminaries

In this section, we present some basic concepts, properties, and notations that we will consider in the development of our work. LetX be a real Banach space with norm ∥·∥, X* its dual space, and 〈·,·〉 denote the duality pairing between X* and X, and let K be a nonempty closed convex subset of X. For a finite subset A of K, we denote by co(A) the convex hull of A. Let 𝒞ℬ(X*) be the family of all nonempty closed and bounded subsets of X*, and let ℋ(·,·) be the Hausdorff metric on 𝒞ℬ(X*) defined for all A,D∈𝒞ℬ(X*) by
(2.1)H(A,D)=max⁡{sup⁡a∈Ad(a,D),sup⁡b∈Dd(b,A)},
where d(a,D)=infd∈D∥a-d∥.Definition 2.1.
A mappingT:X→X* is said to be(i)
monotone if and only if〈T(x)-T(y),x-y〉≥0 for all x,y∈X;(ii)
α-strongly monotone with α>0 if and only if
(2.2)〈T(x)-T(y),x-y〉≥α‖x-y‖2∀x,y∈X;(iii)
L-Lipschitz continuous if there exists L>0 such that
(2.3)‖T(x)-T(y)‖≤L‖x-y‖∀x,y∈X.Definition 2.2.
LetT:X→X* be a bounded linear operator. T is said to be δ-strongly positive if there exists δ>0 such that
(2.4)〈T(x),x〉≥δ‖x‖2,∀x∈X.Remark 2.3.
One can easily see that if a bounded linear operatorT:X→X* is δ-strongly positive, then it is δ-strongly monotone and ∥T∥-Lipschitz continuous, where ∥T∥ is the operator norm of T.Definition 2.4.
LetX1 and X2 be two Banach spaces with respective norm ∥·∥1 and ∥·∥2. A mapping A1:X1×X2→X1 is said to be (β1,ξ2)-Lipschitz continuous if there exist constants β1,ξ2>0 such that
(2.5)‖A1(x1,y1)-A1(x2,y2)‖1≤β1‖x1-x2‖1+ξ2‖y1-y2‖2∀(x1,y1),(x2,y2)∈X1×X2.Definition 2.5.
A mappingη:X×X→X is said to be (i)
affine in the second argument if(2.6)η(y,tx+(1-t)z)=tη(y,x)+(1-t)η(y,z),∀t∈[0,1],∀x,y,z∈K;(ii)
τ-Lipschitz continuous if there exists a constant τ>0 such that
(2.7)‖η(x,y)‖≤τ‖x-y‖,∀x,y∈X.Definition 2.6.
A real-valued functionψ:K×K→ℝ is said to be skew-symmetric if for all x,y∈K,
(2.8)ψ(x,x)-ψ(x,y)-ψ(y,x)+ψ(y,y)≥0.
The skew-symmetric functions have certain properties, see [20], which can be regarded as analogs of the conditions governing the gradient monotonicity and nonnegativity of the second derivative of convex functions.Definition 2.7.
LetF:K×K→ℝ be a real-valued bifunction. Then,  (i)
F is said to be monotone if
(2.9)F(x,y)+F(y,x)≤0,∀x,y∈K;(ii)
F is said to be α-strongly monotone if there exists α>0 such that
(2.10)F(x,y)+F(y,x)≤-α‖x-y‖2,∀x,y∈K;(iii)
F is said to be upper-hemicontinuous if, for all x,y,z∈K, the mapping g:[0,1]→ℝ defined by g(t)=F(tz+(1-t)x,y) is upper-semicontinuous.Remark 2.8.
Clearly, strong monotonicity ofF implies monotonicity of F.
Now, letSF denote the solution set of the equilibrium problem (EP) associated to a bifunction F, that is, SF={x¯∈K:F(x¯,y)≥0∀y∈K}. In many situations, we usually need some more information on the structure of the solution set SF when it is nonempty. As a preliminary result that we will need in the sequel, the following Lemma gives some sufficient conditions which insure that SF is convex and closed.Lemma 2.9.
LetX be a Banach space, and let K be a closed convex subset of X. Let F:K×K→ℝ be a real-valued bifunction such that F(x,x)≥0 for all x∈K. Assume that(i)
F is monotone and upper-hemicontinuous;(ii)
for eachx∈K fixed, the function y↦F(x,y) is convex and lower semicontinuous. Then, the solution set SF is convex and closed whenever it is nonempty.Proof.
