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Research ArticleSystem of Operator Quasi Equilibrium Problems

Suhel Ahmad Khan

Department of Mathematics, BITS-Pilani, Dubai Campus, P.O. Box 345055, Dubai, UAE

Correspondence should be addressed to Suhel Ahmad Khan; [email protected]

Received 24 January 2014; Accepted 4 June 2014; Published 19 June 2014

Academic Editor: Sivaguru Sritharan

Copyright © 2014 Suhel Ahmad Khan.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a system of operator quasi equilibrium problems and system of generalized quasi operator equilibrium problems intopological vector spaces. Using a maximal element theorem for a family of set-valued mappings as basic tool, we derive someexistence theorems for solutions to these problems with and without involvingΦ-condensing mappings.

1. Introduction

In 2002, Domokos and Kolumbán [1] gave an interestinginterpretation of variational inequality and vector variationalinequalities (for short, VVI) in Banach space settings in termsof variational inequalities with operator solutions (for short,OVVI). The notion and viewpoint of OVVI due to DomokosandKolumbán [1] looknew and interesting even though it hasa limitation in application toVVI. Recently, Kazmi and Raouf[2] introduced the operator equilibrium problem which gen-eralizes the notion of OVVI to operator vector equilibriumproblems (for short, OVEP) using the operator solution.Theyderived some existence theorems of solution of OVEP withpseudomonotonicity, without pseudomonotonicity, and with𝐵-pseudomonotonicity. However, they dealt with only thesingle-valued case of the bioperator. It is very natural anduseful to extend a single-valued case to a corresponding set-valued one from both theoretical and practical points of view.

The system of vector equilibrium problems and thesystem of vector quasi equilibriumproblemswere introducedand studied by Ansari et al. [3, 4]. Inspired by above citedwork, in this paper, we consider a system of operator quasiequilibrium problems (for short, SOQEP) in topologicalvector spaces. Using a maximal element theorem for a familyof set-valued mappings according to [5] as basic tool, wederive some existence theorems for solutions to SOQEP withand without involvingΦ-condensing mappings.

Further, we consider a system of generalized quasi oper-ator equilibrium problems (for short, SGQOEP) in topo-logical vector spaces and give some of its special cases and

derive some existence theorems for solutions to SOQEPwith andwithout involvingΦ-condensingmappings by usingwell-known maximal element theorem [5] for a family ofset-valued mappings, and, consequently, we also get someexistence theorems for solutions to a system of operatorequilibrium problems.

2. Preliminaries

Let 𝐼 be an index set, for each 𝑖 ∈ 𝐼, and let𝑋𝑖be a Hausdorff

topological vector space. We denote 𝐿(𝑋𝑖, 𝑌𝑖), the space of

all continuous linear operators from 𝑋𝑖into 𝑌

𝑖, where 𝑌

𝑖is

topological vector space for each 𝑖 ∈ 𝐼. Consider a family ofnonempty convex subsets {𝐾

𝑖}𝑖∈𝐼

with𝐾𝑖in 𝐿(𝑋

𝑖, 𝑌𝑖).

Let

𝑋 = ∏

𝑖∈𝐼

𝑋𝑖,

𝐾 = ∏

𝑖∈𝐼

𝐾𝑖.

(1)

Let 𝐶𝑖: 𝐾 → 2

𝑌𝑖 be a set-valued mapping such that, foreach 𝑓 ∈ 𝐾, 𝐶

𝑖(𝑓) is solid, open, and convex cone such that

0 ∉ 𝐶𝑖(𝑓) and 𝑃

𝑖= ⋂𝑓∈𝐾

𝐶𝑖(𝑓).

For each 𝑖 ∈ 𝐼, let 𝐹𝑖: 𝐾 × 𝐾

𝑖→ 𝑌𝑖be a bifunction and

let 𝐴𝑖: 𝐾 → 2

𝐾𝑖 be a set-valued mapping with nonemptyvalues. We consider the following system of operator quasi

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2014, Article ID 848206, 6 pageshttp://dx.doi.org/10.1155/2014/848206

2 International Journal of Analysis

equilibrium problems (for short, SOQEP). Find 𝑓 ∈ 𝐾 suchthat, for each 𝑖 ∈ 𝐼,

𝑓𝑖∈ 𝐴𝑖(𝑓) , 𝐹

𝑖(𝑓, 𝑔𝑖) ∉ −𝐶

𝑖(𝑓) , ∀𝑔

𝑖∈ 𝐴𝑖(𝑓) . (2)

We remarked that, for the suitable choices of 𝐼, 𝐹𝑖, 𝐾𝑖,

𝑋𝑖, 𝑌𝑖, 𝐶𝑖, and 𝐴

𝑖, SOQEP (2) reduces to the problems

considered and studied by [3–6] and the references therein.Now, we will give the following concepts and results

which are used in the sequel.

Definition 1. Let 𝑀 be a nonempty and convex subset of atopological vector space, and let 𝑍 be a topological vectorspace with a closed and convex cone𝑃with apex at the origin.A vector-valued function 𝜙 : 𝑀 → 𝑍 is said to be as follows:

(i) P-function if and only if ∀𝑥, 𝑦 ∈ 𝑀 and 𝜆 ∈ [0, 1]:

𝜙 (𝜆𝑥 + (1 − 𝜆) 𝑦) ∈ 𝜆𝜙 (𝑥) + (1 − 𝜆) 𝜙 (𝑦) − 𝑃; (3)

(ii) natural P-quasifunction if and only if ∀𝑥, 𝑦 ∈ 𝑀 and𝜆 ∈ [0, 1]:

𝜙 (𝜆𝑥 + (1 − 𝜆) 𝑦) ∈ 𝐶𝑜 {𝜙 (𝑥) , 𝜙 (𝑦)} − 𝑃, (4)

where 𝐶𝑜𝐵 denotes the convex hull of 𝐵;(iii) P-quasifunction if and only if ∀𝛼 ∈ 𝑍 and the set {𝑥 ∈

𝑀 : 𝜙(𝑥) − 𝛼 ∈ −𝑃} is convex.

