# Applications of an HIDS Theorem to the Existence of Fixed Point, Abstract Equilibria and Optimization Problems

Authors: Wei-Shih Du
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

By applying hybrid inclusion and disclusion systems (HIDS), we establish several vectorial variants
of system of Ekeland's variational principle on topological vector spaces, some existence theorems of system of parametric vectorial quasi-equilibrium
problem, and an existence theorem of system of the Stampacchia-type vectorial equilibrium problem. As an application, a vectorial minimization theorem is
also given. Moreover, we discuss some equivalence relations between our vectorial variant of Ekeland's variational principle, common fixed point
theorem, and maximal element theorem.


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## Body

## 1. Introduction

LetX be a nonempty subset of a topological space (t.s., for short), and let f:X×X→ℝ be a function with f(x,x)≥0 for all x∈X. Then the scalar equilibrium problem in the sense of Blum and Oettli [1] is to find x̅∈X such that f(x̅,y)≥0 for all y∈X. The equilibrium problem was extensively investigated and generalized to the vectorial equilibrium problems for single-valued or multivalued maps and contains optimization problems, variational inequalities problems, saddle point problems, the Nash equilibrium problems, fixed point problems, complementary problems, bilevel problems, and semi-infinite problems as special cases and have some applications in mathematical program with equilibrium constraint; for detail one can refer to [1–4] and references therein.The famous Ekeland's variational principle (EVP, for short) [5–7] is a forceful tool in various fields of applied mathematical analysis and nonlinear analysis. A number of generalizations in various different directions of these results for functions defined in metric (or quasimetric) spaces and more general in topological vector spaces have been investigated by several authors in the past; see [8–23] and references therein. It is wellknown that the original EVP is equivalent to Caristi's fixed point theorem, to Takahashi's nonconvex minimization theorem; and to the flower petal theorem for detail, see [14, 16–18, 20] and references therein. EVPs were extended to the vector case by using scalarization method and were applied to the study of efficiency (or approximative efficiency) and others; see, for example, [3, 8, 9, 12, 13, 15, 23].Recently, the author first studied the following mathematical model about hybrid inclusion and disclusion systems (HIDS, for short) [11]. Let I be any index set. For each i∈I, let Yi be a nonempty closed convex subset of a Hausdorff topological vector space (t.v.s., for short) Vi, Hi⊆Yi, Y=∏i∈IYi, Ai:Y⊸Yi, and let Ti:Y⊸Yi be multivalued maps. Hybrid inclusion and disclusion systems (HIDS) are defined as follows:(1.1)(HIDS)Findv=(v)i∈I∈Ysuchthatvi∈Hi,yi∉Ai(v),∀yi∈Ti(v),∀i∈I.In fact, HIDS contains several important problems as special cases. LetX be a nonempty subset of a topological space E, and let u∈X be given. For each i∈I, let Ui and Zi be real t.v.s. with zero vector θUi and θZi, respectively.Example 1.1.
