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

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.

1. Introduction and the Problem Statement

Let 𝐾 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: (EP)Find𝑢∈𝐾suchthat𝐹(𝑢,𝑣)≥0∀𝑣∈𝐾. (1.1) 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, let 𝐼={1,2} be an index set. For each 𝑖∈𝐼, let 𝑋_𝑖 be a Banach space, 𝑋^∗_𝑖 its dual, ⟨⋅,⋅⟩_𝑖 is the duality pairing between 𝑋^∗_𝑖 and 𝑋_𝑖, ‖⋅‖_𝑖 denotes the norm in 𝑋_𝑖 and 𝑋^∗_𝑖, and let 𝐾_𝑖 be a nonempty closed convex subset of 𝑋_𝑖. We will denote by 𝒞ℬ(𝑋^∗_𝑖) the family of all nonempty closed and bounded subsets of 𝑋^∗_𝑖.

For each 𝑖∈𝐼, let 𝑅_𝑖∶𝐾₁⇉𝒞ℬ(𝑋^∗₁) and 𝐴_𝑖∶𝐾₂⇉𝒞ℬ(𝑋^∗₂) be multivalued mappings, 𝑁_𝑖∶𝑋^∗₁×𝑋^∗₂→𝑋^∗_𝑖 and 𝜂_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖 be single-valued mappings, 𝐹_𝑖,𝜓_𝑖∶𝐾_𝑖×𝐾_𝑖→ℝ be two real-valued functions, and let 𝑤_𝑖∈𝑋^∗_𝑖. We consider the following System of Generalized Mixed Equilibrium Problem involving variational-like inequalities (SGMEP).

Find (𝑥₁,𝑥₂)∈𝐾₁×𝐾₂, (𝑢₁,𝑣₁)∈𝑅₁(𝑥₁)×𝐴₁(𝑥₂), and (𝑢₂,𝑣₂)∈𝑅₂(𝑥₁)×𝐴₂(𝑥₂) such that 𝐹₁𝑥₁,𝑦₁+𝑁₁𝑢₁,𝑣₁−𝑤₁,𝜂₁𝑦₁,𝑥₁₁+𝜓₁𝑥₁,𝑦₁−𝜓₁𝑥₁,𝑥₁≥0∀𝑦₁∈𝐾₁;𝐹₂𝑥₂,𝑦₂+𝑁₂𝑢₂,𝑣₂−𝑤₂,𝜂₂𝑦₂,𝑥₂₂+𝜓₂𝑥₂,𝑦₂−𝜓₂𝑥₂,𝑥₂≥0∀𝑦₂∈𝐾₂. (1.2)

1.1. Some Special Cases

(1)If for each 𝑖∈𝐼, 𝑋_𝑖=𝑋^∗_𝑖=𝐻_𝑖 is a Hilbert space, 𝐾_𝑖=𝐻_𝑖, 𝐹_𝑖≡0,  𝑤_𝑖=0 and for each (𝑥,𝑦)∈𝐻₁×𝐻₂, 𝑅_𝑖(𝑥)=𝑥 and 𝐴_𝑖(𝑦)=𝑦, then problem (1.2) reduces to the problem of finding (𝑥₁,𝑥₂)∈𝐻₁×𝐻₂ such that 𝑁₁𝑥₁,𝑥₂,𝜂₁𝑦₁,𝑥₁₁+𝜓₁𝑥₁,𝑦₁−𝜓₁𝑥₁,𝑥₁≥0∀𝑦₁∈𝐻₁;𝑁₂𝑥₁,𝑥₂,𝜂₂𝑦₂,𝑥₂₂+𝜓₂𝑥₂,𝑦₂−𝜓₂𝑥₂,𝑥₂≥0∀𝑦₂∈𝐻₂. (1.3) Problem (1.3) has been introduced and studied by Kazmi and Khan in [16]. (2)If for each 𝑖∈𝐼, 𝑋_𝑖 is a reflexive Banach space, 𝐾_𝑖=𝑋_𝑖 and 𝐹_𝑖≡0, then problem (1.2) reduces to the problem of finding (𝑥₁,𝑥₂)∈𝑋₁×𝑋₂, (𝑢₁,𝑣₁)∈𝑅₁(𝑥₁)×𝐴₁(𝑥₂) and (𝑢₂,𝑣₂)∈𝑅₂(𝑥₁)×𝐴₂(𝑥₂) such that 𝑁₁𝑢₁,𝑣₁−𝑤₁,𝜂₁𝑦₁,𝑥₁₁+𝜓₁𝑥₁,𝑦₁−𝜓₁𝑥₁,𝑥₁≥0∀𝑦₁∈𝑋₁;𝑁₂𝑢₂,𝑣₂−𝑤₂,𝜂₂𝑦₂,𝑥₂₂+𝜓₂𝑥₂,𝑦₂−𝜓₂𝑥₂,𝑥₂≥0∀𝑦₂∈𝑋₂. (1.4) 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 Section 2, 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. Let 𝑋 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 ℋ(𝐴,𝐷)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐷),sup_𝑏∈𝐷𝑑(𝑏,𝐴), (2.1) where 𝑑(𝑎,𝐷)=inf_𝑑∈𝐷‖𝑎−𝑑‖.

Definition 2.1. A mapping 𝑇∶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 𝑇(𝑥)−𝑇𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²∀𝑥,𝑦∈𝑋; (2.2)(iii)𝐿-Lipschitz continuous if there exists 𝐿>0 such that 𝑇(𝑥)−𝑇𝑦≤𝐿𝑥−𝑦∀𝑥,𝑦∈𝑋. (2.3)

Definition 2.2. Let 𝑇∶𝑋→𝑋^∗ be a bounded linear operator. 𝑇 is said to be 𝛿-strongly positive if there exists 𝛿>0 such that ⟨𝑇(𝑥),𝑥⟩≥𝛿‖𝑥‖²,∀𝑥∈𝑋. (2.4)

Remark 2.3. One can easily see that if a bounded linear operator 𝑇∶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. Let 𝑋₁ 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 𝐴₁𝑥₁,𝑦₁−𝐴₁𝑥₂,𝑦₂₁≤𝛽₁𝑥₁−𝑥₂₁+𝜉₂𝑦₁−𝑦₂₂∀𝑥₁,𝑦₁,𝑥₂,𝑦₂∈𝑋₁×𝑋₂. (2.5)

Definition 2.5. A mapping 𝜂∶𝑋×𝑋→𝑋 is said to be (i)affine in the second argument if 𝜂𝑦,𝑡𝑥+(1−𝑡)𝑧=𝑡𝜂𝑦,𝑥+(1−𝑡)𝜂𝑦,𝑧,∀𝑡∈[0,1],∀𝑥,𝑦,𝑧∈𝐾; (2.6)(ii)𝜏-Lipschitz continuous if there exists a constant 𝜏>0 such that 𝜂𝑥,𝑦≤𝜏𝑥−𝑦,∀𝑥,𝑦∈𝑋. (2.7)