Assume thatSF≠∅. Let u¯,v¯∈SF and for λ∈]0,1[ set xλ=λu¯+(1-λ)v¯∈K. Let us show that xλ∈SF. Since u¯,v¯∈SF, it follows
(2.11)F(u¯,y)≥0,F(v¯,y)≥0∀y∈K.
From the monotonicity of F, one deduces
(2.12)F(y,u¯)≤0,F(y,v¯)≤0∀y∈K.
Therefore, from the convexity of F with respect to its second argument, it follows
(2.13)F(y,xλ)≤0,∀y∈K.
Now for t∈]0,1[ and y∈K, set xt=ty+(1-t)xλ∈K. Since F is an equilibrium bifunction and F is convex with respect to its second argument, it follows that
(2.14)0≤F(xt,xt)≤tF(xt,y)+(1-t)F(xt,xλ).
Taking account of relation (2.13) with y=xt, it follows that tF(xt,y)≥0 for all y∈K, and therefore since t>0(2.15)F(xt,y)≥0,∀y∈K.
Since F is upper-hemicontinuous, it follows that
(2.16)F(xλ,y)≥limsup⁡t→0F(xt,y)≥0.
Therefore, xλ∈SF, and hence SF is convex.
Now, let us show thatSF is closed. To this aim, let {xn}⊂SF such that xn→x¯. Let us show that x¯∈SF. One has F(xn,y)≥0 for all y∈K, since F is monotone, it follows that
(2.17)F(y,xn)≤0,∀y∈K.
From the lower semicontinuity of F with respect to the second argument, it follows that
(2.18)F(y,x¯)≤liminf⁡n→+∞F(y,xn)≤0.
By a similar argument as above, one can easily show that F(x¯,y)≥0 for all y∈K. Therefore, x¯∈SF, which completes the proof.In the sequel, we will need the following result that we present in a more general setting and for which we refer to [21].Lemma 2.10.
LetX be a Hausdorff topological vector space and K a closed convex subset of X. Consider two real bifunctions Φ,Ψ:K×K→ℝ such that(i)
for eachx,y∈K, if Ψ(x,y)≤0 then Φ(x,y)≤0;(ii)
for each fixedx∈K, the function y∈K↦Φ(x,y) is lower semicontinuous on every compact subset of K;(iii)
for each finite subsetA of K, one has
(2.19)sup⁡y∈co⁡(A)min⁡x∈AΨ(x,y)≤0;(iv)
coercivity: there exists a nonempty compact convex subsetC of X such that either (a) or (b) in the following holds
(a)
for ally∈K∖C,there  existsx∈Csuch  thatΦ(x,y)>0;(b)
there existsx0∈Csuch  that∀y∈K∖CΨ(x0,y)>0.
Then, there existsy¯∈C such that Φ(x,y¯)≤0 for all x∈K. Furthermore, the set of solution is compact.Remark 2.11.
(
1
) Condition (iii) in Lemma 2.10 is much more related to convexity assumptions of the bifunction Ψ, see Proposition 2.12 in the following.
(
2
) If K is compact, then condition (iv) in Lemma 2.10 can be dropped.
The following proposition gives some sufficient conditions which insure condition (iii) in Lemma2.10.Proposition 2.12.
Suppose that(i)
Ψ(x,x)≤0  for each x∈K;(ii)
for eachy∈K fixed, the set  {x∈K:Ψ(x,y)>0}  is convex.
Then, condition (iii) of Lemma2.10 is staisfied.Proof.
Suppose by contradiction that condition (iii) of Lemma2.10 is not satisfied. Then, there exist x1,…,xn∈K and λ1,…,λn≥0, with ∑i=1n‍λi=1, such that Ψ(xi,∑j=1n‍λjxj)>0.
Therefore, by settingy=∑j=1n‍λjxj, it follows from (ii) that Ψ(y,y)>0, which contradicts assumption (i).As a consequence of Lemma2.10, we obtain the following result on existence of mixed equilibrium problem that we will need in the sequel. We will include its proof for completeness.Lemma 2.13.
LetX be a Banach space and K a closed convex subset of X. Let f,g:K×K→ℝ be two real bifunctions such that(i)
f(x,x)≥0 for all x∈K;f is monotone and upper-hemicontinuous; for each x∈K fixed, the function y↦f(x,y) is convex and lower semicontinuous;(ii)
g(x,x)≥0 for all x∈K; for each y∈K fixed, the function x↦g(x,y) is upper-semicontinuous; for each x∈K fixed, the function y↦g(x,y) is convex and lower semicontinuous;(iii)
coercivity: there exists a nonempty compact convex subsetC of X and y0∈C∩K such that
(2.20)f(x,y0)+g(x,y0)<0for  eachx∈K∖C.Then, there exists x¯∈C∩K such that
(2.21)f(x¯,y)+g(x¯,y)≥0∀y∈K.