Definition 2 (see [7]). Let𝑋 be a topological vector space andlet 𝐿 be a lattice with a minimal element, denoted by 0. Amapping 𝜙 : 2𝑋 → 𝐿 is called a measure of noncompactnessprovided that the following conditions hold for any 𝑀,𝑁 ∈2𝑋:

(i) 𝜙(𝐶𝑜𝑀) = 𝜙(𝑀), where 𝐶𝑜𝑀 denotes the closedconvex hull of𝑀;

(ii) 𝜙(𝑀) = 𝑜 if and only if𝑀 is precompact;(iii) 𝜙(𝑀 ∪𝑁) = max{𝜙(𝑀), 𝜙(𝑁)}.

Definition 3 (see [7]). Let 𝑋 be a topological vector space,𝐷 ⊂ 𝑋, and let 𝜙 be a measure of noncompactness on 𝑋.A set-valued mapping 𝑇 : 𝐷 → 2𝑋 is called 𝜙-condensingprovided that 𝑀 ⊂ 𝐷 with 𝜙(𝑇(𝑀)) ≥ 𝜙(𝑀); then 𝑀 isrelative compact; that is,𝑀 is compact.

Remark 4. Note that every set-valued mapping defined ona compact set is 𝜙-condensing for any measure of noncom-pactness 𝜙. If 𝑋 is locally convex, then a compact set-valuedmapping (i.e., 𝑇(𝐷) is precompact) is 𝜙-condensing for anymeasure of noncompactness 𝜙. Obviously, if 𝑇 : 𝐷 → 2𝑋 is𝜙-condensing and 𝑇 : 𝐷 → 2𝑋 satisfies 𝑇(𝑥) ⊂ 𝑇(𝑥), forall 𝑥 ∈ 𝑋, then 𝑇 is also 𝜙-condensing.

The following maximal element theorems will play keyrole in establishing existence results.

Theorem 5 (see [8]). For each 𝑖 ∈ 𝐼, let 𝐾𝑖be a nonempty

convex subset of a topological vector space 𝑋𝑖and let 𝑆

𝑖, 𝑇𝑖:

𝐾 → 2𝐾𝑖 be the two set-valued mappings. For each 𝑖 ∈ 𝐼,

assume that the following conditions hold:

(a) for all 𝑥 ∈ 𝐾, 𝐶𝑜𝑆𝑖(𝑥) ⊆ 𝑇

𝑖(𝑥);

(b) for all 𝑥 ∈ 𝐾, 𝑥𝑖∉ 𝑇𝑖(𝑥);

(c) for all 𝑦𝑖∈ 𝐾𝑖, 𝑆𝑖

−1(𝑦𝑖) is compactly open 𝐾;

(d) there exist a nonempty compact subset 𝐷 of 𝐾 and anonempty compact convex subset 𝐸

𝑖⊆ 𝐾𝑖, for each 𝑖 ∈

𝐼, such that, for all 𝑥 ∈ 𝐾 \ 𝐷, there exists 𝑖 ∈ 𝐼 suchthat 𝑆

𝑖(𝑥) ∩ 𝐸

𝑖̸= 0.

Then, there exists 𝑥 ∈ 𝐾 such that 𝑆𝑖(𝑥) = 0 for each 𝑖 ∈ 𝐼.

We will use the following particular form of a maximalelement theorem for a family of set-valued mappings due toDeguire et al. [5].

Theorem 6 (see [5]). Let 𝐼 be any index set, for each 𝑖 ∈ 𝐼,let 𝐾𝑖be a nonempty convex subset of a Hausdorff topological

vector space𝑋𝑖, and let 𝑆

𝑖: 𝐾 = ∏

𝑖∈𝐼𝐾𝑖→ 2𝐾𝑖 be a set-valued

mapping. Assume that the following conditions hold:

(i) ∀𝑖 ∈ 𝐼 and ∀𝑥 ∈ 𝐾; 𝑆𝑖(𝑥) is convex;

(ii) ∀𝑖 ∈ 𝐼 and ∀𝑥 ∈ 𝐾; 𝑥𝑖∉ 𝑆𝑖(𝑥), where 𝑥

𝑖is the 𝑖th

component of 𝑥;(iii) ∀𝑖 ∈ 𝐼 and ∀𝑦

𝑖∈ 𝐾𝑖; 𝑆𝑖

−1(𝑦𝑖) is open 𝐾;

(iv) there exist a nonempty compact subset 𝐷 of 𝐾 and anonempty compact convex subset 𝐸

𝑖⊆ 𝐾𝑖, ∀𝑖 ∈ 𝐼 such

that ∀𝑥 ∈ 𝐾 \𝐷 and there exists 𝑖 ∈ 𝐼 such that 𝑆𝑖(𝑥) ∩

𝐸𝑖

̸= 0.

Then, there exists 𝑥 ∈ 𝐾 such that 𝑆𝑖(𝑥) = 0 for each 𝑖 ∈ 𝐼.

Remark 7. If ∀𝑖 ∈ 𝐼, 𝐾𝑖is nonempty, closed, and convex

subset of a locally convex Hausdorff topological vector space𝑋𝑖, then condition (iv) of Theorem 6 can be replaced by the

following condition:(iv)1 the set-valued mapping 𝑆 : 𝐾 → 2𝐾 is defined as

𝑆(𝑥) = ∏𝑖∈𝐼

𝑆𝑖(𝑥), ∀𝑥 ∈ 𝐾, 𝜙-condensing.