Definition 2.6. A real-valued function 𝜓∶𝐾×𝐾→ℝ is said to be skew-symmetric if for all 𝑥,𝑦∈𝐾, 𝜓(𝑥,𝑥)−𝜓𝑥,𝑦−𝜓𝑦,𝑥+𝜓𝑦,𝑦≥0. (2.8)
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. Let 𝐹∶𝐾×𝐾→ℝ be a real-valued bifunction. Then, (i)𝐹 is said to be monotone if 𝐹𝑥,𝑦+𝐹𝑦,𝑥≤0,∀𝑥,𝑦∈𝐾; (2.9)(ii)𝐹 is said to be 𝛼-strongly monotone if there exists 𝛼>0 such that 𝐹𝑥,𝑦+𝐹𝑦,𝑥≤−𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐾; (2.10)(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 of 𝐹 implies monotonicity of 𝐹.
Now, let 𝑆_𝐹 denote the solution set of the equilibrium problem (EP) associated to a bifunction 𝐹, that is, 𝑆_𝐹={𝑥∈𝐾∶𝐹(𝑥,𝑦)≥0∀𝑦∈𝐾}. In many situations, we usually need some more information on the structure of the solution set 𝑆_𝐹 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 𝑆_𝐹 is convex and closed.

Lemma 2.9. Let 𝑋 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 each 𝑥∈𝐾 fixed, the function 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous. Then, the solution set 𝑆_𝐹 is convex and closed whenever it is nonempty.

Proof. Assume that 𝑆_𝐹≠∅. Let 𝑢,𝑣∈𝑆_𝐹 and for 𝜆∈]0,1[ set 𝑥_𝜆=𝜆𝑢+(1−𝜆)𝑣∈𝐾. Let us show that 𝑥_𝜆∈𝑆_𝐹. Since 𝑢,𝑣∈𝑆_𝐹, it follows 𝐹𝑢,𝑦≥0,𝐹𝑣,𝑦≥0∀𝑦∈𝐾. (2.11) From the monotonicity of 𝐹, one deduces 𝐹𝑦,𝑢≤0,𝐹𝑦,𝑣≤0∀𝑦∈𝐾. (2.12) Therefore, from the convexity of 𝐹 with respect to its second argument, it follows 𝐹𝑦,𝑥_𝜆≤0,∀𝑦∈𝐾. (2.13) Now for 𝑡∈]0,1[ and 𝑦∈𝐾, set 𝑥_𝑡=𝑡𝑦+(1−𝑡)𝑥_𝜆∈𝐾. Since 𝐹 is an equilibrium bifunction and 𝐹 is convex with respect to its second argument, it follows that 0≤𝐹𝑥_𝑡,𝑥_𝑡≤𝑡𝐹𝑥_𝑡,𝑦+(1−𝑡)𝐹𝑥_𝑡,𝑥_𝜆. (2.14) Taking account of relation (2.13) with 𝑦=𝑥_𝑡, it follows that 𝑡𝐹(𝑥_𝑡,𝑦)≥0 for all 𝑦∈𝐾, and therefore since 𝑡>0𝐹𝑥_𝑡,𝑦≥0,∀𝑦∈𝐾. (2.15) Since 𝐹 is upper-hemicontinuous, it follows that 𝐹𝑥_𝜆,𝑦≥limsup_𝑡→0𝐹𝑥_𝑡,𝑦≥0. (2.16) Therefore, 𝑥_𝜆∈𝑆_𝐹, and hence 𝑆_𝐹 is convex.
Now, let us show that 𝑆_𝐹 is closed. To this aim, let {𝑥_𝑛}⊂𝑆_𝐹 such that 𝑥_𝑛→𝑥. Let us show that 𝑥∈𝑆_𝐹. One has 𝐹(𝑥_𝑛,𝑦)≥0 for all 𝑦∈𝐾, since 𝐹 is monotone, it follows that 𝐹𝑦,𝑥_𝑛≤0,∀𝑦∈𝐾. (2.17) From the lower semicontinuity of 𝐹 with respect to the second argument, it follows that 𝐹𝑦,𝑥≤liminf_𝑛→+∞𝐹𝑦,𝑥_𝑛≤0. (2.18) By a similar argument as above, one can easily show that 𝐹(𝑥,𝑦)≥0 for all 𝑦∈𝐾. Therefore, 𝑥∈𝑆_𝐹, 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. Let 𝑋 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 each 𝑥,𝑦∈𝐾, if Ψ(𝑥,𝑦)≤0 then Φ(𝑥,𝑦)≤0; (ii)for each fixed 𝑥∈𝐾, the function 𝑦∈𝐾↦Φ(𝑥,𝑦) is lower semicontinuous on every compact subset of 𝐾; (iii)for each finite subset 𝐴 of 𝐾, one has sup_{(𝐴)𝑦∈co}min_𝑥∈𝐴Ψ𝑥,𝑦≤0; (2.19)(iv) coercivity: there exists a nonempty compact convex subset 𝐶 of 𝑋 such that either (a) or (b) in the following holds(a) for all 𝑦∈𝐾⧵𝐶,thereexists𝑥∈𝐶suchthatΦ(𝑥,𝑦)>0;(b)there exists  𝑥₀∈𝐶suchthat∀𝑦∈𝐾⧵𝐶Ψ(𝑥₀,𝑦)>0.
Then, there exists 𝑦∈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 Lemma 2.10.

Proposition 2.12. Suppose that(i)Ψ(𝑥,𝑥)≤0 for each 𝑥∈𝐾;(ii)for each 𝑦∈𝐾 fixed, the set {𝑥∈𝐾∶Ψ(𝑥,𝑦)>0} is convex.
Then, condition (iii) of Lemma 2.10 is staisfied.

Proof. Suppose by contradiction that condition (iii) of Lemma 2.10 is not satisfied. Then, there exist 𝑥₁,…,𝑥_𝑛∈𝐾 and 𝜆₁,…,𝜆_𝑛≥0, with ∑^𝑛_𝑖=1𝜆_𝑖=1, such that Ψ(𝑥_𝑖,∑^𝑛_𝑗=1𝜆_𝑗𝑥_𝑗)>0.
Therefore, by setting 𝑦=∑^𝑛_𝑗=1𝜆_𝑗𝑥_𝑗, it follows from (ii) that Ψ(𝑦,𝑦)>0, which contradicts assumption (i).

