On System of Generalized Vector Quasiequilibrium Problems with Applications

Abstract

We introduce a new system of generalized vector quasiequilibrium problems which includes system of vector quasiequilibrium problems, system of vector equilibrium problems, and vector equilibrium problems, and so forth in literature as special cases. We prove the existence of solutions for this system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of generalized quasi-equilibrium problems and the generalized Debreu-type equilibrium problem for both vector-valued functions and scalar-valued functions.

1. Introduction and Formulations

In the resent years, the vector equilibrium problems have been studied in [1–7] and the references therein which is a unified model of several problems, for instance, vector variational inequality, vector variational-like inequality, vector complementarity problems, vector optimization problems. A comprehensive bibliography on vector equilibrium problems, vector variational inequalities, vector variational-like inequalities and their generalizations can be found in a recent volume [1]. Ansari and Yao [8] and Chiang et al. [9] introduced and studied some vector quasi-equilibrium problems which generalized those quasi-equilibrium problems in [10–17] to the case of vector-valued function. Very recently, the system of vector equilibrium problems was introduced by Ansari et al. [18] with applications in Nash-type equilibrium problem for vector-valued functions. The system of vector quasi-equilibrium problems was introduced by Ansari et al. [19] with applications in Debreu-type equilibrium problem for vector-valued functions. As a generalization of the above models, we introduce a new system of generalized vector quasi-equilibrium problems, that is, a family of generalized quasi-equilibrium problems for vector-valued maps defined on a product set.

Throughout this paper, for a set 𝐴 in a topological space, we denote by co𝐴, int𝐴, co𝐴 the convex hull, interior, and the convex closure of 𝐴, respectively.

Let 𝐼 be an index set. For each 𝑖∈𝐼, let 𝑍_𝑖, 𝐸_𝑖 and let 𝐹_𝑖 be topological vector spaces. Consider two family of nonempty convex subsets {𝑋_𝑖}_𝑖∈𝐼 with 𝑋_𝑖⊆𝐸_𝑖 and {𝑌_𝑖}_𝑖∈𝐼 with 𝑌_𝑖⊆𝐹_𝑖. Let

𝐸=_𝑖∈𝐼𝐸_𝑖,𝑋=_𝑖∈𝐼𝑋_𝑖,𝐹=_𝑖∈𝐼𝐹_𝑖,𝑌=_𝑖∈𝐼𝑌_𝑖. (1.1) An element of the set 𝑋^𝑖=∏_{𝑗∈𝐼⧵𝑖}𝑋_𝑖 will be denoted by 𝑥^𝑖, therefore, 𝑥∈𝑋 will be written as 𝑥=(𝑥^𝑖,𝑥_𝑖)∈𝑋^𝑖×𝑋_𝑖. Similarly, an element of the set 𝑌 will be denoted by 𝑦=(𝑦^𝑖,𝑦_𝑖)∈𝑌^𝑖×𝑌_𝑖. For each 𝑖∈𝐼, let 𝐶_𝑖∶𝑋→2^𝑍𝑖, 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty values, and let 𝑓_𝑖∶𝑋×𝑌×𝑋_𝑖→𝑍_𝑖 be a the vector-valued function. Then the system of generalized vector quasi-equilibrium problems (in Short, SGVQEP) is to find (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼,

𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)∉−int𝐶_𝑖(𝑥),∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (1.2)

Here are some special cases of the (SGVQEP).

(i) For each 𝑖∈𝐼, let 𝜙_𝑖∶𝑋×𝑌→𝑍_𝑖 be a vector-valued function. We define a trifunction 𝑓_𝑖∶𝑋×𝑌×𝑋_𝑖→𝑍_𝑖 as 𝑓_𝑖(𝑥,𝑦,𝑢_𝑖)=𝜙_𝑖(𝑥^𝑖,𝑦,𝑢_𝑖)−𝜙_𝑖(𝑥,𝑦), ∀(𝑥,𝑦,𝑢_𝑖)∈𝑋×𝑌×𝑋_𝑖. Then the (SGVQEP) reduces to the generalized Debreu-type equilibrium problem for vector-valued functions (in short, G-Debreu VEP), which is to find (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼,

𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝜙_𝑖(𝑥^𝑖,𝑦,𝑧_𝑖)−𝜙_𝑖(𝑥,𝑦)∉−int𝐶_𝑖(𝑥),∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (1.3)

(ii) We denote by 𝑅 and 𝑅⁺ the set of real numbers and the set of real nonnegative numbers, respectively. For each 𝑖∈𝐼, if 𝑍_𝑖=𝑅, and 𝐶_𝑖(𝑥)=𝑅⁺ for all 𝑥∈𝑋, then the (SGVQEP) reduces to the system of generalized quasi-equilibrium problems (in short, SGQEP), which is to find (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼,

𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)≥0,∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (1.4)

And the G-Debreu VEP reduces to the generalized Debreu-type equilibrium problem for scalar-valued functions (in short, G-Debreu EP), which is to find (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼,

𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝜙_𝑖(𝑥^𝑖,𝑦,𝑧_𝑖)≥𝜙_𝑖(𝑥,𝑦),∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (1.5)

(iii) Let 𝑌={𝑦}. For each 𝑖∈𝐼, let 𝐷_𝑖(𝑥,𝑦)=𝐴_𝑖(𝑥), 𝑇_𝑖(𝑥,𝑦)={𝑦_𝑖} for all 𝑥∈𝑋, where 𝐴_𝑖∶𝑋→2^𝑋𝑖 is a set-valued map. We define a function 𝜑_𝑖∶𝑋×𝑋_𝑖→𝑍_𝑖 and a function ℎ_𝑖∶𝑋×𝑌→𝑍_𝑖 as 𝜑_𝑖(𝑥,𝑧_𝑖)=𝑓_𝑖(𝑥,𝑦,𝑧_𝑖), forall(𝑥,𝑧_𝑖)∈𝑋×𝑋_𝑖, and ℎ_𝑖(𝑥)=𝜙_𝑖(𝑥,𝑦),forall𝑥∈𝑋, then (SGVQEP) and (G-Debreu VEP), respectively, reduce to the system of vector quasi-equilibrium problems and the (Debreu VEP) introduced by Ansari et al. [19] which contain those mathematical in [18, 20] as special cases. The (SGQEP) reduces to the mathematical models in [21, page 286] and [22, pages 152-153] and the (G-Debreu EP) reduces to the abstract economy in [23, page 345] which contains the noncooperative game in [24] as a special case.

(iv) If the index set 𝐼 is singleton, 𝐷(𝑥,𝑦)=𝐷_𝑖(𝑥), 𝑇(𝑥,𝑦)=𝑇(𝑥), and 𝐶(𝑥)=𝐶, then the (SGVQEP) becomes the implicit vector variational inequality in [9] and the (SGQEP) reduces to the quasi-equilibrium problem investigated in [14–17].

The rest of this paper is arranged in the following manner. The following section deals with some preliminary definitions, notations and results which will be used in the sequel. In Section 3, we establish existence results for a solution to the (SGVQEP) and the (SGQEP) with or without involving Φ-condensing maps by using similar techniques in [19]. In Section 4, as applications of the results of Section 3, we derive some existence results of a solution for the (G-Debreu VEP) and the the (G-Debreu EP).

2. Preliminaries

In order to prove the main results, we need the following definitions.

Definition 2.1 ([19, 25]). Let 𝑀 be a nonempty convex subset of a topological vector space 𝐸 and 𝑍 a real topological space with a closed and convex cone 𝑃 with apex at the origin. A vector-valued function 𝜑∶𝑀→𝑍 is called(i)𝑃-quasifunction if and only if, for all 𝑧∈𝑍, the set {𝑥∈𝑀∶𝜑(𝑥)∈𝑧−𝑃} is convex,(ii)natural 𝑃-quasifunction if and only if, ∀𝑥,𝑦∈𝑀, and 𝜆∈[0,1], 𝜑(𝜆𝑥+(1−𝜆)𝑦)∈co{𝜑(𝑥),𝜑(𝑦)}−𝑃.

Definition 2.2 ([13]). Let 𝑋 and 𝑌 be two topological spaces. 𝑇∶𝑋→2^𝑌 be a set-valued map. Then 𝑇 is said to be upper semicontinuous if the set {𝑥∈𝑋∶𝑇(𝑥)⊆𝑉} is open in 𝑋 for every open subset 𝑉 of 𝑌. Also 𝑇 is said to be lower semicontinuous if the set {𝑥∈𝑋∶𝑇(𝑥)∩𝑉} is open in 𝑋 for every open subset 𝑉 of 𝑌⋅𝑇 is said to have open lower sections if the set 𝑇⁻¹(𝑦)={𝑥∈𝑋∶𝑦∈𝑇(𝑥)} is open in 𝑋 for each 𝑦∈𝑌.

Definition 2.3 ([26]). Let 𝐸 be a Hausdorff topological space and 𝐿 a lattice with least element, denoted by 0. A map Φ∶2^𝐸→𝐿 is a measure of noncompactness provided that the following conditions hold ∀𝑀,𝑁∈2^𝐸:(i)Φ(𝑀)=0 iff 𝑀 is precompact ( i.e., it is relatively compact),(ii)Φ(co𝑀)=Φ(𝑀),(iii)Φ(𝑀∪𝑁)=max{Φ(𝑀),Φ(𝑁)}.

Definition 2.4 ([26]). Let Φ∶2^𝐸→𝐿 be a measure of noncompactness on 𝐸 and 𝑋⊆𝐸. A set-valued map 𝑇∶𝑋→2^𝐸 is called Φ-condensing provided that, if 𝑀⊆𝑋 with Φ(𝑇(𝑀))≥Φ(𝑀), then 𝑀 is relatively compact.