Furthermore, the solution set Sf,g of the mixed equilibrium problem (2.21) is compact and convex.Proof.
The proof is a direct consequence of Lemma2.10 by setting
(2.22)Φ(x,y)=f(x,y)-g(y,x),Ψ(x,y)=-g(y,x)-f(y,x).Remark 2.14.
(
1
) Lemma 2.13 is in fact a slight extension of Theorem 1 in [2] and Theorem 4.5 in [21] where the equilibrium condition f(x,x)=g(x,x)=0 has been relaxed by assuming f(x,x)≥0 and g(x,x)≥0 for all x∈K.
(
2
) If X is a reflexive Banach space endowed with its weak topology σ(X,X*), then the coercivity condition (iii) in Lemma 2.13 can be replaced by the following condition:(iii)′ there  existsy0∈K such  thatlim∥x-y0∥→+∞g(x,y0)/∥x-y0∥=-∞.
Ended, letr1>0, and set B(y0,r1)={x∈X:∥x-y0∥≤r1}. One has B(y0,r1) a convex and σ(X,X*)-compact subset of X. Since X is a reflexive Banach space, f(y0,·) is lower semicontinuous and B(y0,r) is weakly compact, it follows that there exists α0∈ℝ such that f(y0,y)>α0 for all y∈B(y0,r1). Let x∈K∖B(y0,r1), and set
(2.23)y=r1‖x-y0‖x+(1-r1‖x-y0‖)y0.
Sincef(y0,·) is convex, y∈B(y0,r1) and f(y0,y0)≥0, one deduces
(2.24)α0≤r1‖x-y0‖f(y0,x)+(1-r1‖x-y0‖)f(y0,y0)∀x∈K∖B(y0,r1).
It follows that(2.25)f(y0,x)≥‖x-y0‖r1(α0-f(y0,y0))+f(y0,y0).
Thus,(2.26)f(y0,x)≥‖x-y0‖r1(α0-f(y0,y0)),∀x∈K∖B(y0,r1).
Sincef is monotone, it follows from relation (2.26) that for all
(2.27)f(x,y0)+g(x,y0)≤g(x,y0)-α0-f(y0,y0)r1‖x-y0‖.
Sinceg(x,y0)/∥x-y0∥→-∞ when ∥x-y0∥→+∞, then there exists r2>0 such that for x∈K with ∥x-y0∥>r2 one has
(2.28)g(x,y0)-α0-f(y0,y0)r1‖x-y0‖<0.
Taker=max{r1,r2} and set C={x∈X:∥x-y0∥≤r}, then from relations (2.27) and (2.28), one deduces that for each x∈K∖C one has
(2.29)f(x,y0)+g(x,y0)<0.
Hence, condition (iii) in Lemma2.13 is satisfied.
We end this section by the following result related to the Hausdorff metric that we will need in the sequel and for which we refer to [22].Lemma 2.15.
LetE be a complete metric space and R:E⇉𝒞ℬ(E) a set-valued mapping. Then, for any given ɛ>0 and any given x,y∈E and u∈R(x), there exists v∈R(y) such that
(2.30)d(u,v)≤(1+ɛ)H(R(x),R(y)).


## 3. Approximation by an Auxiliary Principle

In order to get approximate solutions for the system (1.2) of generalized mixed equilibrium problem involving generalized mixed variational-like inequality problems (SGMEP), we consider the following auxiliary problem: for i∈I={1,2} and for given mappings Ti:Xi→Xi*, ρi>0, (x1,x2)∈K1×K2, (u1,v1)∈R1(x1)×A1(x2), and (u2,v2)∈R2(x1)×A2(x2),
(3.1)(AP){find(z1,z2)∈K1×K2such  that  for  allyi∈Kiρi[Fi(zi,yi)+〈Ni(ui,vi)-wi,ηi(yi,zi)〉i+ψi(zi,yi)-ψi(zi,zi)]ρi+〈Ti(yi-zi),zi-xi〉i≥0.In this section, we give some existence results of solutions for the auxiliary problem (AP). The results obtained will be needed in the sequel to generate a unified algorithm to approach solutions of the system (1.2) under some weaker assumptions in comparison with some known results in literature.Theorem 3.1.