As a consequence of Lemma 2.10, 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. Let 𝑋 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 subset 𝐶 of 𝑋 and 𝑦₀∈𝐶∩𝐾 such that 𝑓𝑥,𝑦₀+𝑔𝑥,𝑦₀<0foreach𝑥∈𝐾⧵𝐶. (2.20)Then, there exists 𝑥∈𝐶∩𝐾 such that 𝑓𝑥,𝑦+𝑔𝑥,𝑦≥0∀𝑦∈𝐾. (2.21) Furthermore, the solution set 𝑆_𝑓,𝑔 of the mixed equilibrium problem (2.21) is compact and convex.

Proof. The proof is a direct consequence of Lemma 2.10 by setting Φ𝑥,𝑦=𝑓𝑥,𝑦−𝑔𝑦,𝑥,Ψ𝑥,𝑦=−𝑔𝑦,𝑥−𝑓𝑦,𝑥. (2.22)

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  exists 𝑦₀∈𝐾 such thatlim_{‖𝑥−𝑦0‖→+∞}𝑔(𝑥,𝑦₀)/‖𝑥−𝑦₀‖=−∞.
Ended, let 𝑟₁>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 𝑟𝑦=¹𝑥−𝑦₀𝑟𝑥+1−¹𝑥−𝑦₀𝑦₀. (2.23)
Since 𝑓(𝑦₀,⋅) is convex, 𝑦∈𝐵(𝑦₀,𝑟₁) and 𝑓(𝑦₀,𝑦₀)≥0, one deduces 𝛼₀≤𝑟₁𝑥−𝑦₀𝑓𝑦₀+𝑟,𝑥1−¹𝑥−𝑦₀𝑓𝑦₀,𝑦₀∀𝑥∈𝐾⧵𝐵𝑦₀,𝑟₁. (2.24)
It follows that 𝑓𝑦₀,𝑥≥𝑥−𝑦₀𝑟₁𝛼₀−𝑓𝑦₀,𝑦₀+𝑓𝑦₀,𝑦₀. (2.25)
Thus, 𝑓𝑦₀,𝑥≥𝑥−𝑦₀𝑟₁𝛼₀−𝑓𝑦₀,𝑦₀,∀𝑥∈𝐾⧵𝐵𝑦₀,𝑟₁. (2.26)
Since 𝑓 is monotone, it follows from relation (2.26) that for all 𝑓𝑥,𝑦₀+𝑔𝑥,𝑦₀≤𝑔𝑥,𝑦₀−𝛼₀−𝑓𝑦₀,𝑦₀𝑟₁𝑥−𝑦₀. (2.27)
Since 𝑔(𝑥,𝑦₀)/‖𝑥−𝑦₀‖→−∞ when ‖𝑥−𝑦₀‖→+∞, then there exists 𝑟₂>0 such that for 𝑥∈𝐾 with ‖𝑥−𝑦₀‖>𝑟₂ one has 𝑔𝑥,𝑦₀−𝛼₀−𝑓𝑦₀,𝑦₀𝑟₁𝑥−𝑦₀<0. (2.28)
Take 𝑟=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 𝑓𝑥,𝑦₀+𝑔𝑥,𝑦₀<0. (2.29)
Hence, condition (iii) in Lemma 2.13 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. Let 𝐸 be a complete metric space and 𝑅∶𝐸⇉𝒞ℬ(𝐸) a set-valued mapping. Then, for any given 𝜀>0 and any given 𝑥,𝑦∈𝐸 and 𝑢∈𝑅(𝑥), there exists 𝑣∈𝑅(𝑦) such that 𝑑(𝑢,𝑣)≤(1+𝜀)ℋ𝑅(𝑥),𝑅𝑦. (2.30)

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 𝑇_𝑖∶𝑋_𝑖→𝑋^∗_𝑖, 𝜌_𝑖>0, (𝑥₁,𝑥₂)∈𝐾₁×𝐾₂, (𝑢₁,𝑣₁)∈𝑅₁(𝑥₁)×𝐴₁(𝑥₂), and (𝑢₂,𝑣₂)∈𝑅₂(𝑥₁)×𝐴₂(𝑥₂), ()𝑧APfind₁,𝑧₂∈𝐾₁×𝐾₂suchthatforall𝑦_𝑖∈𝐾_𝑖𝜌_𝑖𝐹_𝑖𝑧_𝑖,𝑦_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑧_𝑖𝑖+𝜓_𝑖𝑧_𝑖,𝑦_𝑖−𝜓_𝑖𝑧_𝑖,𝑧_𝑖+𝑇_𝑖𝑦_𝑖−𝑧_𝑖,𝑧_𝑖−𝑥_𝑖𝑖≥0. (3.1)

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 each 𝑖∈𝐼={1,2}, let 𝑋_𝑖 be a Banach space and 𝐾_𝑖 a nonempty closed convex subset of 𝑋_𝑖, 𝐹_𝑖,𝜓_𝑖∶𝐾_𝑖×𝐾_𝑖→ℝ two real-valued bifunctions, 𝑇_𝑖∶𝑋_𝑖→𝑋^∗_𝑖 a bounded linear operator, 𝜌_𝑖>0 and 𝜂_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖 single-valued mappings such that (i)𝐹_𝑖(𝑥,𝑥)≥0 for all 𝑥∈𝐾_𝑖; 𝐹_𝑖 is monotone and upper-hemicontinuous; for each 𝑥∈𝐾_𝑖 fixed, the function 𝑦↦𝐹_𝑖(𝑥,𝑦) is convex and lower semicontinuous; (ii)𝜂_𝑖 is affine in the first argument and continuous in the second argument such that 𝜂_𝑖𝑥,𝑦+𝜂_𝑖𝑦,𝑥=0,∀𝑥,𝑦∈𝐾_𝑖; (3.2)(iii)𝜓_𝑖 is skew symmetric and continuous; for each 𝑦∈𝐾_𝑖 fixed, the function 𝑥↦𝜓_𝑖(𝑥,𝑦) is convex; (iv)𝑇_𝑖 is 𝛿_𝑖-strongly positive; (v)coercivity: for each 𝑥_𝑖∈𝐾_𝑖, 𝑢_𝑖∈𝑋^∗₁ and 𝑣_𝑖∈𝑋^∗₂, there exists a nonempty compact convex subset 𝐶^𝑖_{𝑥𝑖,𝑢𝑖,𝑣𝑖} of 𝑋_𝑖 and 𝑦⁰_𝑖∈𝐶^𝑖_{𝑥𝑖,𝑢𝑖,𝑣𝑖}∩𝐾_𝑖 such that ∀𝑧_𝑖∈𝐾_𝑖⧵𝐶^𝑖_{𝑥𝑖,𝑢𝑖,𝑣𝑖}𝜌_𝑖𝐹_𝑖𝑧_𝑖,𝑦⁰_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦⁰_𝑖,𝑧_𝑖𝑖+𝜓_𝑖𝑧_𝑖,𝑦⁰_𝑖−𝜓𝑧_𝑖,𝑧_𝑖+𝑇_𝑖𝑦⁰_𝑖−𝑧_𝑖,𝑧_𝑖−𝑥_𝑖<0. (3.3)
Then, the auxiliary problem (AP) has a unique solution.