3. Main Result

Throughout this paper, unless otherwise stated, for any indexset 𝐼 and for each 𝑖 ∈ 𝐼, let 𝑌

𝑖be a topological vector space

and let 𝐾 = ∏𝑖∈𝐼

𝐾𝑖, 𝐶𝑖: 𝐾 → 2

𝑌𝑖 be a set-valued mappingsuch that, for each 𝑓 ∈ 𝐾, 𝐶

𝑖(𝑓) is proper, solid, open, and

convex cone such that 0 ∉ 𝐶𝑖(𝑓) and 𝑃

𝑖= ⋂𝑓∈𝐾

𝐶𝑖(𝑓). We

denote 𝐿(𝑋𝑖, 𝑌𝑖), the space of all continuous linear operators

from𝑋𝑖into 𝑌

𝑖. We also assume that ∀𝑖 ∈ 𝐼, 𝐴

𝑖: 𝐾 → 2

𝐾𝑖 isa set-valued mapping such that ∀𝑓 ∈ 𝐾,𝐴

𝑖(𝑓) is nonempty

and convex,𝐴−1(𝑔𝑖) is open in𝐾,𝑓

𝑖∈ 𝐾𝑖, and the set𝐹

𝑖: {𝑓 ∈

𝐾 : 𝑓𝑖∈ 𝐴𝑖(𝑓)} is closed in𝐾, where 𝑓

𝑖is the 𝑖th component

of 𝑓.Now, we have the following existence result for SOQEP

(2).

Theorem 8. For each 𝑖 ∈ 𝐼, let 𝐾𝑖be nonempty and convex

subset of aHausdorff topological vector space𝑋𝑖and let𝐹

𝑖: 𝐾×

𝐾𝑖→ 𝑌𝑖be a bifunction. Suppose that the following conditions

hold:

International Journal of Analysis 3

(i) ∀𝑖 ∈ 𝐼 and ∀𝑓 ∈ 𝐾, 𝐹𝑖(𝑓, 𝑓𝑖) ∉ −𝐶

𝑖(𝑓), where 𝑓

𝑖is the

𝑖th component of 𝑓;(ii) ∀𝑖 ∈ 𝐼 and ∀𝑓 ∈ 𝐾; the vector-valued function 𝑔

𝑖→

𝐹𝑖(𝑓, 𝑔𝑖) is natural 𝑃

𝑖-quasifunction;

(iii) ∀𝑖 ∈ 𝐼 and ∀𝑔𝑖∈ 𝐾𝑖; the set {𝑓 ∈ 𝐾 : 𝐹

𝑖(𝑓, 𝑔𝑖) ∉

−𝐶𝑖(𝑓)} is closed in 𝐾;

(iv) there exist a nonempty compact subset 𝑁 of 𝐾 and anonempty compact convex subset𝐵

𝑖of𝐾𝑖, for each 𝑖 ∈ 𝐼

such that ∀𝑓 ∈ 𝐾 \ 𝑁; there exists 𝑖 ∈ 𝐼 and 𝑔𝑖∈ 𝐵𝑖

such that 𝑔𝑖∈ 𝐴𝑖(𝑓) and 𝐹

𝑖(𝑓, 𝑔𝑖) ∈ −𝐶

𝑖(𝑓).

Then SOQEP (2) has a solution.

Proof. Let us define, for each given 𝑖 ∈ 𝐼, a set-valuedmapping 𝑇

𝑖: 𝐾 → 2

𝐾𝑖 by

𝑇𝑖(𝑓) = {𝑔

𝑖∈ 𝐾𝑖: 𝐹𝑖(𝑓, 𝑔𝑖) ∈ −𝐶

𝑖(𝑓)} , ∀𝑓 ∈ 𝐾. (5)

First, we claim that ∀𝑖 ∈ 𝐼 and 𝑓 ∈ 𝐾, 𝑇𝑖(𝑓) is convex. Fix an

arbitrary 𝑖 ∈ 𝐼 and 𝑓 ∈ 𝐾. Let 𝑔𝑖,1, 𝑔𝑖,2

∈ 𝑇𝑖(𝑓) and 𝜆 ∈ [0, 1];

then we have

𝐹𝑖(𝑓, 𝑔𝑖𝑗) ∈ −𝐶

𝑖(𝑓) , for 𝑗 = 1, 2. (𝑖)

Since𝐹𝑖(𝑓, ⋅) is natural𝑃

𝑖-quasifunction, there exists𝜇 ∈ [0, 1]

such that

𝐹𝑖(𝑓, 𝜆𝑔

𝑖,1+ (1 − 𝜆) 𝑔𝑖,2) ∈ 𝜇𝐹𝑖 (𝑓, 𝑔𝑖,1)

+ (1 − 𝜇) 𝐹𝑖(𝑓, 𝑔𝑖,2) − 𝑃𝑖.

(𝑖𝑖)

From the inclusion of (𝑖) and (𝑖𝑖), we get

𝐹𝑖(𝑓, 𝜆𝑔

𝑖,1+ (1 − 𝜆) 𝑔𝑖,2) ∈ −𝐶𝑖 (𝑓) − 𝐶𝑖 (𝑓) − 𝑃𝑖 ⊆ 𝐶𝑖 (𝑓) .

(6)

Hence, 𝜆𝑔𝑖,1+(1−𝜆)𝑔

𝑖,2∈ 𝑇𝑖(𝑓) and therefore𝑇

𝑖(𝑓) is convex.

Since 𝑖 ∈ 𝐼 and 𝑓 ∈ 𝐾 are arbitrary, 𝑇𝑖(𝑓) is convex, ∀𝑓 ∈ 𝐾

and ∀𝑖 ∈ 𝐼.Hence, our claim is then verified.Now ∀𝑖 ∈ 𝐼 and ∀𝑔

𝑖∈ 𝐾𝑖; the complement of 𝑇

𝑖

−1(𝑔𝑖) in

𝐾 can be defined as

[𝑇𝑖

−1(𝑔𝑖)]𝑐

= {𝑓 ∈ 𝐾 : 𝐹𝑖(𝑓, 𝑔𝑖) ∉ −𝐶

𝑖(𝑓)} . (7)

From condition (iii) of the above theorem, [𝑇𝑖

−1(𝑔𝑖)]𝑐 will be

closed in 𝐾.Suppose that ∀𝑖 ∈ 𝐼 and ∀𝑓 ∈ 𝐾; we define another set-

valued mapping𝑀𝑖: 𝐾 → 2

𝐾𝑖 by

𝑀𝑖(𝑓) =

{

{

{

𝐴𝑖(𝑓) ∩ 𝑇

𝑖(𝑓) if 𝑓 ∈ F

𝑖

𝐴𝑖(𝑓) ; if 𝑓 ∈ 𝐾 \F

𝑖.