Remark 2.5. Note that every set-valued map defined on a compact set is Φ-condensing for any measure of noncompactness Φ. If 𝐸 is locally convex and 𝑇∶𝑋→2^𝐸 is a compact set-valued map (i.e., 𝑇(𝑋) is precompact), then 𝑇 is Φ-condensing for any measure of noncompactness Φ. It is clear that if 𝑇∶𝑋→2^𝐸 is Φ-condensing and 𝑇^∗∶𝑋→2^𝐸 satisfies 𝑇^∗(𝑥)⊆𝑇(𝑥)∀𝑥∈𝑋, then 𝑇^∗ is also Φ-condensing.
We will use the following particular forms of two maximal element theorems for a family of set-valued maps due to Deguire et al. [27, Theorem  7] and Chebbi and Florenzano [28, Corollary  4].

Lemma 2.6 ([19, 27]). Let {𝑋_𝑖}_𝑖∈𝐼 be a family of nonempty convex subsets where each 𝑋_𝑖 is contained in a Hausdorff topological vector space 𝐸_𝑖, For each 𝑖∈𝐼, let 𝑆_𝑖∶𝑋→2^𝑋𝑖 be a set-valued map such that (i)for each 𝑖∈𝐼, 𝑆_𝑖(𝑥) is convex,(ii)for each 𝑥∈𝑋, 𝑥_𝑖∉𝑆_𝑖(𝑥),(iii)for each 𝑦_𝑖∈𝑋_𝑖, 𝑆_𝑖⁻¹(𝑦_𝑖) is open in 𝑋.(iv)there exist a nonempty compact subset 𝑁 of 𝑋 and a nonempty compact convex subset 𝐵_𝑖 of 𝑋_𝑖 for each 𝑖∈𝐼 such that for each 𝑥∈𝑋⧵𝑁 there exists 𝑖∈𝐼 satisfying 𝑆_𝑖(𝑥)∩𝐵_𝑖≠∅. Then there exists 𝑥∈𝑋 such that 𝑆_𝑖(𝑥)=∅ for all 𝑖∈𝐼.

Lemma 2.7 ([19, 28]). Let 𝐼 be any index set and {𝑋_𝑖}_𝑖∈𝐼 be a family of nonempty, closed and convex subsets where each 𝑋_𝑖 is contained in a locally convex Hausdorff topological vector space 𝐸_𝑖. For each 𝑖∈𝐼, let 𝑆_𝑖∶𝑋→2^𝑋𝑖 be a set-valued map. Assume that the set-valued map 𝑆∶𝑋→2^𝑋 defined as 𝑆(𝑥)=∏_𝑖∈𝐼𝑆_𝑖(𝑥), ∀𝑥∈𝑋, is Φ-condensing and the conditions (i), (ii), (iii) of Lemma 2.6 hold. Then there exists 𝑥∈𝑋 such that 𝑆_𝑖(𝑥)=∅ for all 𝑖∈𝐼.

3. Existence Results

An existence result of a solution for the system of generalized vector quasi-equilibrium problems with or without Φ-condensing maps are will shown in this section.

Theorem 3.1. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑍_𝑖 be a topological vector space, let 𝐸_𝑖 and 𝐹_𝑖 be two Hausdorff topological vector spaces, let 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty and convex subsets, let 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, and the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌 and let 𝑓_𝑖∶𝑋×𝑌×𝑋_𝑖→𝑍_𝑖 be a vector-valued function. For each 𝑖∈𝐼, let 𝐶_𝑖∶𝑋→2^𝑍𝑖 be a set-valued map such that 𝐶_𝑖(𝑥) be a proper closed and convex cone with apex at the origin and int𝐶_𝑖(𝑥)≠∅ for all 𝑥∈𝑋 and 𝑃_𝑖=∩_𝑥∈𝑋𝐶_𝑖(𝑥). Assume that(i)for all 𝑥=(𝑥^𝑖,𝑥_𝑖)∈𝑋, for all 𝑦∈𝑌, 𝑓_𝑖(𝑥,𝑦,𝑥_𝑖)∉−int𝐶_𝑖(𝑥);(ii)for each (𝑥,𝑦)∈𝑋×𝑌, 𝑧_𝑖↦𝑓_𝑖(𝑥,𝑦,𝑧_𝑖) is natural 𝑃_𝑖-quasifunction;(iii)for all 𝑧_𝑖∈𝑋_𝑖, the set {(𝑥,𝑦)∈𝑋×𝑌∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)∉−int𝐶_𝑖(𝑥)} is closed in 𝑋×𝑌;(iv)there exist nonempty and compact subsets 𝑁⊆𝑋 and 𝐾⊆𝑌 and nonempty, compact and convex subsets 𝐵_𝑖⊆𝑋_𝑖, 𝐴_𝑖⊆𝑌_𝑖 for each 𝑖∈𝐼 such that ∀(𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦)∈𝑋×𝑌⧵𝑁×𝐾∃𝑖∈𝐼 and ∃𝑢_𝑖∈𝐵_𝑖, 𝑣_𝑖∈𝐴_𝑖 satisfying 𝑢_𝑖∈𝐷_𝑖(𝑥,𝑦), 𝑣_𝑖∈𝑇_𝑖(𝑥,𝑦) and 𝑓_𝑖(𝑥,𝑦,𝑢_𝑖)∈−int𝐶_𝑖(𝑥).Then, there exists (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼, 𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)∉−int𝐶_𝑖(𝑥),∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (3.1) That is, the solution set of the (SGVQEP) is nonempty.