Proof. The proof of this lemma is a direct application of Lemma 2.13 by considering, for each 𝑖∈𝐼={1,2}, the bifunctions 𝑓_𝑖,𝑔_𝑖∶𝐾_𝑖×𝐾_𝑖→ℝ defined by 𝑓_𝑖𝑧_𝑖,𝑦_𝑖=𝜌_𝑖𝐹_𝑖𝑧_𝑖,𝑦_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑧_𝑖𝑖+𝜓_𝑖𝑧_𝑖,𝑦_𝑖−𝜓_𝑖𝑧_𝑖,𝑧_𝑖𝑔_𝑖𝑧_𝑖,𝑦_𝑖=𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑧_𝑖,𝑧_𝑖−𝑥_𝑖𝑖, (3.4) 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 𝑧¹_𝑖 and 𝑧²_𝑖, then for 𝑖∈𝐼 and for all 𝑧∈𝐾_𝑖, we have 𝜌_𝑖𝐹_𝑖𝑧¹_𝑖,𝑧+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑧,𝑧¹_𝑖𝑖+𝜓_𝑖𝑧¹_𝑖,𝑧−𝜓_𝑖𝑧¹_𝑖,𝑧¹_𝑖+𝑇_𝑖(𝑧)−𝑇_𝑖𝑧¹_𝑖,𝑧¹_𝑖−𝑥_𝑖𝑖𝜌≥0,_𝑖𝐹_𝑖𝑧²_𝑖,𝑧+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑧,𝑧²_𝑖𝑖+𝜓_𝑖𝑧²_𝑖,𝑧−𝜓_𝑖𝑧²_𝑖,𝑧²_𝑖+𝑇_𝑖(𝑧)−𝑇_𝑖𝑧²_𝑖,𝑧²_𝑖−𝑥_𝑖𝑖≥0. (3.5)(3.6) Take 𝑧=𝑧²_𝑖 in relation (3.5) and 𝑧=𝑧¹_𝑖 in relation (3.6) and adding the two inequalities, one obtain 𝜌_𝑖𝐹_𝑖𝑧¹_𝑖+𝐹_𝑖𝑧²_𝑖,𝑧¹_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑧¹_𝑖,𝑧²_𝑖+𝜂_𝑖𝑧²_𝑖,𝑧¹_𝑖𝑖𝜓_𝑖𝑧¹_𝑖,𝑧²_𝑖+𝜓_𝑖𝑧²_𝑖,𝑧¹_𝑖−𝜓_𝑖𝑧¹_𝑖,𝑧¹_𝑖−𝜓_𝑖𝑧²_𝑖,𝑧²_𝑖≥𝑇_𝑖𝑧²_𝑖−𝑧¹_𝑖,𝑧²_𝑖−𝑧¹_𝑖𝑖. (3.7) Since for each 𝑖∈𝐼, 𝐹_𝑖 is monotone, 𝜓_𝑖 is skew symmetric, and 𝑇_𝑖 is 𝛿_𝑖-strongly positive, it follows that 0≥𝑇_𝑖𝑧²_𝑖−𝑧¹_𝑖,𝑧²_𝑖−𝑧¹_𝑖𝑖≥𝛿_𝑖𝑧¹_𝑖−𝑧²_𝑖²_𝑖. (3.8) Therefore, 𝑧¹_𝑖=𝑧²_𝑖 for 𝑖∈𝐼={1,2}, which completes the proof.

Theorem 3.2. For each 𝑖∈𝐼={1,2}, let 𝑋_𝑖 be a reflexive Banach space and 𝐾_𝑖 a nonempty closed convex subset of 𝑋_𝑖, 𝐹_𝑖,𝜓_𝑖∶𝐾_𝑖×𝐾_𝑖→ℝ two real-valued bifunctions, 𝑇_𝑖∶𝑋_𝑖→𝑋^∗_𝑖 a bounded linear operator, 𝜌_𝑖>0 and 𝜂_𝑖∶𝑋_𝑖×𝑋_𝑖→𝑋_𝑖 single-valued mappings such that (i)𝐹_𝑖(𝑥,𝑥)≥0 for all 𝑥∈𝐾_𝑖; 𝐹_𝑖 is monotone and upper-hemicontinuous; for each 𝑥∈𝐾_𝑖 fixed, the function 𝑦↦𝐹_𝑖(𝑥,𝑦) is convex and lower semicontinuous; (ii)𝜂_𝑖 is affine in the first argument and continuous in the second argument such that 𝜂_𝑖𝑥,𝑦+𝜂_𝑖𝑦,𝑥=0,∀𝑥,𝑦∈𝐾_𝑖; (3.9)(iii)𝜓_𝑖 is skew symmetric and continuous; for each 𝑦∈𝐾_𝑖 fixed, the function 𝑥↦𝜓_𝑖(𝑥,𝑦) is convex;(iv)𝑇_𝑖 is 𝛿_𝑖-strongly positive linear operator.
Then, the auxiliary problem (AP) has a unique solution.