For eachi∈I, let Fi:X×Yi→ℝ and Gi:Y×Yi→ℝ be functions. If Hi and Ai are defined as follows:
(1.2)Hi={yi∈Yi:Fi(u,yi)≤0},Ai(y)={zi∈Yi:Gi(y,zi)≤0},
then HIDS will reduce to the following system of hybrid scalar equilibrium problem (P1):
(P1)
Findv=(v)i∈I∈Y such that Fi(u,vi)≤0 and Gi(v,yi)>0 for all yi∈Ti(v) and for all i∈I.Example 1.2.
For eachi∈I, let Fi:X×Yi⊸Ui and Gi:Y×Yi⊸Zi be multivalued maps with nonempty values, and let Ci and Di be nonempty subsets of Ui and Zi, respectively. If Hi and Ai are defined as follows:
(1.3)Hi={yi∈Yi:Fi(u,yi)∩(-Ci∖{θUi})=∅},Ai(y)={zi∈Yi:Gi(y,zi)∩(-Di∖{θZi})=∅},
then HIDS will reduce to the following problem (P2), which is an abstract equilibrium problem:
(P2)
Findv=(v)i∈I∈Y such that Fi(u,vi)∩(-Ci∖{θUi})=∅ and Gi(v,yi)∩(-Di∖{θZi})≠∅ for all yi∈Ti(v) and for all i∈I.Example 1.3.
For eachi∈I, let Fi:Yi⊸Ui and Gi:Y×Yi⊸Zi be multivalued maps. If Hi and Ai are defined as follows:
(1.4)Hi={yi∈Yi:yi∈Fi(yi)},Ai(y)={zi∈Yi:y∉Gi(y,zi)},
then HIDS will reduce to the following fixed point problem (P3):
(P3)
Findv=(v)i∈I∈Y such that vi∈Fi(vi) and v∈Gi(v,yi) for all yi∈Ti(v) for all i∈I.Example 1.4.
For eachi∈I, let Fi:X×Yi⊸Ui and Gi:Y×Yi⊸Zi be multivalued maps with nonempty values. If Hi and Ai are defined as follows:
(1.5)Hi={yi∈Yi:θUi∉Fi(u,yi)},Ai(y)={zi∈Yi:θZi∉Gi(y,zi)},
then HIDS will reduce to the following system of mixed type of parametric variational inclusion and disclusion problem (P4):
(P4)
Findv=(v)i∈I∈Y such that θUi∉Fi(u,vi) and θZi∈Gi(v,yi) for all yi∈Ti(v) and for all i∈I.In this paper, we study the existence theorems of system of parametric vectorial quasi-equilibrium problems, vectorial variants of system of Ekeland's variational principle (VSEVP, for short), and the existence theorems of system of the Stampacchia-type vectorial equilibrium problem by using an HIDS theorem established by the author [11]. Our results improve and generalize some theorems in [19] to the vector case. Till now, to my knowledge, there are extremely few results about Stampacchia-type vectorial equilibrium problem in the literature. The existence of  VSEVP and the Stampacchia-type vectorial equilibrium problem are established by applying the HIDS theorem without the use of any scalarization method. So our results are completely different from [3, 8, 9, 12, 13, 15, 23]. As an application, a vectorial minimization theorem is also proved. Moreover, we prove some equivalence relations between our VSEVP, common fixed point theorem, and maximal element theorem.