For eachi∈I={1,2}, let Xi be a Banach space and Ki a nonempty closed convex subset of Xi, Fi,ψi:Ki×Ki→ℝ two real-valued bifunctions, Ti:Xi→Xi* a bounded linear operator, ρi>0 and ηi:Xi×Xi→Xi single-valued mappings such that  (i)
Fi(x,x)≥0 for all x∈Ki; Fi is monotone and upper-hemicontinuous; for each x∈Ki fixed, the function y↦Fi(x,y) is convex and lower semicontinuous;(ii)
ηi is affine in the first argument and continuous in the second argument such that
(3.2)ηi(x,y)+ηi(y,x)=0,∀x,y∈Ki;(iii)
ψi is skew symmetric and continuous; for each y∈Ki fixed, the function x↦ψi(x,y) is convex;(iv)
Ti is δi-strongly positive;(v)
coercivity: for eachxi∈Ki, ui∈X1* and vi∈X2*, there exists a nonempty compact convex subset Cxi,ui,vii of Xi and yi0∈Cxi,ui,vii∩Ki such that ∀zi∈Ki∖Cxi,ui,vii(3.3)ρi[Fi(zi,yi0)+〈Ni(ui,vi)-wi,ηi(yi0,zi)〉i+ψi(zi,yi0)-ψ(zi,zi)]+〈Ti(yi0-zi),zi-xi〉<0.
Then, the auxiliary problem (AP) has a unique solution.Proof.
The proof of this lemma is a direct application of Lemma2.13 by considering, for each i∈I={1,2}, the bifunctions fi,gi:Ki×Ki→ℝ defined by
(3.4)fi(zi,yi)=ρi[Fi(zi,yi)+〈Ni(ui,vi)-wi,ηi(yi,zi)〉i+ψi(zi,yi)-ψi(zi,zi)]gi(zi,yi)=〈Ti(yi)-Ti(zi),zi-xi〉i,
where (x1,x2)∈K1×K2, (u1,v1)∈R1(x1)×A1(x2), and (u2,v2)∈R2(x1)×A2(x2) are given. We need only to show that the solution is unique. To this aim, suppose that problem (AP) has two solutions zi1 and zi2, then for i∈I and for all z∈Ki, we have
(3.5)ρi[Fi(zi1,z)+〈Ni(ui,vi)-wi,ηi(z,zi1)〉i+ψi(zi1,z)-ψi(zi1,zi1)]+〈Ti(z)-Ti(zi1),zi1-xi〉i≥0,(3.6)ρi[Fi(zi2,z)+〈Ni(ui,vi)-wi,ηi(z,zi2)〉i+ψi(zi2,z)-ψi(zi2,zi2)]+〈Ti(z)-Ti(zi2),zi2-xi〉i≥0.
Take z=zi2 in relation (3.5) and z=zi1 in relation (3.6) and adding the two inequalities, one obtain
(3.7)ρi[Fi(zi1)+Fi(zi2,zi1)+〈Ni(ui,vi)-wi,ηi(zi1,zi2)+ηi(zi2,zi1)〉iψi(zi1,zi2)ρiFi+ψi(zi2,zi1)-ψi(zi1,zi1)-ψi(zi2,zi2)]≥〈Ti(zi2-zi1),zi2-zi1〉i.
Since for each i∈I, Fi is monotone, ψi is skew symmetric, and Ti is δi-strongly positive, it follows that
(3.8)0≥〈Ti(zi2-zi1),zi2-zi1〉i≥δi‖zi1-zi2‖i2.
Therefore, zi1=zi2 for i∈I={1,2}, which completes the proof.Theorem 3.2.
For eachi∈I={1,2}, let Xi be a reflexive Banach space and Ki a nonempty closed convex subset of Xi, Fi,ψi:Ki×Ki→ℝ two real-valued bifunctions, Ti:Xi→Xi*  a bounded linear operator, ρi>0 and ηi:Xi×Xi→Xi  single-valued mappings such that (i)
Fi(x,x)≥0 for all x∈Ki; Fi is monotone and upper-hemicontinuous; for each x∈Ki fixed, the function y↦Fi(x,y) is convex and lower semicontinuous;(ii)
ηi is affine in the first argument and continuous in the second argument such that
(3.9)ηi(x,y)+ηi(y,x)=0,∀x,y∈Ki;(iii)
ψi is skew symmetric and continuous; for each y∈Ki fixed, the function x↦ψi(x,y) is convex;(iv)
Ti is δi-strongly positive linear operator.
Then, the auxiliary problem (AP) has a unique solution.Proof.