Proof. For each 𝑖∈𝐼 and given (𝑥₁,𝑥₂)∈𝐾₁×𝐾₂, (𝑢₁,𝑣₁)∈𝑅₁(𝑥₁)×𝐴₁(𝑥₂), and (𝑢₂,𝑣₂)∈𝑅₂(𝑥₁)×𝐴₂(𝑥₂), define the following bifunctions 𝑓_𝑖,𝑔_𝑖∶𝐾_𝑖×𝐾_𝑖→ℝ by 𝑓_𝑖𝑧_𝑖,𝑦_𝑖=𝜌_𝑖𝐹_𝑖𝑧_𝑖,𝑦_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑧_𝑖𝑖+𝜓_𝑖𝑦_𝑖,𝑧_𝑖−𝜓_𝑖𝑧_𝑖,𝑧_𝑖,𝑔_𝑖𝑧_𝑖,𝑦_𝑖=𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑧_𝑖,𝑧_𝑖−𝑥_𝑖𝑖. (3.10) 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 𝑣⁰_𝑖∈𝐾_𝑖 one has 𝑔_𝑖(𝑢_𝑖,𝑣⁰_𝑖)/‖𝑣⁰_𝑖−𝑢_𝑖‖_𝑖→−∞ when ‖𝑣⁰_𝑖−𝑢_𝑖‖_𝑖→+∞. Let 𝑣⁰_𝑖∈𝐾_𝑖. Then, 𝑔_𝑖𝑢_𝑖,𝑣⁰_𝑖=𝑇_𝑖𝑣⁰_𝑖−𝑇_𝑖𝑢_𝑖,𝑢_𝑖−𝑥_𝑖𝑖=𝑇_𝑖𝑣⁰_𝑖−𝑢_𝑖,𝑢_𝑖−𝑣⁰_𝑖𝑖+𝑇_𝑖𝑣⁰_𝑖−𝑢_𝑖,𝑣⁰_𝑖−𝑥_𝑖𝑖=−𝑇_𝑖𝑣⁰_𝑖−𝑢_𝑖,𝑣⁰_𝑖−𝑢_𝑖𝑖+𝑇_𝑖𝑣⁰_𝑖−𝑢_𝑖,𝑣⁰_𝑖−𝑥_𝑖𝑖≤−𝛿𝑣⁰_𝑖−𝑢_𝑖²_𝑖+𝑇_𝑖𝑣⁰_𝑖−𝑢_𝑖𝑖𝑣⁰_𝑖−𝑥_𝑖𝑖. (3.11) Therefore, 𝑔_𝑖𝑢_𝑖,𝑣⁰_𝑖𝑣⁰_𝑖−𝑢_𝑖𝑖≤−𝛿𝑣⁰_𝑖−𝑢_𝑖𝑖+𝑇_𝑖𝑣⁰_𝑖−𝑥_𝑖𝑖. (3.12) It follows that 𝑔_𝑖(𝑢_𝑖,𝑣⁰_𝑖)/‖𝑣⁰_𝑖−𝑢_𝑖‖_𝑖→−∞ when ‖𝑣⁰_𝑖−𝑢_𝑖‖_𝑖→+∞ which completes the proof.

Remark 3.3. Theorem 3.2 improves recent results given by Ding [18, Theorem 3.1] since the bifunction 𝐹_𝑖 is not needed to be 𝛿_𝑖-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{𝑦_𝑖∈𝐾_𝑖,𝜓_𝑖(𝑦_𝑖,𝑦_𝑖)<∞}=int𝐾_𝑖≠∅; in our approach, this assumption is not needed.
Theorem 3.2 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 (𝑥⁰₁,𝑥⁰₂)∈𝐾₁×𝐾₂, (𝑢⁰₁,𝑣⁰₁)∈𝑅₁(𝑥⁰₁)×𝐴₁(𝑥⁰₂), and (𝑢⁰₂,𝑣⁰₂)∈𝑅₂(𝑥⁰₁)×𝐴₂(𝑥⁰₂). By Theorem 3.1, the auxiliary problem (AP) has a unique solution (𝑥¹₁,𝑥¹₂)∈𝐾₁×𝐾₂, that is, for each 𝑖∈𝐼, we have 𝜌¹_𝑖𝐹_𝑖𝑥¹_𝑖,𝑦_𝑖+𝑁_𝑖𝑢⁰_𝑖,𝑣⁰_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥¹_𝑖𝑖+𝜓_𝑖𝑥¹_𝑖,𝑦_𝑖−𝜓_𝑖𝑥¹_𝑖,𝑥¹_𝑖+⟨𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑥¹_𝑖,𝑥¹_𝑖−𝑥⁰_𝑖⟩_𝑖≥0∀𝑦_𝑖∈𝐾_𝑖. (3.13) Since for each 𝑖∈𝐼, 𝑢⁰_𝑖∈𝑅_𝑖(𝑥⁰_𝑖)∈𝒞ℬ(𝑋^∗₁), and 𝑣⁰_𝑖∈𝐴_𝑖(𝑥⁰₂)∈𝒞ℬ(𝑋^∗₂), by Lemma 2.15, there exist 𝑢¹_𝑖∈𝑅_𝑖(𝑥¹₁) and 𝑣¹_𝑖∈𝐴_𝑖(𝑥¹₂) such that 𝑢¹_𝑖−𝑢⁰_𝑖1≤(1+1)ℋ₁𝑅_𝑖𝑥¹₁,𝑅_𝑖𝑥⁰₁𝑣¹_𝑖−𝑣⁰_𝑖2≤(1+1)ℋ₂𝐴_𝑖𝑥¹₂,𝐴_𝑖𝑥⁰₂, (3.14) where ℋ₁ and ℋ₂ are the Hausdorff metrics on 𝒞ℬ(𝑋^∗₁) and 𝒞ℬ(𝑋^∗₂), respectively.
By using Theorem 3.1 again, the auxiliary problem (AP) has a unique solution (𝑥²₁,𝑥²₂)∈𝐾₁×𝐾₂ such that 𝜌²_𝑖𝐹_𝑖𝑥²_𝑖,𝑦_𝑖+𝑁_𝑖𝑢¹_𝑖,𝑣¹_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥²_𝑖𝑖+𝜓_𝑖𝑥²_𝑖,𝑦_𝑖−𝜓_𝑖𝑥²_𝑖,𝑥²_𝑖+𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑥²_𝑖,𝑥²_𝑖−𝑥¹_𝑖𝑖≥0,∀𝑦_𝑖∈𝐾_𝑖. (3.15) By induction, we can construct an iterative algorithm to compute the approximate solution for the system (1.2) as follows: for given (𝑥⁰₁,𝑥⁰₂)∈𝐾₁×𝐾₂, (𝑢⁰₁,𝑣⁰₁)∈𝑅₁(𝑥⁰₁)×𝐴₁(𝑥⁰₂), and (𝑢⁰₂,𝑣⁰₂)∈𝑅₂(𝑥⁰₁)×𝐴₂(𝑥⁰₂), there exist sequences {𝑥^𝑛₁},{𝑥^𝑛₂},{𝑢^𝑛₁},{𝑢^𝑛₂},{𝑣^𝑛₁}, and {𝑣^𝑛₂} such that for each 𝑖∈𝐼𝑢_𝑖^𝑛+1−𝑢^𝑛_𝑖1≤11+ℋ𝑛+1¹𝑅_𝑖𝑥₁^𝑛+1,𝑅_𝑖𝑥^𝑛₁,𝑢^𝑛_𝑖∈𝑅_𝑖𝑥^𝑛₁,𝑣_𝑖^𝑛+1−𝑣^𝑛_𝑖2≤11+ℋ𝑛+1²𝐴_𝑖𝑥₂^𝑛+1,𝐴_𝑖𝑥^𝑛₂,𝑣^𝑛_𝑖∈𝐴_𝑖𝑥^𝑛₂,𝜌_𝑖^𝑛+1𝐹_𝑖𝑥_𝑖^𝑛+1,𝑦_𝑖+𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1_𝑖+𝜓_𝑖𝑥_𝑖^𝑛+1,𝑦_𝑖−𝜓_𝑖𝑥_𝑖^𝑛+1,𝑥_𝑖^𝑛+1+𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑥_𝑖^𝑛+1,𝑥_𝑖^𝑛+1−𝑥^𝑛_𝑖𝑖≥0∀𝑦_𝑖∈𝐾_𝑖,𝑛=0,1,2,…. (3.16)
The following convergence analysis is presented for the algorithm above.