(8)

Then, it is clear that ∀𝑖 ∈ 𝐼 and ∀𝑓 ∈ 𝐾,𝑀𝑖(𝑓) is convex,

because 𝐴(𝑓) and 𝑇𝑖(𝑓) are both convex. Now, by condition

(i), 𝑓𝑖∉ 𝑀𝑖(𝑓). Since ∀𝑖 ∈ 𝐼 and ∀𝑔

𝑖∈ 𝐾𝑖,

𝑀𝑖

−1(𝑔𝑖)

= (𝐴𝑖

−1(𝑔𝑖) ∩ 𝑇𝑖

−1(𝑔𝑖))⋃((𝐾 \F

𝑖) ∩ 𝐴𝑖

−1(𝑔𝑖))

(9)

is open in 𝐾, because 𝐴𝑖

−1(𝑔𝑖), 𝑇𝑖

−1(𝑔𝑖) and 𝐾 \F

𝑖are open

in 𝐾.Condition (iv) of Theorem 6 is followed from condition

(iv). Hence, by fixed pointTheorem 6, there exists𝑓 ∈ 𝐾 suchthat 𝑀

𝑖(𝑓) = 0, ∀𝑖 ∈ 𝐼. Since ∀𝑖 ∈ 𝐼 and ∀𝑓 ∈ 𝐾,𝐴

𝑖(𝑓)

is nonempty, we have 𝐴𝑖(𝑓) ∩ 𝑇

𝑖(𝑓) = 0, ∀𝑖 ∈ 𝐼. Therefore,

∀𝑖 ∈ 𝐼, 𝑓𝑖∈ 𝐴𝑖(𝑓) and 𝐹

𝑖(𝑓, 𝑔𝑖) ∈ −𝐶

𝑖(𝑓), ∀𝑔

𝑖∈ 𝐴𝑖(𝑓).

This completes the proof.

Now, we establish an existence result for SOQEP (2)involving 𝜙-condensing maps.

Theorem 9. For each 𝑖 ∈ 𝐼, let 𝐾𝑖be a nonempty, closed, and

convex subset of a locally convex Hausdorff topological vectorspace 𝑋

𝑖, suppose that 𝐹

𝑖: 𝐾 × 𝐾

𝑖→ 𝑌𝑖is a bifunction, and

let the set-valued mapping 𝐴 = ∏𝑖∈𝐼

𝐴𝑖: 𝐾 → 2

𝐾 definedas 𝐴(𝑓) = ∏

𝑖∈𝐼𝐴𝑖(𝑓), ∀𝑓 ∈ 𝐾 be 𝜙-condensing. Assume that

conditions (i), (ii), and (iii) of Theorem 8 hold. Then SOQEP(2) has a solution.

Proof. In view of Remark 7, it is sufficient to show that theset-valued mapping 𝑆 : 𝐾 → 2𝐾 defined as 𝑆(𝑓) =∏𝑖∈𝐼

𝑆𝑖(𝑓), ∀𝑓 ∈ 𝐾, is 𝜙-condensing, where 𝑆

𝑖s are the same

as defined in the proof of Theorem 8. By the definition of𝑆𝑖, 𝑆𝑖(𝑓) ⊆ 𝐴

𝑖(𝑓), ∀𝑖 ∈ 𝐼 and ∀𝑓 ∈ 𝐾 and therefore 𝑆(𝑓) ⊆

𝐴(𝑓), ∀𝑓 ∈ 𝐾. Since 𝐴 is 𝜙-condensing, by Remark 7, wehave 𝑆 being also 𝜙-condensing.


4. System of Generalized Quasi OperatorEquilibrium Problem

Throughout this section, unless otherwise stated, let 𝐼 be anyindex set. For each 𝑖 ∈ 𝐼, let 𝑋

𝑖be a Hausdorff topological

vector space. We denote 𝐿(𝑋𝑖, 𝑌𝑖), the space of all continuous

linear operators from𝑋𝑖into𝑌

𝑖, where𝑌

𝑖is topological vector

space for each 𝑖 ∈ 𝐼 and for each 𝑖 ∈ 𝐼; let 𝑃𝑖⊂ 𝑌𝑖be a closed,

pointed, and convex conewith int𝑃𝑖

̸= 0, where int 𝑃𝑖denotes

the interior of set 𝑃𝑖, ∀𝑖 ∈ 𝐼. Consider a family of nonempty

convex subsets {𝐾𝑖}𝑖∈𝐼

with𝐾𝑖in 𝐿(𝑋

𝑖, 𝑌𝑖). Let, for each 𝑖 ∈ 𝐼,

a bifunction 𝐹𝑖: 𝐾 × 𝐾

𝑖→ 𝑌𝑖and two set-valued mappings

𝐴𝑖, 𝐵𝑖: 𝐾 → 2

𝐾𝑖 be with nonempty values.Let 𝑒𝑖be the unit vector in 𝑌

𝑖, for each 𝑖 ∈ 𝐼, and also

𝛼𝑖𝑒𝑖, 𝛽𝑖𝑒𝑖∈ 𝑌𝑖such that 𝛼

𝑖𝑒𝑖̸<𝑃𝑖

𝛽𝑖𝑒𝑖, where 𝛼

𝑖, 𝛽𝑖∈ R are two

real numbers such that 𝛼𝑖≤ 𝛽𝑖.