## 2. Preliminaries

LetA and B be nonempty sets. A multivalued map T:A⊸B is a function from A to the power set 2B of B. We denote T(A)=⋃{T(x):x∈A} and let T-:B⊸A be defined by the condition that x∈T-(y) if and only if y∈T(x). We recall that a point x∈A is a maximal element of T:A→2B if T(x)=∅. Let X be a linear space with zero vector θ. A nonempty subset C of X is called a convex cone if C+C⊆C and λC⊆C for all λ≥0. A cone C in X is pointed if C∩(-C)={θ}. Let Z be a real t.v.s., and let D be a proper convex cone in Z, and A⊆Z. A point y̅∈A is called a vectorial minimal point of A with respect to D if for any y∈A, y-y̅∉-D∖{θ}. The set of vectorial minimal point of A is denoted by MinDA. The convex hull of A and the closure of A are denoted by coA and clA, respectively.Definition 2.1.
LetX and Y be linear spaces, and let C be a proper convex cone in Y. A map f:X⊸Y is called C-convex if for any x1,x2∈X and λ∈[0,1], one has
(2.1)λf(x1)+(1-λ)f(x2)-f(λx1+(1-λ)x2)⊆C.
Clearly, iff1 and f2 are C-convex and α≥0, then αf1 and f1+f2 are C-convex.Definition 2.2.
LetX be a nonempty convex subset of a vector space E, let Y be a nonempty convex subset of a vector space V, and let Z be a real t.v.s. Let F:X×Y⊸Z and C:X⊸Z be multivalued maps such that for each x∈X, C(x) is a nonempty closed convex cone. For each fixed x∈X, y⊸F(x,y) is called C(x)-quasiconvex if for any y1, y2∈Y and λ∈[0,1], one has either
(2.2)F(x,y1)⊆F(x,λy1+(1-λ)y2)+C(x),
or
(2.3)F(x,y2)⊆F(x,λy1+(1-λ)y2)+C(x).
Now, we define the concept of vectorial upper and lower semicontinuous on t.v.s.Definition 2.3.
LetX be a t.s., let Y be a t.v.s. with zero vector θ, and let C≠{θ} be a pointed convex cone in Y. A map f:X→Y is said to be (i)
vectorial lower semicontinuous with respect toC (C-v.l.s.c., for short) at x0∈X if for any a∈C∖{θ}, there exists an open neighborhood N(x0) of x0 such that f(y)-f(x0)+a∈C∖{θ} for all y∈N(x0),(ii)
vectorial upper semicontinuous with respect toC (C-v.u.s.c., for short) at x0∈X if for any a∈C∖{θ}, there exists an open neighborhood N(x0) of x0 such that f(x0)-f(y)+a∈C∖{θ} for all y∈N(x0).
The functionf is called   C-v.l.s.c. (resp., C-v.u.s.c.) on X if f is C-v.l.s.c. (resp., C-v.u.s.c.) at every point of X.Proposition 2.4.
LetX be a t.s., let Y be a t.v.s with zero vector θ, and let C≠{θ} be a pointed convex cone in Y. Let f, g:X→Y be maps and γ≥0. Then the following hold: (a)
f is C-v.u.s.c. (resp., C-v.l.s.c.) on X⇔-f is C-v.l.s.c. (resp., C-v.u.s.c.) on X;(b)
iff and g are C-v.u.s.c. (resp., C-v.l.s.c.) on X, then γf and f+g are C-v.u.s.c. (resp., C-v.l.s.c.) on X;(c)
iff is C-v.u.s.c. on X, then
(i)
{x∈X:λ-f(x)∈C∖{θ}} is open in X for all λ∈Y,(ii)