Proof. For each 𝑖∈𝐼, let us define a set-valued map 𝑄_𝑖∶𝑋×𝑌→2^𝑋𝑖 by 𝑄_𝑖(𝑥,𝑦)={𝑧_𝑖∈𝑋_𝑖∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)∈−int𝐶_𝑖(𝑥)},∀(𝑥,𝑦)∈𝑋×𝑌. (3.2) Then, ∀𝑖∈𝐼 and ∀(𝑥,𝑦)∈𝑋×𝑌, 𝑄_𝑖(𝑥,𝑦) is a convex set.
To prove it, let us fix arbitrary 𝑖∈𝐼 and (𝑥,𝑦)∈𝑋×𝑌. Let 𝑧_𝑖1,𝑧_𝑖2∈𝑄_𝑖(𝑥,𝑦) and 𝜆∈[0,1], then we have
𝑓_𝑖(𝑥,𝑦,𝑧_𝑖𝑗)∈−int𝐶_𝑖(𝑥),for𝑗=1,2. (3.3) Since 𝑓_𝑖(𝑥,𝑦,⋅) is natural 𝑃_𝑖-quasifunction, ∃𝜇∈[0,1] such that 𝑓_𝑖(𝑥,𝑦,𝜆𝑧_𝑖1+(1−𝜆)𝑧_𝑖2)∈𝜇𝑓_𝑖(𝑥,𝑦,𝑧_𝑖1)+(1−𝜇)𝑓_𝑖(𝑥,𝑦,𝑧_𝑖2)−𝑃_𝑖. (3.4) From (3.3) and (3.4), we get 𝑓_𝑖(𝑥,𝑦,𝜆𝑧_𝑖1+(1−𝜆)𝑧_𝑖2)∈−int𝐶_𝑖(𝑥)−int𝐶_𝑖(𝑥)−𝑃_𝑖⊆−int𝐶_𝑖(𝑥). (3.5)
Hence 𝜆𝑧_𝑖1+(1−𝜆)𝑧_𝑖2∈𝑄_𝑖(𝑥,𝑦) and, therefore, 𝑄_𝑖(𝑥,𝑦) is convex.
It follows from condition (i) that, for each 𝑖∈𝐼 and for all (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦)∈𝑋×𝑌,
𝑥_𝑖∉𝑄_𝑖(𝑥,𝑦). (3.6)
It follows from condition (iii) that for each 𝑖∈𝐼 and each 𝑧_𝑖∈𝑋_𝑖, the set
𝑄_𝑖⁻¹(𝑧_𝑖)={(𝑥,𝑦)∈𝑋×𝑌∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)∈−int𝐶_𝑖(𝑥)} (3.7) is open in 𝑋_𝑖. That is, 𝑄_𝑖 has open lower sections on 𝑋×𝑌. For each 𝑖∈𝐼, we also define another set-valued map 𝑆_𝑖∶𝑋×𝑌→2^{𝑋𝑖×𝑌𝑖} by 𝑆_𝑖(𝑥,𝑦)=[𝐷_𝑖(𝑥,𝑦)∩𝑄_𝑖(𝑥,𝑦)]×𝑇_𝑖(𝑥,𝑦),if(𝑥,𝑦)∈𝑊_𝑖,𝐷_𝑖(𝑥,𝑦)×𝑇_𝑖(𝑥,𝑦),if(𝑥,𝑦)∉𝑊_𝑖. (3.8) Then, it is clear that ∀𝑖∈𝐼 and ∀(𝑥,𝑦)∈𝑋×𝑌, 𝑆_𝑖(𝑥,𝑦) is convex, and (𝑥_𝑖,𝑦_𝑖)∉𝑆_𝑖(𝑥,𝑦). Since ∀𝑖∈𝐼 and ∀(𝑢_𝑖,𝑣_𝑖)∈𝑋_𝑖×𝑌_𝑖, 𝑆_𝑖⁻¹(𝑢_𝑖,𝑣_𝑖)=[𝑄_𝑖⁻¹(𝑢_𝑖)∩(𝐷_𝑖⁻¹(𝑢_𝑖))∩(𝑇_𝑖⁻¹(𝑣_𝑖))]∪[(𝑋×𝑌⧵𝑊_𝑖)∩(𝐷_𝑖⁻¹(𝑢_𝑖))∩(𝑇_𝑖⁻¹(𝑣_𝑖))], (3.9) and 𝐷_𝑖⁻¹(𝑢_𝑖), 𝑇_𝑖⁻¹(𝑣_𝑖), 𝑄_𝑖⁻¹(𝑢_𝑖), and 𝑋×𝑌⧵𝑊_𝑖 are open in 𝑋×𝑌, we have 𝑆_𝑖⁻¹(𝑢_𝑖,𝑣_𝑖) is open in 𝑋×𝑌.
From condition (iv), there exist a nonempty and compact subset 𝑁×𝐾⊆𝑋×𝑌 and a nonempty, compact, and convex subset 𝐵_𝑖×𝐴_𝑖⊆𝑋_𝑖×𝑌_𝑖 for each 𝑖∈𝐼 such that ∀(𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦)∈𝑋×𝑌⧵𝑁×𝐾∃𝑖∈𝐼 and ∃(𝑢_𝑖,𝑣_𝑖)∈𝑆_𝑖(𝑥,𝑦)∩(𝐵_𝑖×𝐴_𝑖). Hence, by Lemma 2.6, ∃(𝑥,𝑦)∈𝑋×𝑌 such that 𝑆_𝑖(𝑥,𝑦)=∅,∀𝑖∈𝐼. Since ∀𝑖∈𝐼 and ∀(𝑥,𝑦)∈𝑋×𝑋, 𝐷_𝑖(𝑥,𝑦) and 𝑇_𝑖(𝑥,𝑦) are nonempty, we have (𝑥,𝑦)∈𝑊_𝑖 and 𝐷_𝑖(𝑥,𝑦)∩𝑄_𝑖(𝑥,𝑦)=∅, ∀𝑖∈𝐼. This implies (𝑥,𝑦)∈𝑊_𝑖 and 𝐷_𝑖(𝑥,𝑦)∩𝑄_𝑖(𝑥,𝑦)=∅, ∀𝑖∈𝐼. Therefore, ∀𝑖∈𝐼,
𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦),𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)∉−int𝐶_𝑖(𝑥),∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (3.10) That is, the solution set of the (SGVQEP) is nonempty.