For eachi∈I and given (x1,x2)∈K1×K2, (u1,v1)∈R1(x1)×A1(x2), and (u2,v2)∈R2(x1)×A2(x2), define the following bifunctions fi,gi:Ki×Ki→ℝ by
(3.10)fi(zi,yi)=ρi[Fi(zi,yi)+〈Ni(ui,vi)-wi,ηi(yi,zi)〉i+ψi(yi,zi)-ψi(zi,zi)],gi(zi,yi)=〈Ti(yi)-Ti(zi),zi-xi〉i.
One can easily see that conditions (i)–(iii) above imply conditions (i) and (ii) of Lemma 2.13. In order to get the conclusion, we need only to show that the coercivity condition (iii) of Lemma 2.13 is satisfied. To this aim, taking into account Remark 2.14 (2), we need only to show that for some vi0∈Ki one has gi(ui,vi0)/∥vi0-ui∥i→-∞ when ∥vi0-ui∥i→+∞. Let vi0∈Ki. Then,
(3.11)gi(ui,vi0)=〈Ti(vi0)-Ti(ui),ui-xi〉i=〈Ti(vi0-ui),ui-vi0〉i+〈Ti(vi0-ui),vi0-xi〉i=-〈Ti(vi0-ui),vi0-ui〉i+〈Ti(vi0-ui),vi0-xi〉i≤-δ‖vi0-ui‖i2+‖Ti‖‖vi0-ui‖i‖vi0-xi‖i.
Therefore,
(3.12)gi(ui,vi0)‖vi0-ui‖i≤-δ‖vi0-ui‖i+‖Ti‖‖vi0-xi‖i.
It follows that gi(ui,vi0)/∥vi0-ui∥i→-∞ when ∥vi0-ui∥i→+∞ which completes the proof.Remark 3.3.
Theorem3.2 improves recent results given by Ding [18, Theorem 3.1] since the bifunction Fi is not needed to be δi-Lipschitz continuous and weakly upper semicontinuous with respect to the first argument. We mention also that all the results obtained in [18] are under the assumption int{yi∈Ki,ψi(yi,yi)<∞}=intKi≠∅; in our approach, this assumption is not needed.
Theorem3.2 shows that the auxiliary problem (AP) has a unique solution; we can define the following general iterative method to approach the solution of system (1.2) of generalized mixed equilibrium problem involving variational-like inequalities (SGMEP).Algorithm 3.4.
For a given(x10,x20)∈K1×K2, (u10,v10)∈R1(x10)×A1(x20), and (u20,v20)∈R2(x10)×A2(x20). By Theorem 3.1, the auxiliary problem (AP) has a unique solution (x11,x21)∈K1×K2, that is, for each i∈I, we have
(3.13)ρi1[Fi(xi1,yi)+〈Ni(ui0,vi0)-wi,ηi(yi,xi1)〉i+ψi(xi1,yi)-ψi(xi1,xi1)]ρi1+〈Ti(yi)-Ti(xi1),xi1-xi0〉i≥0∀yi∈Ki.
Since for each i∈I, ui0∈Ri(xi0)∈𝒞ℬ(X1*), and vi0∈Ai(x20)∈𝒞ℬ(X2*), by Lemma 2.15, there exist ui1∈Ri(x11) and vi1∈Ai(x21) such that
(3.14)‖ui1-ui0‖1≤(1+1)H1(Ri(x11),Ri(x10))‖vi1-vi0‖2≤(1+1)H2(Ai(x21),Ai(x20)),
where ℋ1 and ℋ2 are the Hausdorff metrics on 𝒞ℬ(X1*) and 𝒞ℬ(X2*), respectively.
By using Theorem3.1 again, the auxiliary problem (AP) has a unique solution (x12,x22)∈K1×K2 such that
(3.15)ρi2[Fi(xi2,yi)+〈Ni(ui1,vi1)-wi,ηi(yi,xi2)〉i+ψi(xi2,yi)-ψi(xi2,xi2)]ρi2+〈Ti(yi)-Ti(xi2),xi2-xi1〉i≥0,∀yi∈Ki.
By induction, we can construct an iterative algorithm to compute the approximate solution for the system (1.2) as follows: for given (x10,x20)∈K1×K2, (u10,v10)∈R1(x10)×A1(x20), and (u20,v20)∈R2(x10)×A2(x20), there exist sequences {x1n},{x2n},{u1n},{u2n},{v1n}, and {v2n} such that for each i∈I(3.16)‖uin+1-uin‖1≤(1+1n+1)H1(Ri(x1n+1),Ri(x1n)),uin∈Ri(x1n),‖vin+1-vin‖2≤(1+1n+1)H2(Ai(x2n+1),Ai(x2n)),vin∈Ai(x2n),ρin+1[Fi(xin+1,yi)+〈Ni(uin,vin)-wi,ηi(yi,xin+1)〉i+ψi(xin+1,yi)-ψi(xin+1,xin+1)]ρin+1+〈Ti(yi)-Ti(xin+1),xin+1-xin〉i≥0∀yi∈Ki,n=0,1,2,….