Theorem 3.5. Under the hypotheses of Theorem 3.2, further assume that for each 𝑖∈𝐼,(i)𝐹_𝑖 is 𝜎_𝑖-strongly monotone and upper-hemicontinuous; 𝑁_𝑖 is (𝛽_𝑖,𝜉_𝑖)-mixed Lipschitz continuity; 𝑅_𝑖 is 𝑘_𝑖-ℋ₁-Lipschitz continuous, and 𝐴_𝑖 is 𝜇_𝑖-ℋ₂-Lipschitz continuous;(ii)𝜂_𝑖 is 𝜏_𝑖-Lipschitz continuous; (iii)the sequence {𝜌^𝑛}_𝑛∈ℕ of positive real numbers is increasing and lim_𝑛→+∞𝜌_𝑛=+∞.
Furthermore, assume that the following condition holds: 𝐶₁Λ=max𝜃₁+𝜃₂,𝜗₁+𝜗₂𝜃<1,𝑤ℎ𝑒𝑟𝑒₁=𝜏₁𝛽₁𝑘₁𝜎₁,𝜃₂=𝜏₂𝛽₂𝑘₂𝜎₂,𝜗₁=𝜏₁𝜉₁𝜇₁𝜎₁,𝜗₂=𝜏₂𝜉₂𝜇₂𝜎₂. (3.17) Then, the sequences {𝑥^𝑛₁},{𝑥^𝑛₂},{𝑢^𝑛₁},{𝑢^𝑛₂},{𝑣^𝑛₁}, and {𝑣^𝑛₂} generated by Algorithm 3.4 converge strongly to 𝑥₁,𝑥₂,𝑢₁,𝑣₁,𝑢₂, and 𝑣₂, respectively, where (𝑢₁,𝑣₁)∈𝑇₁(𝑥₁)×𝐴₁(𝑥₂),(𝑢₂,𝑣₂)∈𝑇₂(𝑥₁)×𝐴₂(𝑥₂), and (𝑥₁,𝑥₂,𝑢₁,𝑣₁,𝑢₂,𝑣₂) is a solution of the System of Generalized Mixed Equilibrium Problem involving variational-like in equalities (1.2).