Remark 3.2. (1) The condition (iii) of Theorem 3.1 is satisfied if the following conditions hold ∀𝑖∈𝐼:
(a)𝐶_𝑖∶𝑋→2^𝑍𝑖 is a set-valued map such that int𝐶_𝑖(𝑥)≠∅ for each 𝑥∈𝑋 and the set-valued map 𝑀_𝑖=𝑍_𝑖⧵(−int𝐶_𝑖)∶𝑋→2^𝑍𝑖 is upper semicontinuous;(b)for all 𝑧_𝑖∈𝑋_𝑖, the map (𝑥,𝑦)↦𝑓_𝑖(𝑥,𝑦,𝑧_𝑖) is continuous on 𝑋×𝑌;
(2) If ∀𝑖∈𝐼, and ∀𝑥∈𝑋, 𝐶_𝑖(𝑥)=𝐶_𝑖, a (fixed) proper, closed and convex cone in 𝑍_𝑖, then the condition (ii) and (iii) of Theorem 3.1 can be replaced, respectively, by the following conditions:
(c)∀𝑖∈𝐼, the vector-valued function ∀(𝑥,𝑦)∈𝑋×𝑌, 𝑧_𝑖↦𝑓_𝑖(𝑥,𝑦,𝑧_𝑖) is 𝐶_𝑖-quasifunction;(d)∀𝑖∈𝐼, ∀𝑧_𝑖∈𝑋_𝑖, the map (𝑥,𝑦)↦𝑓_𝑖(𝑥,𝑦,𝑧_𝑖) is 𝐶_𝑖-upper semicontinuous on 𝑋×𝑌;
(3) Theorem 3.1 extends and generalizes in [19, Theorem  2], [20, Theorem  2.1] and [18, Theorem  2.1] in several ways.
(4) If ∀𝑖∈𝐼, 𝑋_𝑖 is a nonempty, compact and convex subset of a Hausdorff topological vector space 𝐸_𝑖, then the conclusion of Theorem 3.1 holds without condition (iv).

Theorem 3.3. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑍_𝑖 be a topological vector space, let 𝐸_𝑖 and 𝐹_𝑖 be two locally convex Hausdorff topological vector spaces, let 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty, closed and convex subsets, let 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌 and 𝑓_𝑖∶𝑋×𝑌×𝑋_𝑖→𝑍_𝑖 be a vector-valued function. For each 𝑖∈𝐼, let 𝐶_𝑖∶𝑋→2^𝑍𝑖 be a set-valued map such that 𝐶_𝑖(𝑥) be a proper closed and convex cone with apex at the origin and int𝐶_𝑖(𝑥)≠∅ for all 𝑥∈𝑋 and 𝑃_𝑖=∩_𝑥∈𝑋𝐶_𝑖(𝑥). Assume that the set-valued map 𝐷×𝑇=(∏_𝑖∈𝐼𝐷_𝑖×∏_𝑖∈𝐼𝑇_𝑖)∶𝑋×𝑌→2^𝑋×𝑌 defined as (𝐷×𝑇)(𝑥,𝑦)=∏_𝑖∈𝐼𝐷_𝑖(𝑥,𝑦)×∏_𝑖∈𝐼𝑇_𝑖(𝑥,𝑦), ∀(𝑥,𝑦)∈𝑋×𝑌, is Φ-condensing and for each 𝑖∈𝐼, the conditions (i), (ii) and (iii) of Theorem 3.1 hold. Then the solution set of the (SGVQEP) is nonempty.