The following convergence analysis is presented for the algorithm above.Theorem 3.5.
Under the hypotheses of Theorem3.2, further assume that for each i∈I,(i)
Fi is σi-strongly monotone and upper-hemicontinuous; Ni is (βi,ξi)-mixed Lipschitz continuity; Ri is ki-ℋ1-Lipschitz continuous, and Ai is μi-ℋ2-Lipschitz continuous;(ii)
ηi is τi-Lipschitz continuous;(iii)
the sequence{ρn}n∈ℕ of positive real numbers is increasing and limn→+∞ρn=+∞.
Furthermore, assume that the following condition holds:(3.17)(C1){Λ=max⁡{θ1+θ2,ϑ1+ϑ2}<1,whereθ1=τ1β1k1σ1,θ2=τ2β2k2σ2,ϑ1=τ1ξ1μ1σ1,ϑ2=τ2ξ2μ2σ2.
Then, the sequences {x1n},{x2n},{u1n},{u2n},{v1n}, and {v2n} generated by Algorithm 3.4 converge strongly to x¯1,x¯2,u¯1,v¯1,u¯2, and v¯2, respectively, where (u¯1,v¯1)∈T1(x¯1)×A1(x¯2),(u¯2,v¯2)∈T2(x¯1)×A2(x¯2), and (x¯1,x¯2,u¯1,v¯1,u¯2,v¯2) is a solution of the System of Generalized Mixed Equilibrium Problem involving variational-like in equalities (1.2).Proof.
By the definition of Algorithm3.4, we have for i∈I,
(3.18)ρin[Fi(xin,yi)+〈Ni(uin-1,vin-1)-wi,ηi(yi,xin)〉i+ψi(xin,yi)-ψi(xin,xin)]ρin+〈Ti(yi)-Ti(xin),xin-xin-1〉i≥0∀yi∈Ki,(3.19)ρin+1[Fi(xin+1,yi)+〈Ni(uin,vin)-wi,ηi(yi,xin+1)〉i+ψi(xin+1,yi)-ψi(xin+1,xin+1)]ρin+〈Ti(yi)-Ti(xin+1),xin+1-xin〉i≥0∀yi∈Ki.
By considering yi=xin+1 in relation (3.18) and yi=xin in relation (3.19), and taking into account ηi(xin+1,xin)=-ηi(xin,xin+1), one obtains by adding the two inequalities
(3.20)Fi(xin+1,xin)+Fi(xin,xin+1)+〈Ni(uin-1,vin-1)-Ni(uin,vin),ηi(xin+1,xin)〉i(xin,yi)+ψi(xin,xin+1)+ψi(xin+1,xin)-ψi(xin,xin)-ψi(xin+1,xin+1)(xin,yi)+1ρin〈Ti(xin+1-xin),xin-xin-1〉i(xin,yi)-1ρin+1〈Ti(xin-xin+1),xin-xin+1〉i≥0.
Since Fi is σi-monotone, ψi is skew-symmetric, and Ti is δi-strongly positive and ∥Ti∥-Lispchitz continuous, it follows from relation (3.20) that
(3.21)‖Ni(uin-1,vin-1)-Ni(uin,vin)‖i‖ηi(xin+1,xin)‖+‖Ti‖ρin‖xin+1-xin‖i‖xin-xin-1‖i‖Ti‖≥[σi+δiρin+1]‖xin-xin+1‖i2.
Note that N1 is (β1,ξ1)-mixed Lipschitz continuous, R1 is k1-ℋ1-Lipschitz continuous, and A1 is μ1-ℋ2-Lipschitz continuous, it follows by Algorithm 3.4(3.22)‖Ni(uin-1,vin-1)-Ni(uin,vin)‖i≤β1‖uin-1-uin‖1+ξ1‖vin-1-vin‖2≤β1(1+1n)H1(R1(x1n-1),R1(x1n))+ξ1(1+1n)H2(R1(x2n-1),R1(x2n))≤β1k1(1+1n)‖x1n-1-x1n‖1+ξ1μ1(1+1n)‖x2n-1-x2n‖2.