Proof. By the definition of Algorithm 3.4, we have for 𝑖∈𝐼, 𝜌^𝑛_𝑖𝐹_𝑖𝑥^𝑛_𝑖,𝑦_𝑖+𝑁_𝑖𝑢_𝑖^𝑛−1,𝑣_𝑖^𝑛−1−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥^𝑛_𝑖𝑖+𝜓_𝑖𝑥^𝑛_𝑖,𝑦_𝑖−𝜓_𝑖𝑥^𝑛_𝑖,𝑥^𝑛_𝑖+𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑥^𝑛_𝑖,𝑥^𝑛_𝑖−𝑥_𝑖^𝑛−1_𝑖≥0∀𝑦_𝑖∈𝐾_𝑖,𝜌_𝑖^𝑛+1𝐹_𝑖𝑥_𝑖^𝑛+1,𝑦_𝑖+𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1_𝑖+𝜓_𝑖𝑥_𝑖^𝑛+1,𝑦_𝑖−𝜓_𝑖𝑥_𝑖^𝑛+1,𝑥_𝑖^𝑛+1+𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑥_𝑖^𝑛+1,𝑥_𝑖^𝑛+1−𝑥^𝑛_𝑖𝑖≥0∀𝑦_𝑖∈𝐾_𝑖. (3.18)(3.19) By considering 𝑦_𝑖=𝑥_𝑖^𝑛+1 in relation (3.18) and 𝑦_𝑖=𝑥^𝑛_𝑖 in relation (3.19), and taking into account 𝜂_𝑖(𝑥_𝑖^𝑛+1,𝑥^𝑛_𝑖)=−𝜂_𝑖(𝑥^𝑛_𝑖,𝑥_𝑖^𝑛+1), one obtains by adding the two inequalities 𝐹_𝑖𝑥_𝑖^𝑛+1,𝑥^𝑛_𝑖+𝐹_𝑖𝑥^𝑛_𝑖,𝑥_𝑖^𝑛+1+𝑁_𝑖𝑢_𝑖^𝑛−1,𝑣_𝑖^𝑛−1−𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖,𝜂_𝑖𝑥_𝑖^𝑛+1,𝑥^𝑛_𝑖𝑖+𝜓_𝑖𝑥^𝑛_𝑖,𝑥_𝑖^𝑛+1+𝜓_𝑖𝑥_𝑖^𝑛+1,𝑥^𝑛_𝑖−𝜓_𝑖𝑥^𝑛_𝑖,𝑥^𝑛_𝑖−𝜓_𝑖𝑥^𝑛+1_i,𝑥_𝑖^𝑛+1+1𝜌^𝑛_𝑖𝑇_𝑖𝑥_𝑖^𝑛+1−𝑥^𝑛_𝑖,𝑥^𝑛_𝑖−𝑥_𝑖^𝑛−1_𝑖−1𝜌_𝑖^𝑛+1𝑇_𝑖(𝑥^𝑛_𝑖−𝑥_𝑖^𝑛+1),𝑥^𝑛_𝑖−𝑥_𝑖^𝑛+1_𝑖≥0. (3.20) Since 𝐹_𝑖 is 𝜎_𝑖-monotone, 𝜓_𝑖 is skew-symmetric, and 𝑇_𝑖 is 𝛿_𝑖-strongly positive and ‖𝑇_𝑖‖-Lispchitz continuous, it follows from relation (3.20) that 𝑁_𝑖𝑢_𝑖^𝑛−1,𝑣_𝑖^𝑛−1−𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖𝑖𝜂_𝑖𝑥_𝑖^𝑛+1,𝑥^𝑛_𝑖+𝑇_𝑖𝜌^𝑛_𝑖𝑥_𝑖^𝑛+1−𝑥^𝑛_𝑖𝑖𝑥^𝑛_𝑖−𝑥_𝑖^𝑛−1_𝑖≥𝜎_𝑖+𝛿_𝑖𝜌_𝑖^𝑛+1𝑥^𝑛_𝑖−𝑥_𝑖^𝑛+12_𝑖. (3.21) 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 3.4𝑁_𝑖𝑢_𝑖^𝑛−1,𝑣_𝑖^𝑛−1−𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖𝑖≤𝛽₁𝑢_𝑖^𝑛−1−𝑢^𝑛_𝑖1+𝜉₁𝑣_𝑖^𝑛−1−𝑣^𝑛_𝑖2≤𝛽₁11+𝑛ℋ₁𝑅₁𝑥₁^𝑛−1,𝑅₁𝑥^𝑛₁+𝜉₁11+𝑛ℋ₂𝑅₁𝑥₂^𝑛−1,𝑅₁𝑥^𝑛₂≤𝛽₁𝑘₁11+𝑛𝑥₁^𝑛−1−𝑥^𝑛₁₁+𝜉₁𝜇₁11+𝑛𝑥₂^𝑛−1−𝑥^𝑛₂₂. (3.22) Since 𝜂₁ is 𝜏₁-Lipschitz continuous, it follows from relation (3.21) considered for 𝑖=1 that 𝜎₁+𝛿₁𝜌₁^𝑛+1𝑥^𝑛₁−𝑥₁^𝑛+1₁≤𝜏₁𝛽₁𝑘₁11+𝑛+𝑇₁𝜌^𝑛₁𝑥₁^𝑛−1−𝑥^𝑛₁₁+𝜏₁𝜉₁𝜇₁11+𝑛𝑥₂^𝑛−1−𝑥^𝑛₂₂. (3.23) Hence, 𝑥^𝑛₁−𝑥₁^𝑛+1₁≤𝜃_1𝑛𝑥₁^𝑛−1−𝑥^𝑛₁₁+𝜗_1𝑛𝑥₂^𝑛−1−𝑥^𝑛₂₂, (3.24) with 𝜃_1𝑛=𝜌₁^𝑛+1𝜏₁𝛽₁𝑘₁(1+1/𝑛)+𝜌₁^𝑛+1𝑇₁/𝜌^𝑛₁𝜌₁^𝑛+1𝜎₁+𝛿₁,𝜗_1𝑛=𝜌₁^𝑛+1𝜏₁𝜉₁𝜇₁(1+1/𝑛)𝜌₁^𝑛+1𝜎₁+𝛿₁. (3.25) Similarly, by the assumptions on 𝐹₂, 𝑁₂, 𝜂₂, 𝑅₂, 𝐴₂, and 𝑇₂, it follows from relation (3.21) considered for 𝑖=2 that 𝑥^𝑛₂−𝑥₂^𝑛+1₂≤𝜃_2𝑛𝑥₁^𝑛−1−𝑥^𝑛₁₁+𝜗_2𝑛𝑥₂^𝑛−1−𝑥^𝑛₂₂, (3.26) with 𝜃_2𝑛=𝜌₂^𝑛+1𝜏₂𝛽₂𝑘₂(1+1/𝑛)𝜌₂^𝑛+1𝜎₂+𝛿₂,𝜗_2𝑛=𝜌₂^𝑛+1𝜏₂𝜉₂𝜇₂(1+1/𝑛)+𝜌₂^𝑛+1𝑇₂/𝜌^𝑛₂𝜌₂^𝑛+1𝜎₂+𝛿₂. (3.27) Adding the inequalities (3.24) and (3.26), we obtain 𝑥^𝑛₁−𝑥₁^𝑛+1₁+𝑥^𝑛₂−𝑥₂^𝑛+1₂≤𝜃_1𝑛+𝜃_2𝑛𝑥₁^𝑛−1−𝑥^𝑛₁₁+𝜗_1𝑛+𝜗_2𝑛𝑥₂^𝑛−1−𝑥^𝑛₂₂≤Λ_𝑛𝑥₁^𝑛−1−𝑥^𝑛₁₁+𝑥₂^𝑛−1−𝑥^𝑛₂₂, (3.28) where Λ_𝑛=max{𝜃_1𝑛+𝜃_2𝑛,𝜗_1𝑛+𝜗_2𝑛}.
On 𝑋₁×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 𝑥^𝑛₁,𝑥^𝑛₂−𝑥₁^𝑛+1,𝑥₂^𝑛+1_∗≤Λ_𝑛𝑥₁^𝑛−1,𝑥₂^𝑛−1−𝑥^𝑛₁,𝑥^𝑛_2∗. (3.29) Taking account of the assumptions, it is easy to see that Λ_𝑛→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 Λ_𝑛≤Λ₀<1 for all 𝑛≥𝑛₀. Therefore, it follows from (3.29) that 𝑥^𝑛₁,𝑥^𝑛₂−𝑥₁^𝑛+1,𝑥₂^𝑛+1_∗≤Λ₀𝑥₁^𝑛−1,𝑥₂^𝑛−1−𝑥^𝑛₁,𝑥^𝑛_2∗. (3.30) This implies that {(𝑥^𝑛₁,𝑥^𝑛₂)} is a Cauchy sequence in 𝑋₁×𝑋₂. Thus, (𝑥^𝑛₁,𝑥^𝑛₂) converges strongly to some (𝑥₁,𝑥₂)∈𝐾₁×𝐾₂.
By Algorithm 3.4 and the Lipschitz continuity assumption on 𝑅₁,𝑅₂, 𝐴₁, and 𝐴₂, we have 𝑢_𝑖^𝑛+1−𝑢^𝑛_𝑖1≤11+ℋ𝑛+1¹𝑅_𝑖𝑥₁^𝑛+1,𝑅_𝑖𝑥^𝑛₁≤𝑘_𝑖11+𝑥𝑛+1^1𝑛+1−𝑥^𝑛₁₁,𝑣_𝑖^𝑛+1−𝑣^𝑛_𝑖2≤11+ℋ𝑛+1²𝐴_𝑖𝑥₂^𝑛+1,𝐴_𝑖𝑥^𝑛₂≤𝜇_𝑖11+𝑥𝑛+1^2𝑛+1−𝑥^𝑛₂₂. (3.31) It follows, for each 𝑖∈𝐼, that {𝑢^𝑛_𝑖} is a Cauchy sequence in 𝑋^∗₁ and {𝑣^𝑛_𝑖} is a Cauchy sequence in 𝑋^∗₂. Thus, there exists (𝑢_𝑖,𝑣_𝑖)∈𝑋^∗₁×𝑋^∗₂ such that (𝑢^𝑛_𝑖,𝑥^𝑛_𝑖) converges strongly to (𝑢_𝑖,𝑣_𝑖).