Now, we consider the system of generalized quasi operatorequilibrium problems (for short, SGQOEP). Find 𝑓 ∈ 𝐾 suchthat, for each 𝑖 ∈ 𝐼,

𝑓𝑖∈ 𝐵𝑖(𝑓) , 𝛼

𝑖𝑒𝑖̸<𝑃𝑖𝐹𝑖(𝑓, 𝑔𝑖) ̸<𝑃𝑖𝛽𝑖𝑒𝑖; ∀𝑔

𝑖∈ 𝐴𝑖(𝑓) .

(10)

4.1. Special Cases

(I) If 𝐴𝑖= 𝐵𝑖, ∀𝑖, then SGQOEP (10) reduces to finding

of 𝑓 ∈ 𝐾 such that, for each 𝑖 ∈ 𝐼,

𝛼𝑖𝑒𝑖̸<𝑃𝐹𝑖(𝑓, 𝑔𝑖) ̸<𝑃𝛽𝑖𝑒𝑖; ∀𝑔

𝑖∈ 𝐴𝑖(𝑓) . (11)


(II) If, in Case (I), we take 𝑌𝑖= R, then 𝑃

𝑖= R+and

𝑒𝑖= 1; then problem (10) reduces to the system of

generalized quasi operator equilibriumproblemswithlower and upper bounds (for short, SGQOEPLUB).Find 𝑓 ∈ 𝐾 such that, for each 𝑖 ∈ 𝐼,

𝑓𝑖∈ 𝐵𝑖(𝑓) , 𝛼

𝑖≤ 𝐹𝑖(𝑓, 𝑔𝑖) ≤ 𝛽𝑖; ∀𝑔

𝑖∈ 𝐴𝑖(𝑓) . (12)

Now, we establish the existence result for SGQOEP (10).

Theorem 10. For each 𝑖 ∈ 𝐼, let 𝐾𝑖be a nonempty convex

subset of a topological vector space𝑋𝑖and𝐹𝑖, 𝑃𝑖, 𝑄𝑖: 𝐾×𝐾

𝑖→

𝐾𝑖are the bifunctions, 𝐵

𝑖: 𝐾 → 2

𝐾𝑖 is a set-valued mappingsuch that the set 𝐹

𝑖= {𝑓 ∈ 𝐾 : 𝑓

𝑖∈ 𝐵𝑖(𝑓)} is compactly closed,

𝐴𝑖: 𝐾 → 2

𝐾𝑖 is a set-valued mapping with nonempty valuessuch that, for each 𝑔

𝑖∈ 𝐾𝑖, 𝐴𝑖

−1(𝑔𝑖) is compactly open in 𝐾,

and ∀𝑖 ∈ 𝐼, 𝑒𝑖∈ 𝑌𝑖are the unit vector such that 𝛼

𝑖𝑒𝑖̸<𝑃𝑖𝛽𝑖𝑒𝑖,

where 𝛼𝑖, 𝛽𝑖∈ R are two real numbers such that 𝛼

𝑖≤ 𝛽𝑖. For

each 𝑖 ∈ 𝐼, assume that the following conditions hold:

(i) for all 𝑓 ∈ 𝐾, 𝐶𝑜𝐴𝑖(𝑓) ⊆ 𝐵

𝑖(𝑓);

(ii) for all 𝑓 ∈ 𝐾, 𝛼𝑖𝑒𝑖>𝑃𝑖𝑃𝑖(𝑓, 𝑓𝑖) or 𝑄

𝑖(𝑓, 𝑓𝑖)>𝑃𝑖𝛽𝑖𝑒𝑖;

(iii) for all𝑓 ∈ 𝐾 and for every nonempty finite subset𝑁𝑖⊆

{𝑔𝑖∈ 𝐾𝑖: 𝐹𝑖(𝑓, 𝑔𝑖)<𝑃𝑖𝛼𝑖𝑒𝑖or 𝐹𝑖(𝑓, 𝑔𝑖)>𝑃𝑖𝛽𝑖𝑒𝑖}, we have

𝐶𝑜𝑁𝑖⊆ {𝑔𝑖∈ 𝐾𝑖: 𝑃𝑖(𝑓, 𝑔𝑖) ̸<𝑃𝑖𝛼𝑖𝑒𝑖and 𝑄

𝑖(𝑓, 𝑔𝑖) ̸>𝑃𝑖𝛽𝑖𝑒𝑖} ;

(13)

(iv) for all 𝑔𝑖∈ 𝐾𝑖, the set {𝑓 ∈ 𝐾 : 𝛽

𝑖𝑒𝑖̸<𝑃𝑖𝐹𝑖(𝑓, 𝑔𝑖) ̸<𝑃𝑖𝛼𝑖𝑒𝑖

is compactly closed in 𝐾;(v) there exist a nonempty compact subset 𝐷 of 𝐾 and a

nonempty compact convex subset 𝐸𝑖⊂ 𝐾𝑖, for each 𝑖 ∈

𝐼, such that, for all𝑓 ∈ 𝐾\𝐷, there exists 𝑖 ∈ 𝐼 such that𝑔𝑖∈ 𝐸𝑖satisfying 𝑔

𝑖∈ 𝐴𝑖(𝑓) and either 𝐹

𝑖(𝑓, 𝑔𝑖)𝑃𝑖

<

𝛼𝑖𝑒𝑖or 𝐹𝑖(𝑓, 𝑔𝑖)>𝑃𝑖𝛽𝑖𝑒𝑖.

Then the problem SGQOEP (10) has a solution.

Proof. For each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾, define two set-valuedmappings 𝐺

𝑖, 𝐻𝑖: 𝐾 → 2

𝐾𝑖 by

𝐺𝑖(𝑓) = {𝑔

𝑖∈ 𝐾𝑖: 𝐹𝑖(𝑓, 𝑔𝑖) <𝑃𝑖𝛼𝑖𝑒𝑖or 𝐹𝑖(𝑓, 𝑔𝑖) >𝑃𝑖𝛽𝑖𝑒𝑖} ,

𝐻𝑖(𝑓) = {𝑔

𝑖∈ 𝐾𝑖: 𝑃𝑖(𝑓, 𝑔𝑖)𝑃𝑖

̸< 𝛼𝑖𝑒𝑖and 𝑄

𝑖(𝑓, 𝑔𝑖) ̸>𝑃𝑖𝛽𝑖𝑒𝑖} .