{x∈X:λ-f(x)∉C∖{θ}} is closed in X for all λ∈Y;(d)
iff is C-v.l.s.c. on X, then
(iii)
{x∈X:f(x)-λ∈C∖{θ}} is open in X for all λ∈Y,(iv)
{x∈X:f(x)-λ∉C∖{θ}} is closed in X for all λ∈Y.Proof.
Clearly, (a) and (b) hold from definition. To prove (c), it suffices to show (i). Suppose thatf is C-v.u.s.c. on X. Let λ∈Y and x0∈{x∈X:λ-f(x)∈C∖{θ}}. Then α:=λ-f(x0)∈C∖{θ}. Since f is C-v.u.s.c. at x0, there exists an open neighborhood N(x0) of x0 such that
(2.4)λ-f(y)=f(x0)-f(y)+α∈C∖{θ}
for all y∈N(x0). Hence {x∈X:λ-f(x)∈C∖{θ}} is an open set in X and (i) is proved. Obviously, (ii) is immediate from (i). It is easy to see that conclusion (d) follows from (a) and (c).Remark 2.5.
LetX be a t.v.s. and α∈X with α≠0X, where 0X is the origin of X. Let f:X→Lα={γα:γ∈ℝ} be a map. Hence in conclusion (c) (resp., (d)) of Proposition 2.4, we have
(2.5)fisC-v.u.s.c. onX⟺(i)⟺(ii)(resp.,fisC-v.l.s.c.on⁡X⟺(iii)⟺(iv)).
In particular, if X=ℝ (the set of real numbers) and C=[0,∞), then the C-v.l.s.c. (resp. C-v.u.s.c.) function f:X→ℝ is l.s.c. (resp. u.s.c.) in usual.
The concept ofC-vectorial ℓ-function and C-vectorial quasi-distance on topological spaces are introduced as follows.Definition 2.6.
LetX and E be t.v.s., let θ be the zero vector of E, and let C≠{θ}, a pointed convex cone in E. A map p:X×X→E is called (a)
aC-vectorialℓ-function (ℓC-function, for short) if the following are satisfied:
(VL1)
p(x,x)∈C for all x∈X;(VL2)
for anyx∈X, p(x,·) is C-convex;(VL3)
for anyy∈X, p(·,y) is C-v.u.s.c.(b)
aC-vectorial  quasi-distance if the following are satisfied:
(VQD1)
p(x,x)∈C for all x∈X;(VQD2)
p(x,y)+p(y,z)∈p(x,z)+C for any x,y,z∈X;(VQD3)
for anyx∈X, p(x,·) is C-convex and C-v.l.s.c.;(VQD4)
for anyy∈X, p(·,y) is C-v.u.s.c.
IfE=(-∞,∞] and letting C=[0,∞) be in (a) and (b), then the function p:X×X→(-∞,∞] is called a ℓ-function and quasi-distance, respectively, introduced by Lin and Du [19]. For examples and results of ℓ-function and quasi-distance, one can see [19].Remark 2.7.
(a) Obviously, aC-vectorial quasi-distance is a ℓC-function, but the reverse is not true;
(b)  ifp1 and p2 are C-vectorial quasi-distances (resp., ℓC-functions) and α≥0, then αp1 and p1+p2 are C-vectorial quasi-distances (resp., ℓC-functions);
(c)  iff:X→E is a C-v.l.s.c. and C-convex function, then the function p:X×X→E defined by p(x,y)=f(y)-f(x) is a C-vectorial quasi-distance.Lemma 2.8 . (see [24, 25]).
LetX and Y be the Hausdorff topological spaces, and let T:X⊸Y be a multivalued map. Then T is l.s.c. at x∈X if and only if for any y∈T(x) and for any net {xα} in X converging to x, there exists a subnet {xϕ(λ)}λ∈Λ of {xα} and a net {yλ}λ∈Λ with yλ→y such that yλ∈T(xϕ(λ)) for all λ∈Λ.