Noting that 𝑢^𝑛₁∈𝑅₁(𝑥^𝑛₁), it follows that 𝑑𝑢₁,𝑅₁𝑥^𝑛₁≤𝑢₁−𝑢^𝑛₁₁+𝑑𝑢^𝑛₁,𝑅₁𝑥^𝑛₁+ℋ₁𝑅₁𝑥^𝑛₁,𝑅₁𝑥₁≤𝑢₁−𝑢^𝑛₁₁+𝑘₁𝑥₁−𝑥^𝑛₁₁⟶0as𝑛⟶∞. (3.32) Hence, we must have 𝑢₁∈𝑅₁(𝑥₁). Similarly, one can show that 𝑢₂∈𝑅₂(𝑥₁),  𝑣₁∈𝐴₁(𝑥₂), and 𝑣₂∈𝐴₂(𝑥₂). By Algorithm 3.4, we have that, for all 𝑖∈𝐼, 𝜌_𝑖^𝑛+1𝐹_𝑖𝑥_𝑖^𝑛+1,𝑦_𝑖+𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1_𝑖+𝜓𝑥_𝑖^𝑛+1,𝑦_𝑖−𝜓𝑥_𝑖^𝑛+1,𝑥_𝑖^𝑛+1+𝑇_𝑖𝑦_𝑖−𝑇_𝑖𝑥_𝑖^𝑛+1,𝑥_𝑖^𝑛+1−𝑥^𝑛_𝑖≥0∀𝑦_𝑖∈𝐾_𝑖,𝑛=0,1,2,… (3.33) Since 𝑁_𝑖 is (𝛽_𝑖,𝜉_𝑖)-mixed Lipschitz continuous and 𝜂_𝑖 is continuous in the second argument, one has, for each 𝑦_𝑖∈𝐾_𝑖, 𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1_𝑖−𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖≤𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑁_𝑖𝑢_𝑖,𝑣_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1−𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖≤𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑁_𝑖𝑢_𝑖,𝑣_𝑖𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1−𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖. (3.34) Hence, 𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1_𝑖−𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖⟶0as𝑛⟶∞. (3.35) From the monotonicity of 𝐹_𝑖 and the linearity of 𝑇_𝑖, one has 𝑁_𝑖𝑢^𝑛_𝑖,𝑣^𝑛_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1_𝑖−𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖+𝜓_𝑖𝑥_𝑖^𝑛+1,𝑦_𝑖−𝜓_𝑖𝑥_𝑖^𝑛+1,𝑥_𝑖^𝑛+1+𝑇_𝑖𝜌_𝑖^𝑛+1𝑦_𝑖−𝑥_𝑖^𝑛+1𝑥_𝑖^𝑛+1−𝑥^𝑛_𝑖≥𝐹_𝑖𝑦_𝑖,𝑥_𝑖^𝑛+1−𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖∀𝑦_𝑖∈𝐾_𝑖,∀𝑛=0,1,2,… (3.36) Taking into account result (3.35) and since, for each 𝑖∈𝐼, 𝐹_𝑖(𝑦,⋅) is a lower semicontinuous function, 𝜓_𝑖 is continuous, and the sequence {𝑥^𝑛_𝑖} converges strongly to 𝑥_𝑖, it follows from relation (3.36) by passing to the limit when 𝑛 goes to infinity that 𝜓_𝑖𝑦_𝑖,𝑥_𝑖−𝜓_𝑖𝑥_𝑖,𝑥_𝑖≥𝐹_𝑖𝑦_𝑖,𝑥_𝑖−𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖. (3.37) Now, for each 𝑖∈𝐼 and for 𝑡∈(0,1] and 𝑦_𝑖∈𝐾_𝑖, set 𝑦_𝑖(𝑡)=𝑡𝑦_𝑖+(1−𝑡)𝑥_𝑖. Since 𝐾_𝑖 is convex, then 𝑦_𝑖(𝑡)∈𝐾_𝑖 for 𝑡∈(0,1]; it follows that 𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖(𝑡),𝑥_𝑖𝑖+𝜓_𝑖𝑦_𝑖(𝑡),𝑥_𝑖−𝜓_𝑖𝑥_𝑖,𝑥_𝑖≥𝐹_𝑖𝑦_𝑖(𝑡),𝑥_𝑖. (3.38) Taking into account the convexity of 𝜓_𝑖(⋅,𝑥_𝑖), the fact that 𝑁_𝑖 is affine with respect to the first argument and 𝜂_𝑖(𝑥_𝑖,𝑥_𝑖)=0, one has 𝐹_𝑖𝑦_𝑖(𝑡),𝑥_𝑖≤𝑡𝜓_𝑖𝑦_𝑖,𝑥_𝑖−𝜓_𝑖𝑥_𝑖,𝑥_𝑖+𝑡𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖. (3.39) On the other hand, since 𝐹_𝑖(𝑥_𝑖,𝑥_𝑖)=0 for each 𝑥_𝑖∈𝐾_𝑖 and 𝐹_𝑖(𝑦_𝑖,⋅) is convex, it follows that 0≤𝐹_𝑖𝑦_𝑖(𝑡),𝑦_𝑖(𝑡)≤𝑡𝐹_𝑖𝑦_𝑖(𝑡),𝑦_𝑖+(1−𝑡)𝐹𝑦_𝑖(𝑡),𝑥_𝑖. (3.40) Hence, from relation (3.39), one has 0≤𝑡𝐹_𝑖𝑦_𝑖(𝑡),𝑦_𝑖+(1−𝑡)𝜓_𝑖𝑦_𝑖,𝑥_𝑖−𝜓_𝑖𝑥_𝑖,𝑥_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖. (3.41) Therefore, for all 𝑡∈]0,1], one has 0≤𝐹_𝑖𝑦_𝑖(𝑡),𝑦_𝑖+(1−𝑡)𝜓_𝑖𝑦_𝑖,𝑥_𝑖−𝜓_𝑖𝑥_𝑖,𝑥_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖. (3.42) Since for each 𝑦_𝑖∈𝐾_𝑖, 𝑥↦𝐹_𝑖(𝑥,𝑦) is upper hemicontinuous, it follows by passing to limit when 𝑡→0 in the previous inequality that 𝐹_𝑖𝑥_𝑖,𝑦_𝑖+𝑁_𝑖𝑢_𝑖,𝑣_𝑖−𝑤_𝑖,𝜂_𝑖𝑦_𝑖,𝑥_𝑖𝑖+𝜓_𝑖𝑥_𝑖,𝑦_𝑖−𝜓_𝑖𝑥_𝑖,𝑥_𝑖≥0∀𝑦_𝑖∈𝐾_𝑖, (3.43) 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)Theorems 3.2 and 3.5 improve recent results given by Ding [18, Theorem 3.1, Theorem 4.1], since the bifunction 𝐹_𝑖 is not needed to be 𝛿_𝑖-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{𝑦_𝑖∈𝐾_𝑖,𝜓_𝑖(𝑦_𝑖,𝑦_𝑖)<∞}=int𝐾_𝑖≠∅; in our approach, this assumption is not needed;(3)in the special case where 𝐹_𝑖≡0, the results obtained improve those obtained by Ding and Wang [17], since the assumptions on the bifunctions 𝜓_𝑖 have been relaxed.