Since η1 is τ1-Lipschitz continuous, it follows from relation (3.21) considered for i=1 that
(3.23)[σ1+δ1ρ1n+1]‖x1n-x1n+1‖1≤[τ1β1k1(1+1n)+‖T1‖ρ1n]‖x1n-1-x1n‖1+τ1ξ1μ1(1+1n)‖x2n-1-x2n‖2.
Hence,
(3.24)‖x1n-x1n+1‖1≤θ1n‖x1n-1-x1n‖1+ϑ1n‖x2n-1-x2n‖2,
with
(3.25)θ1n=[ρ1n+1τ1β1k1(1+1/n)+ρ1n+1‖T1‖/ρ1n][ρ1n+1σ1+δ1],ϑ1n=ρ1n+1τ1ξ1μ1(1+1/n)[ρ1n+1σ1+δ1].
Similarly, by the assumptions on F2, N2, η2, R2, A2, and T2, it follows from relation (3.21) considered for i=2 that
(3.26)‖x2n-x2n+1‖2≤θ2n‖x1n-1-x1n‖1+ϑ2n‖x2n-1-x2n‖2,
with
(3.27)θ2n=[ρ2n+1τ2β2k2(1+1/n)][ρ2n+1σ2+δ2],ϑ2n=ρ2n+1τ2ξ2μ2(1+1/n)+ρ2n+1‖T2‖/ρ2n[ρ2n+1σ2+δ2].
Adding the inequalities (3.24) and (3.26), we obtain
(3.28)‖x1n-x1n+1‖1+‖x2n-x2n+1‖2≤(θ1n+θ2n)‖x1n-1-x1n‖1+(ϑ1n+ϑ2n)‖x2n-1-x2n‖2≤Λn(‖x1n-1-x1n‖1+‖x2n-1-x2n‖2),
where Λn=max{θ1n+θ2n,ϑ1n+ϑ2n}.
OnX1×X2, let us consider the norm ∥·∥* defined by ∥(x,y)∥*=∥x∥1+∥y∥2 for all (x,y)∈X1×X2, that is, (X1×X2,∥·∥*) is a Banach space. It follows from relation (3.28) that
(3.29)‖(x1n,x2n)-(x1n+1,x2n+1)‖*≤Λn‖(x1n-1,x2n-1)-(x1n,x2n)‖*.
Taking account of the assumptions, it is easy to see that Λn→Λ as n→∞. From condition (C1) (relation (3.17)), we know that 0<Λ<1. Hence, there exist Λ0∈(0,1) and n0>0 such that Λn≤Λ0<1 for all n≥n0. Therefore, it follows from (3.29) that
(3.30)‖(x1n,x2n)-(x1n+1,x2n+1)‖*≤Λ0‖(x1n-1,x2n-1)-(x1n,x2n)‖*.
This implies that {(x1n,x2n)} is a Cauchy sequence in X1×X2. Thus, (x1n,x2n) converges strongly to some (x¯1,x¯2)∈K1×K2.
By Algorithm3.4 and the Lipschitz continuity assumption on R1,R2, A1, and A2, we have
(3.31)‖uin+1-uin‖1≤(1+1n+1)H1(Ri(x1n+1),Ri(x1n))≤ki(1+1n+1)‖x1n+1-x1n‖1,‖vin+1-vin‖2≤(1+1n+1)H2(Ai(x2n+1),Ai(x2n))≤μi(1+1n+1)‖x2n+1-x2n‖2.
It follows, for each i∈I, that {uin} is a Cauchy sequence in X1* and {vin} is a Cauchy sequence in X2*. Thus, there exists (u¯i,v¯i)∈X1*×X2* such that (uin,xin) converges strongly to (u¯i,v¯i).
Noting thatu1n∈R1(x1n), it follows that
(3.32)d(u¯1,R1(x1n))≤‖u¯1-u1n‖1+d(u1n,R1(x1n))+H1(R1(x1n),R1(x¯1))≤‖u¯1-u1n‖1+k1‖x¯1-x1n‖1⟶0asn⟶∞.