Proof. In view of Lemma 2.7 and the proof of Theorem 3.1, it is sufficient to show that the set-valued map 𝑆∶𝑋×𝑌→2^𝑋×𝑌 defined as 𝑆(𝑥,𝑦)=∏_𝑖∈𝐼𝑆_𝑖(𝑥,𝑦),forall(𝑥,𝑦)∈𝑋×𝑌, is Φ-condensing, where 𝑆_𝑖's are the same as in the proof of Theorem 3.1. By the definition of 𝑆_𝑖, 𝑆_𝑖(𝑥,𝑦)⊆𝐷_𝑖(𝑥,𝑦)×𝑇_𝑖(𝑥,𝑦) for all (𝑥,𝑦)∈𝑋×𝑌 and for each 𝑖∈𝐼, and therefore 𝑆(𝑥,𝑦)⊆𝐷(𝑥,𝑦)×𝑇(𝑥,𝑦) for all (𝑥,𝑦)∈𝑋×𝑌. Since 𝐷×𝑇 is Φ-condensing, by Remark 2.5, we have 𝑆 is also Φ-condensing.

By Theorem 3.1 and Remark 3.2, we can easily get the following result.

Corollary 3.4. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝐸_𝑖 and 𝐹_𝑖 be two Hausdorff topological vector spaces, let 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty and convex subsets, let 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, let the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌, and 𝑓_𝑖∶𝑋×𝑌×𝑋_𝑖→𝑅 be a function. Assume that(i)for all 𝑥=(𝑥^𝑖,𝑥_𝑖)∈𝑋, for all 𝑦∈𝑌, 𝑓_𝑖(𝑥,𝑦,𝑥_𝑖)≥0;(ii)for each (𝑥,𝑦)∈𝑋×𝑌, 𝑧_𝑖↦𝑓_𝑖(𝑥,𝑦,𝑧_𝑖) is quasiconvex;(iii)for all 𝑧_𝑖∈𝑋_𝑖, the set {(𝑥,𝑦)∈𝑋×𝑌∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)≥0} is closed in 𝑋×𝑌;(iv)there exist nonempty and compact subsets 𝑁⊆𝑋 and 𝐾⊆𝑌 and nonempty, compact and convex subsets 𝐵_𝑖⊆𝑋_𝑖, 𝐴_𝑖⊆𝑌_𝑖 for each 𝑖∈𝐼 such that ∀(𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦)∈𝑋×𝑌⧵𝑁×𝐾∃𝑖∈𝐼 and ∃𝑢_𝑖∈𝐵_𝑖, 𝑣_𝑖∈𝐴_𝑖 satisfying 𝑢_𝑖∈𝐷_𝑖(𝑥,𝑦), 𝑣_𝑖∈𝑇_𝑖(𝑥,𝑦) and 𝑓_𝑖(𝑥,𝑦,𝑢_𝑖)<0.Then, there exists (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼, 𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝑓_𝑖(𝑥,𝑦,𝑧_𝑖)≥0,∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (3.11) That is, the solution set of the (SGQEP) is nonempty.

By Theorem 3.3, we can easily get the following result.

Corollary 3.5. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑍_𝑖 be a topological vector space, let 𝐸_𝑖 and 𝐹_𝑖 be two locally convex Hausdorff topological vector spaces, let 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty, closed and convex subsets, let 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌 and 𝑓_𝑖∶𝑋×𝑌×𝑋_𝑖→𝑅 be a function. Assume that the set-valued map 𝐷×𝑇=(∏_𝑖∈𝐼𝐷_𝑖×∏_𝑖∈𝐼𝑇_𝑖)∶𝑋×𝑌→2^𝑋×𝑌 defined as (𝐷×𝑇)(𝑥,𝑦)=∏_𝑖∈𝐼𝐷_𝑖(𝑥,𝑦)×∏_𝑖∈𝐼𝑇_𝑖(𝑥,𝑦), ∀(𝑥,𝑦)∈𝑋×𝑌, is Φ-condensing and for each 𝑖∈𝐼, the conditions (i), (ii) and (iii) of Corollary 3.4 hold. Then the solution set of the (SGQEP) is nonempty.

Remark 3.6. Theorem 3.3 is a generalization of [19, Theorem  3]. Corollaries 3.4 and 3.5 extend and generalize the main results in [10–17].

4. Applications

In this section, we present some existence of a solution for the (G-Debreu VEP) and the (G-Debreu EP).