(14)

Condition (iii) implies that, for each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾,𝐶𝑜𝐺𝑖(𝑓) ⊆ 𝐻

𝑖(𝑓).

From condition (ii), we have 𝑓𝑖∉ 𝐻𝑖(𝑓) for all 𝑓 ∈ 𝐾 and

for each 𝑖 ∈ 𝐼.Thus, for each 𝑖 ∈ 𝐼 and for all 𝑔

𝑖∈ 𝐾𝑖,

𝐺−1

𝑖(𝑔𝑖) = {𝑓 ∈ 𝐾 : 𝐹

𝑖(𝑓, 𝑔𝑖) <𝑃𝑖𝛼𝑖𝑒𝑖or 𝐹𝑖(𝑓, 𝑔𝑖) >𝑃𝑖𝛽𝑖𝑒𝑖} .

(15)

We have complement of 𝐺−1𝑖(𝑔𝑖) in𝐾:

[𝐺−1

𝑖(𝑔𝑖)]𝑐

= {𝛽𝑖𝑒𝑖̸<𝑃𝑖𝐹𝑖(𝑓, 𝑔𝑖) ̸<𝑃𝑖𝛼𝑖𝑒𝑖} , (16)

which is compactly closed by virtue of condition (iv). There-fore, for each 𝑖 ∈ 𝐼 and for all 𝑔

𝑖∈ 𝐾𝑖, 𝐺−1

𝑖(𝑔𝑖) is compactly

open in𝐾.For each 𝑖 ∈ 𝐼, define two set-valued mappings 𝑆

𝑖, 𝑇𝑖:

𝐾 → 2𝐾𝑖 by

𝑆𝑖(𝑓) =

{

{

{

𝐺𝑖(𝑓) ∩ 𝐴

𝑖(𝑓) ; if 𝑓 ∈ F

𝑖


𝑖,

𝑇𝑖(𝑓) =

{

{

{

𝐻𝑖(𝑓) ∩ 𝐵

𝑖(𝑓) ; if 𝑓 ∈ F

𝑖

𝐵𝑖(𝑓) ; if 𝑓 ∈ 𝐾 \F

𝑖.

(17)

Thus, for each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾,𝐶𝑜𝐺𝑖(𝑓) ⊆ 𝐻

𝑖(𝑓) and

in view of condition (i), we obtain 𝐶𝑜𝑆𝑖(𝑓) ⊆ 𝑇

𝑖(𝑓). It is easy

to see that

𝑆𝑖

−1(𝑔𝑖) = (𝐴

𝑖

−1(𝑔𝑖) ∩ 𝐺𝑖

−1(𝑔𝑖))⋃((𝐾 \F

𝑖) ∩ 𝐴𝑖

−1(𝑔𝑖))

(18)

for each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾. Thus, for each 𝑖 ∈ 𝐼 and forall𝐺𝑖

−1(𝑔𝑖), 𝐴𝑖

−1(𝑔𝑖) and𝐾\F

𝑖are compactly open in𝐾. We

have 𝑆𝑖

−1(𝑔𝑖) being compactly open in 𝐾. Also 𝑓

𝑖∉ 𝑇𝑖(𝑓) for

all 𝑓 ∈ 𝐾 and for each 𝑖 ∈ 𝐼.Then, byTheorem 5, there exists 𝑓 ∈ 𝐾 such that 𝑆

𝑖(𝑓) =

0 for each 𝑖 ∈ 𝐼. If 𝑓 ∈ 𝐾 \F𝑖, then 𝐴

𝑖(𝑓) = 𝑆

𝑖(𝑓) = 0, which

contradicts the fact that 𝐴𝑖(𝑓) is nonempty for each 𝑖 ∈ 𝐼

and for all 𝑓 ∈ 𝑋. Hence, 𝑓 ∈ F𝑖, for each 𝑖 ∈ 𝐼. Therefore,

𝑓𝑖∈ 𝐵𝑖(𝑓) and 𝐺

𝑖(𝑓) ∩ 𝐴

𝑖(𝑓) = 0, for all 𝑖 ∈ 𝐼. Thus, for each

𝑖 ∈ 𝐼, 𝑓𝑖∈ 𝐵𝑖(𝑓) and 𝛽

𝑖𝑒𝑖̸<𝑃𝑖𝐹𝑖(𝑓, 𝑔𝑖) ̸<𝑃𝑖𝛼𝑖𝑒𝑖for all 𝑔



Now, we establish an existence result for SGQOEP (10)involving 𝜙-condensing maps.

Theorem 11. For each 𝑖 ∈ 𝐼, assume that conditions (i)–(iv)of Theorem 10. hold. Let 𝜙 be a measure of noncompactness on∏𝑖∈𝐼

𝑋𝑖. Further, assume that the set-valuedmapping 𝐵 : 𝐾 →

2𝐾 defined as 𝐾

𝑖is a nonempty, closed, and convex subset of a

locally convex Hausdorff topological vector space 𝑋𝑖and 𝐹

𝑖:

𝐾×𝐾𝑖→ 𝑌𝑖is a bifunction and let the set-valuedmapping𝐴 =

∏𝑖∈𝐼

𝐴𝑖: 𝐾 → 2

𝐾 defined as 𝐵(𝑓) = ∏𝑖∈𝐼

𝐵𝑖(𝑓), ∀𝑓 ∈ 𝐾 be

𝜙-condensing. Then, there exists a solution 𝑓 ∈ 𝐾 of SGQOEP(10).

Proof. In view of Remark 7, it is sufficient to show that theset-valued mapping 𝑇 : 𝐾 → 2𝐾 defined as 𝑇(𝑓) =∏𝑖∈𝐼

𝑇𝑖(𝑓), ∀𝑓 ∈ 𝐾, is 𝜙-condensing, where 𝑇

𝑖s are the same

as defined in the proof of Theorem 10. By the definition of𝑇𝑖, 𝑇𝑖(𝑓) ⊆ 𝐵


𝐴(𝑓), ∀𝑓 ∈ 𝐾. Since𝐵 is𝜙-condensing, by Remark 7, we have𝑇 being also 𝜙-condensing.