Hence, we must have u¯1∈R1(x¯1). Similarly, one can show that u¯2∈R2(x¯1),  v¯1∈A1(x¯2), and v¯2∈A2(x¯2). By Algorithm 3.4, we have that, for all i∈I,
(3.33)ρin+1[Fi(xin+1,yi)+〈Ni(uin,vin)-wi,ηi(yi,xin+1)〉i+ψ(xin+1,yi)-ψ(xin+1,xin+1)]ρin+1+〈Ti(yi)-Ti(xin+1),xin+1-xin〉≥0∀yi∈Ki,n=0,1,2,…
Since Ni is (βi,ξi)-mixed Lipschitz continuous and ηi is continuous in the second argument, one has, for each yi∈Ki,
(3.34)|〈Ni(uin,vin)-wi,ηi(yi,xin+1)〉i-〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i|yi,≤|〈Ni(uin,vin)-Ni(u¯i,v¯i),ηi(yi,xin+1)〉i|+|〈Ni(u¯i,v¯i)-wi,ηi(yi,xin+1)-ηi(yi,x¯i)〉i|yi,≤‖Ni(uin,vin)-Ni(u¯i,v¯i)‖‖ηi(yi,xin+1)‖+|〈Ni(u¯i,v¯i)-wi,ηi(yi,xin+1)-ηi(yi,x¯i)〉i|.
Hence,
(3.35)|〈Ni(uin,vin)-wi,ηi(yi,xin+1)〉i-〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i|⟶0asn⟶∞.
From the monotonicity of Fi and the linearity of Ti, one has
(3.36)〈Ni(uin,vin)-wi,ηi(yi,xin+1)〉i-〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉iNi+ψi(xin+1,yi)-ψi(xin+1,xin+1)+‖Ti‖ρin+1‖yi-xin+1‖‖xin+1-xin‖Ni≥Fi(yi,xin+1)-〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i∀yi∈Ki,∀n=0,1,2,…
Taking into account result (3.35) and since, for each i∈I, Fi(y,·) is a lower semicontinuous function, ψi is continuous, and the sequence {xin} converges strongly to x¯i, it follows from relation (3.36) by passing to the limit when n goes to infinity that
(3.37)ψi(yi,x¯i)-ψi(x¯i,x¯i)≥Fi(yi,x¯i)-〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i.
Now, for each i∈I and for t∈(0,1] and yi∈Ki, set yi(t)=tyi+(1-t)x¯i. Since Ki is convex, then yi(t)∈Ki for t∈(0,1]; it follows that
(3.38)〈Ni(u¯i,v¯i)-wi,ηi(yi(t),x¯i)〉i+ψi(yi(t),x¯i)-ψi(x¯i,x¯i)≥Fi(yi(t),x¯i).
Taking into account the convexity of ψi(·,x¯i), the fact that Ni is affine with respect to the first argument and ηi(x¯i,x¯i)=0, one has
(3.39)Fi(yi(t),x¯i)≤t(ψi(yi,x¯i)-ψi(x¯i,x¯i))+t〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i.
On the other hand, since Fi(xi,xi)=0 for each xi∈Ki and Fi(yi,·) is convex, it follows that
(3.40)0≤Fi(yi(t),yi(t))≤tFi(yi(t),yi)+(1-t)F(yi(t),x¯i).
Hence, from relation (3.39), one has
(3.41)0≤t[Fi(yi(t),yi)+(1-t)(ψi(yi,x¯i)-ψi(x¯i,x¯i)+〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i)].
Therefore, for all t∈]0,1], one has
(3.42)0≤Fi(yi(t),yi)+(1-t)(ψi(yi,x¯i)-ψi(x¯i,x¯i)+〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i).
Since for each yi∈Ki, x↦Fi(x,y) is upper hemicontinuous, it follows by passing to limit when t→0 in the previous inequality that
(3.43)Fi(x¯i,yi)+〈Ni(u¯i,v¯i)-wi,ηi(yi,x¯i)〉i+ψi(x¯i,yi)-ψi(x¯i,x¯i)≥0∀yi∈Ki,
which completes the proof.


## 4. Commentaries

In conclusion, the approach used in this paper lets us improve and extend some new results in literature related to the problem studied. To be more precise,(1)
Theorems3.2 and 3.5 improve recent results given by Ding [18, Theorem 3.1, Theorem 4.1], since the bifunction Fi is not needed to be δi-Lipschitz continuous nor weakly upper semicontinuous with respect to the first argument in all steps of the procedure, firstly when dealing with the existence of solutions for the auxiliary problem and secondly when studying the convergence of the algorithm;(2)
we mention also that all the results obtained in [18] are under the assumption int{yi∈Ki,ψi(yi,yi)<∞}=intKi≠∅; in our approach, this assumption is not needed;(3)
in the special case whereFi≡0, the results obtained improve those obtained by Ding and Wang [17], since the assumptions on the bifunctions ψi have been relaxed.



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*Source: 843486-2012-07-19.xml*