Theorem 4.1. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑍_𝑖 be a topological vector space, let 𝐸_𝑖 and 𝐹_𝑖 be two Hausdorff topological vector spaces, let 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty and convex subsets, let 𝐶_𝑖∶𝑋→2^𝑍𝑖 be a set-valued map such that 𝐶_𝑖(𝑥) is a proper, closed and convex cone with apex at the origin and int𝐶_𝑖(𝑥)≠∅ for each 𝑥∈𝑋 and 𝑃_𝑖=∩_𝑥∈𝑋𝐶_𝑖(𝑥), 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌 and 𝜙_𝑖 be a bifunction from 𝑋×𝑌 into 𝑍_𝑖. For each 𝑖∈𝐼, assume that(i)𝑀_𝑖=𝑍_𝑖⧵(−int𝐶_𝑖)∶𝑋→2^𝑍𝑖 is upper semicontinuous;(ii)For all 𝑥^𝑖∈𝑋^𝑖 and 𝑦∈𝑌, 𝑧_𝑖↦𝜙_𝑖(𝑥^𝑖,𝑦,𝑧_𝑖) is natural 𝑃_𝑖-quasifunction, where 𝑃_𝑖=∩_𝑥∈𝑋𝐶_𝑖(𝑥);(iii)𝜙_𝑖 is continuous on 𝑋×𝑌;(iv)there exist nonempty and compact subsets 𝑁⊆𝑋 and 𝐾⊆𝑌 and nonempty, compact and convex subsets 𝐵_𝑖⊆𝑋_𝑖, 𝐴_𝑖⊆𝑌_𝑖 for each 𝑖∈𝐼 such that ∀(𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦)∈𝑋×𝑌⧵𝑁×𝐾∃𝑖∈𝐼 and ∃𝑢_𝑖∈𝐵_𝑖, 𝑣_𝑖∈𝐴_𝑖 satisfying 𝑢_𝑖∈𝐷_𝑖(𝑥,𝑦), 𝑣_𝑖∈𝑇_𝑖(𝑥,𝑦) and 𝜙_𝑖(𝑥^𝑖,𝑦,𝑢_𝑖)−𝜙_𝑖(𝑥,𝑦)∈−int𝐶_𝑖(𝑥).Then, there exists (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼, 𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝜙_𝑖(𝑥^𝑖,𝑦,𝑧_𝑖)−𝜙_𝑖(𝑥,𝑦)∉−int𝐶_𝑖(𝑥),∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (4.1) That is, the solution set of the (G-Debreu VEP) is nonempty.

Proof. For each 𝑖∈𝐼, we define a trifunction 𝑓_𝑖∶𝑋×𝑌×𝑋_𝑖 as 𝑓_𝑖(𝑥,𝑦,𝑢_𝑖)=𝜙_𝑖(𝑥^𝑖,𝑦,𝑢_𝑖)−𝜙_𝑖(𝑥,𝑦),∀(𝑥,𝑦,𝑢_𝑖)∈𝑋×𝑌×𝑋_𝑖. (4.2) Since 𝜙_𝑖(𝑥^𝑖,𝑦,⋅) is natural 𝑃_𝑖 quasi-function, by [19, Remark  2], for all 𝑢_𝑖1,𝑢_𝑖2∈𝑋_𝑖 and for all 𝜆∈[0,1],∃𝛼∈[0,1] such that 𝜙_𝑖(𝑥^𝑖,𝑦,𝜆𝑢_𝑖1+(1−𝜆)𝑢_𝑖2)∈𝛼𝜙_𝑖(𝑥^𝑖,𝑦,𝑢_𝑖1)+(1−𝛼)𝜙_𝑖(𝑥^𝑖,𝑦,𝑢_𝑖2)−𝑃_𝑖, (4.3) Hence 𝑓_𝑖(𝑥,𝑦,𝜆𝑢_𝑖1+(1−𝜆)𝑢_𝑖2)∈𝛼𝑓_𝑖(𝑥,𝑦,𝑢_𝑖1)+(1−𝛼)𝑓_𝑖(𝑥,𝑦,𝑢_𝑖2)−𝑃_𝑖. (4.4) Hence, for all (𝑥,𝑦)∈𝑋×𝑌, 𝑓_𝑖(𝑥,𝑦,⋅) is natural 𝑃_𝑖 quasifunction.
By condition (iii), we know that for all 𝑧_𝑖∈𝑋_𝑖, the map (𝑥,𝑦)↦𝑓_𝑖(𝑥,𝑦,𝑧_𝑖) is continuous on 𝑋×𝑌. So it follows from Remark 3.2 that condition (iii) of Theorem 3.1 holds. It is easy to verify that the other conditions of Theorem 3.1 are satisfied. By Theorem 3.1, we know that the conclusion holds.

Similarly, by Theorem 3.3, Corollaries 3.4 and 3.5, respectively, we have the following results.