Next, we derive the existence result for the solution ofSGQOEPLUB (12).

International Journal of Analysis 5

Corollary 12. For each 𝑖 ∈ 𝐼, let 𝐾𝑖be a nonempty convex

subset of a topological vector space𝑋𝑖and𝐹𝑖, 𝐿𝑖, 𝑄𝑖: 𝐾×𝐾

𝑖→

R are the bifunctions, 𝐵𝑖: 𝐾 → 2

𝐾𝑖 is a set-valued mappingsuch that the set F

𝑖= {𝑓 ∈ 𝐾 : 𝑓

𝑖∈ 𝐵𝑖(𝑓)} is compactly

closed, 𝐴𝑖: 𝐾 → 2

𝐾𝑖 is a set-valued mapping with nonemptyvalues such that, for each 𝑔

𝑖∈ 𝐾𝑖, 𝐴𝑖

−1(𝑔𝑖) is compactly open

in 𝐾, and 𝛼𝑖, 𝛽𝑖∈ R are two real numbers such that 𝛼

𝑖≤ 𝛽𝑖.

For each 𝑖 ∈ 𝐼, assume that the following conditions hold:

(i) for all 𝑓 ∈ 𝐾, 𝐶𝑜𝐴𝑖(𝑓) ⊆ 𝐵

𝑖(𝑓);

(ii) for all 𝑓 ∈ 𝐾, 𝐿𝑖(𝑓, 𝑓𝑖) < 𝛼𝑖or 𝑄𝑖(𝑓, 𝑓𝑖) > 𝛽𝑖;

(iii) for all𝑓 ∈ 𝐾 and for every nonempty finite subset𝑁𝑖⊆

{𝑔𝑖∈ 𝐾𝑖: 𝐹𝑖(𝑓, 𝑔𝑖) < 𝛼𝑖or 𝐹𝑖(𝑓, 𝑔𝑖) > 𝛽𝑖}, we have

𝐶𝑜𝑁𝑖⊆ {𝑔𝑖∈ 𝐾𝑖: 𝐿𝑖(𝑓, 𝑔𝑖) ≥ 𝛼𝑖and 𝑄

𝑖(𝑓, 𝑔𝑖) ≤ 𝛽𝑖} ; (19)

(iv) for all 𝑔𝑖∈ 𝐾𝑖, the set {𝑓 ∈ 𝐾 : 𝛼

𝑖≤ 𝐹𝑖(𝑓, 𝑔𝑖) ≤ 𝛽𝑖} is

compactly closed in 𝐾;(v) there exist a nonempty compact subset 𝐷 of 𝐾 and a

nonempty compact convex subset 𝐸𝑖⊂ 𝐾𝑖, for each 𝑖 ∈

𝐼, such that, for all𝑓 ∈ 𝐾\𝐷, there exists 𝑖 ∈ 𝐼 such that𝑔𝑖∈ 𝐸𝑖satisfying 𝑔

𝑖∈ 𝐴𝑖(𝑓) and either 𝐹

𝑖(𝑓, 𝑔𝑖) < 𝛼𝑖

or 𝐹𝑖(𝑓, 𝑔𝑖) > 𝛽𝑖.

Then the problem SGQOEPLUB (12) has a solution.

Proof. For each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾, define two set-valuedmappings 𝐺

𝑖, 𝐻𝑖: 𝐾 → 2

𝐾𝑖 by

𝐺𝑖(𝑓) = {𝑔

𝑖∈ 𝐾𝑖: 𝐹𝑖(𝑓, 𝑔𝑖) < 𝛼𝑖or 𝐹𝑖(𝑓, 𝑔𝑖) > 𝛽𝑖} ,

𝐻𝑖(𝑓) = {𝑔

𝑖∈ 𝐾𝑖: 𝐿𝑖(𝑓, 𝑔𝑖) ≥ 𝛼𝑖and 𝑄

𝑖(𝑓, 𝑔𝑖) ≤ 𝛽𝑖} .

(20)

Condition (iii) implies that, for each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾,𝐶𝑜𝐺𝑖(𝑓) ⊆ 𝐻

𝑖(𝑓).

From condition (ii), we have 𝑓𝑖∉ 𝐻𝑖(𝑓) for all 𝑓 ∈ 𝐾 and

for each 𝑖 ∈ 𝐼.Thus, for each 𝑖 ∈ 𝐼 and for all 𝑔

𝑖∈ 𝐾𝑖,

𝐺−1

𝑖(𝑔𝑖) = {𝑓 ∈ 𝐾 : 𝐹

𝑖(𝑓, 𝑔𝑖) < 𝛼𝑖or 𝐹𝑖(𝑓, 𝑔𝑖) > 𝛽𝑖} . (21)

We have complement of 𝐺−1𝑖(𝑔𝑖) in𝐾:

[𝐺−1

𝑖(𝑔𝑖)]𝑐

= {𝛼𝑖≤ 𝐹𝑖(𝑓, 𝑔𝑖) ≤ 𝛽𝑖} , (22)

which is compactly closed by virtue of condition (iv). There-fore, for each 𝑖 ∈ 𝐼 and for all 𝑔

𝑖∈ 𝐾𝑖, 𝐺−1

𝑖(𝑔𝑖) is compactly

open in𝐾.For each 𝑖 ∈ 𝐼, define two set-valued mappings 𝑆

𝑖, 𝑇𝑖:

𝐾 → 2𝐾

𝑖by

𝑆𝑖(𝑓) =

{

{

{

𝐺𝑖(𝑓) ∩ 𝐴


𝑖


𝑖,

𝑇𝑖(𝑓) =

{

{

{

𝐻𝑖(𝑓) ∩ 𝐵


𝑖

𝐵𝑖(𝑓) ; if 𝑓 ∈ 𝐾 \F

𝑖.