Theorem 4.2. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑍_𝑖 be a topological vector space, let 𝐸_𝑖 and 𝐹_𝑖 be two locally convex Hausdorff topological vector spaces, let 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty, closed and convex subsets, let 𝐶_𝑖∶𝑋→2^𝑍𝑖 be a set-valued map such that 𝐶_𝑖(𝑥) is a proper, closed and convex cone with apex at the origin and int𝐶_𝑖(𝑥)≠∅ for each 𝑥∈𝑋 and 𝑃_𝑖=∩_𝑥∈𝑋𝐶_𝑖(𝑥), 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌 and 𝜑_𝑖∶𝑋×𝑌→𝑍_𝑖 be a vector-valued function. Assume that the set-valued map 𝐷×𝑇=(∏_𝑖∈𝐼𝐷_𝑖×∏_𝑖∈𝐼𝑇_𝑖)∶𝑋×𝑌→2^𝑋×𝑌 defined as (𝐷×𝑇)(𝑥,𝑦)=∏_𝑖∈𝐼𝐷_𝑖(𝑥,𝑦)×∏_𝑖∈𝐼𝑇_𝑖(𝑥,𝑦), ∀(𝑥,𝑦)∈𝑋×𝑌, is Φ-condensing and (i), (ii), and (iii) of Theorem 4.1 hold. Then, the solution set of the (G-Debreu VEP) is nonempty.

Theorem 4.3. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty and convex subsets, let 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌 and 𝜙_𝑖 be a bifunction from 𝑋×𝑌 into 𝑅. For each 𝑖∈𝐼, assume that(i)for all 𝑥^𝑖∈𝑋^𝑖 and 𝑦∈𝑌, 𝑧_𝑖↦𝜙_𝑖(𝑥^𝑖,𝑦,𝑧_𝑖) is quasiconvex;(ii)𝜙_𝑖 is continuous on 𝑋×𝑌;(iii)there exist nonempty and compact subsets 𝑁⊆𝑋 and 𝐾⊆𝑌 and nonempty, compact and convex subsets 𝐵_𝑖⊆𝑋_𝑖, 𝐴_𝑖⊆𝑌_𝑖 for each 𝑖∈𝐼 such that ∀(𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦)∈𝑋×𝑌⧵𝑁×𝐾∃𝑖∈𝐼 and ∃𝑢_𝑖∈𝐵_𝑖, 𝑣_𝑖∈𝐴_𝑖 satisfying 𝑢_𝑖∈𝐷_𝑖(𝑥,𝑦), 𝑣_𝑖∈𝑇_𝑖(𝑥,𝑦) and 𝜙_𝑖(𝑥^𝑖,𝑦,𝑢_𝑖)<𝜙_𝑖(𝑥,𝑦).Then, there exists (𝑥,𝑦)=(𝑥^𝑖,𝑥_𝑖,𝑦^𝑖,𝑦_𝑖) in 𝑋×𝑌 such that for each 𝑖∈𝐼, 𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦),𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)∶𝜙_𝑖(𝑥^𝑖,𝑦,𝑧_𝑖)≥𝜙_𝑖(𝑥,𝑦),∀𝑧_𝑖∈𝐷_𝑖(𝑥,𝑦). (4.5) That is, the solution set of the (G-Debreu EP) is nonempty.

Theorem 4.4. Let 𝐼 be any index set. For each 𝑖∈𝐼, let 𝐸_𝑖 and 𝐹_𝑖 be two locally convex Hausdorff topological vector spaces, 𝑋_𝑖⊆𝐸_𝑖 and 𝑌_𝑖⊆𝐹_𝑖 be nonempty, closed and convex subsets, let 𝐷_𝑖∶𝑋×𝑌→2^𝑋𝑖 and 𝑇_𝑖∶𝑋×𝑌→2^𝑌𝑖 be set-valued maps with nonempty convex values and open lower sections, the set 𝑊_𝑖={(𝑥,𝑦)∈𝑋×𝑌∶𝑥_𝑖∈𝐷_𝑖(𝑥,𝑦)and𝑦_𝑖∈𝑇_𝑖(𝑥,𝑦)} be closed in 𝑋×𝑌 and 𝜑_𝑖∶𝑋×𝑌→𝑅 be a function. Assume that the set-valued map 𝐷×𝑇=(∏_𝑖∈𝐼𝐷_𝑖×∏_𝑖∈𝐼𝑇_𝑖)∶𝑋×𝑌→2^𝑋×𝑌 defined as (𝐷×𝑇)(𝑥,𝑦)=∏_𝑖∈𝐼𝐷_𝑖(𝑥,𝑦)×∏_𝑖∈𝐼𝑇_𝑖(𝑥,𝑦), ∀(𝑥,𝑦)∈𝑋×𝑌, is Φ-condensing and (i), and (ii) of Theorem 4.3 hold. Then, the solution set of the (G-Debreu EP) is nonempty.

Remark 4.5. Theorem 4.1 extends and generalizes [19, Theorem  5] and [20, Theorems  3.1,  3.6 and Corollaries  3.2,  3.3, and  3.5]. Theorem 4.2 extends and generalizes [19, Theorem  6]. Theorems 4.3 and 4.4 are generalizations of [20, Corollaries  3.5 and  3.7] and the corresponding results in [21–24].

Acknowledgment

This research was supported by the National Natural Science Foundation of China (Grant no. 10771228 and Grant no. 10831009).