(23)

Thus, for each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾,𝐶𝑜𝐺𝑖(𝑓) ⊆ 𝐻

𝑖(𝑓) and

in view of condition (i), we obtain 𝐶𝑜𝑆𝑖(𝑓) ⊆ 𝑇

𝑖(𝑓). It is easy

to see that

𝑆𝑖

−1(𝑔𝑖) = (𝐴

𝑖

−1(𝑔𝑖) ∩ 𝐺𝑖

−1(𝑔𝑖))⋃((𝐾 \F

𝑖) ∩ 𝐴𝑖

−1(𝑔𝑖))

(24)

for each 𝑖 ∈ 𝐼 and for all 𝑓 ∈ 𝐾. Thus, for each 𝑖 ∈ 𝐼 and forall𝐺𝑖

−1(𝑔𝑖), 𝐴𝑖

−1(𝑔𝑖) and𝐾\F

𝑖are compactly open in𝐾. We

have 𝑆𝑖

−1(𝑔𝑖) being compactly open in 𝐾. Also 𝑓

𝑖∉ 𝑇𝑖(𝑓) for

all 𝑓 ∈ 𝐾 and for each 𝑖 ∈ 𝐼.Then, byTheorem 5, there exists 𝑓 ∈ 𝐾 such that 𝑆

𝑖(𝑓) =

0 for each 𝑖 ∈ 𝐼. If 𝑓 ∈ 𝐾 \F𝑖, then 𝐴

𝑖(𝑓) = 𝑆

𝑖(𝑓) = 0, which

contradicts the fact that 𝐴𝑖(𝑓) is nonempty for each 𝑖 ∈ 𝐼

and for all 𝑓 ∈ 𝑋. Hence, 𝑓 ∈ F𝑖, for each 𝑖 ∈ 𝐼. Therefore,

𝑓𝑖∈ 𝐵𝑖(𝑓) and 𝐺

𝑖(𝑓) ∩ 𝐴

𝑖(𝑓) = 0, for all 𝑖 ∈ 𝐼. Thus, for each

𝑖 ∈ 𝐼, 𝑓𝑖∈ 𝐵𝑖(𝑓) and 𝛼

𝑖≤ 𝐹𝑖(𝑓, 𝑔𝑖) ≤ 𝛽

𝑖for all 𝑔



Now, we establish an existence result for SGQOEPLUB(12) involving 𝜙-condensing maps.

Theorem 13. For each 𝑖 ∈ 𝐼, assume that conditions (i)–(iv)of Corollary 12 hold. Let 𝜙 be a measure of noncompactnesson ∏𝑖∈𝐼

𝑋𝑖. Further, assume that the set-valued mapping 𝐵 :

𝐾 → 2𝐾 defined as𝐾

𝑖is a nonempty, closed, and convex subset

of a locally convex Hausdorff topological vector space 𝑋𝑖and

𝐹𝑖: 𝐾×𝐾

𝑖→ R is a bifunction and let the set-valuedmapping

𝐴 = ∏𝑖∈𝐼

𝐴𝑖: 𝐾 → 2

𝐾 defined as 𝐵(𝑓) = ∏𝑖∈𝐼

𝐵𝑖(𝑓), ∀𝑓 ∈

𝐾, be 𝜙-condensing. Then, there exists a solution 𝑓 ∈ 𝐾 ofSGQOEPLUB (12).

Proof. In view of Remark 7, it is sufficient to show that theset-valued mapping 𝑇 : 𝐾 → 2𝐾 defined as 𝑇(𝑓) =∏𝑖∈𝐼

𝑇𝑖(𝑓), ∀𝑓 ∈ 𝐾, is 𝜙-condensing, where 𝑇𝑖𝑠 are the same

as defined in the proof of Theorem 10. By the definition of𝑇𝑖, 𝑇𝑖(𝑓) ⊆ 𝐵


𝐴(𝑓), ∀𝑓 ∈ 𝐾. Since𝐵 is𝜙-condensing, by Remark 7, we have𝑇 being also 𝜙-condensing.


Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

References

[1] A. Domokos and J. Kolumbán, “Variational inequalities withoperator solutions,” Journal of Global Optimization, vol. 23, no.1, pp. 99–110, 2002.

[2] K. R. Kazmi and A. Raouf, “A class of operator equilibriumproblems,” Journal of Mathematical Analysis and Applications,vol. 308, no. 2, pp. 554–564, 2005.

[3] Q. H. Ansari, W. K. Chan, and X. Q. Yang, “The system ofvector quasi-equilibrium problems with applications,” Journalof Global Optimization, vol. 29, no. 1, pp. 45–57, 2004.


[4] Q. H. Ansari, S. Schaible, and J. C. Yao, “System of vector equi-librium problems and its applications,” Journal of OptimizationTheory and Applications, vol. 107, no. 3, pp. 547–557, 2000.

[5] P. Deguire, K. K. Tan, and G. X.-Z. Yuan, “The study of maximalelements, fixed points for 𝐿

𝑆-majorized mappings and their

applications to minimax and variational inequalities in producttopological spaces,” Nonlinear Analysis: Theory, Methods &Applications, vol. 37, no. 7, pp. 933–951, 1999.

[6] S. Al-Homidan andQ.H.Ansari, “Systems of quasi-equilibriumproblems with lower and upper bounds,” Applied MathematicsLetters, vol. 20, no. 3, pp. 323–328, 2007.

[7] P. M. Fitzpatrick and W. V. Petryshyn, “Fixed point theoremsfor multivalued noncompact acyclic mappings,” Pacific Journalof Mathematics, vol. 54, no. 2, pp. 17–23, 1974.

[8] L.-J. Lin andQ. H. Ansari, “Collective fixed points andmaximalelements with applications to abstract economies,” Journal ofMathematical Analysis andApplications, vol. 296, no. 2, pp. 455–472, 2004.

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