## 3. The Existence of System of VSEVP and Abstract Equilibrium Problems

The following existence theorem for the solution of HIDS was established in [11].Theorem 3.1 (HIDS theorem [11]).
LetI be any index set. For each i∈I, let Yi be a nonempty closed convex subset of a Hausdorff t.v.s. Vi. Let X be a nonempty subset of a topological space E, Y=∏i∈IYi and u∈X. For each i∈I, let Hi be a nonempty closed subset of Yi, let Ai:Y⊸Yi be a multivalued map, and let Ti:Y⊸Yi be a multivalued map with nonempty values. For each i∈I, suppose that the following conditions are satisfied:(i)
for eachy=(yi)i∈I∈Y,yi∉Ai(y);(ii)
for eachy∈Y, coTi(y)⊆Hi and Ai(y) is convex;(iii)
for eachzi∈Yi, Ti-(zi) and Ai-(zi) are open in Y;(iv)
there exist a nonempty compact subsetK of Y and a nonempty compact convex subset Mi of Yi for each i∈I such that for each y∈Y∖K there exist j∈I and zj∈Mj∩Tj(y)∩Aj(y).
Then there existsv=(vi)i∈I∈Y such that for each i∈I, vi∈Hi and yi∉Ai(v) for all yi∈Ti(v).Example 3.2.
LetX and Y be Hausdorff t.v.s., let U be a real t.v.s. with its zero vector θ, and u∈X.
(a) LetF:X×Y⊸U be a multivalued map with nonempty values such that there exists w=w(u)∈Y such that θ∈F(u,w) and the map y⊸F(u,y) is closed. Then it is easy to see that H={y∈Y:θ∈F(u,y)} is a nonempty closed subset of Y.
(b) LetG:Y×Y⊸U be a multivalued map with nonempty values and W be a nonempty open set in U. Suppose that(i)
for eachy∈Y,θ∉G(y,y)+W,(ii)
for eachy∈Y,G(y,·) is {θ}-quasiconvex and for each z∈Y, G(·,z) is l.s.c.
LetA:Y⊸Y be defined by
(3.1)A(y)={z∈Y:θ∈G(y,z)+W}.
Theny∉A(y) for each y∈Y. We claim that A-(z) is open in Y for each z∈Y. Indeed, let z∈Y be given, and let y∈cl(Y∖A-(z)). Then there exists a net {yα}α∈Λ in Y∖A-(z) such that yα→y. Thus we have θ∉G(yα,z)+W or G(yα,z)⊆U∖W. Clearly, y∈Y. For any w∈G(y,z), since G(·,z) is l.s.c. at y and yα→y, by Lemma 2.8, there exists a net {wα} with wα→w such that wα∈G(yα,z)⊆U∖W. Since U∖W is closed, w∈U∖W. So G(y,z)⊆U∖W. It implies cl(Y∖A-(z))=Y∖A-(z), and hence A-(z) is open in Y. Next, we show that for each y∈Y, A(y) is convex. Let a, b∈A(y). Then θ∈G(y,a)+W and θ∈G(y,b)+W. For any λ∈[0,1], let eλ:=λa+(1-λ)b∈Y. Suppose to the contrary that there exists λ0∈(0,1) such that θ∉G(y,eλ0)+W. By the {θ}-quasiconvexity of G(y,·), either
(3.2)θ∈G(y,a)+W⊆G(y,eλ0)+W
or
(3.3)θ∈G(y,b)+W⊆G(y,eλ0)+W,
which leads to a contradiction. Hence for each y∈Y, A(y) is convex.
Applying Theorem3.1, we establish the following existence theorem of system of parametric vectorial quasi-equilibrium problem.Theorem 3.3.
LetI be any index set. For each i∈I, let Xi be a nonempty Hausdorff t.v.s., and let Ci≠{θ} be a pointed convex cone in a t.v.s. E with zero vector θ. Let X=∏i∈IXi. For each i∈I, let pi, qi:Xi×Xi→E be Ci-vectorial quasi-distances and let Ti:X⊸Xi be a multivalued map with nonempty values. Let u=(ui)i∈I∈X with pi(ui,ui)=qi(ui,ui)=θ for all i∈I. For each i∈I, suppose that the following conditions are satisfied:(i)
for eachy∈X, coTi(y)⊆{xi∈Xi:qi(ui,xi)∉Ci∖{θ}} and Ti-(zi) is open for all zi∈Xi;(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y) such that pj(yj,zj)∈-Cj∖{θ}.
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has (a)
qi(ui,vi)∉Ci∖{θ},(b)
pi(vi,xi)∉-Ci∖{θ} for all xi∈Ti(v).Proof.
For eachi∈I, define Fi:X×Xi⊸E and Gi:X×X×Xi⊸E by
(3.4)Fi(x,yi)=-qi(xi,yi)+Ci∖{θ},∀(x,yi)∈X×Xi,(3.5)G(t,x,yi)=pi(xi,yi)+Ci∖{θ},∀(t,x,yi)∈X×X×Xi,
respectively. Thus, for each i∈I, we let
(3.6)Hi:={xi∈Xi:θ∉Fi(u,xi)}={xi∈Xi:qi(ui,xi)∉Ci∖{θ}},
and let Ai:X⊸Xi be defined by
(3.7)Ai(x)={yi∈Xi:θ∈Gi(u,x,yi)}={yi∈Xi:pi(xi,yi)∈-Ci∖{θ}},∀x=(xi)i∈I∈X.
Clearly, for each i∈I, θ∉Gi(u,x,xi) for all x=(xi)i∈I∈X, and hence xi∉Ai(x) for all x=(xi)i∈I∈X. By the Ci-vectorial lower semicontinuity of qi(ui,·), Hi is a nonempty closed subset of Xi. By (i), for each x∈X, coTi(x)⊆Hi. For each yi∈Xi, by the vectorial upper semicontinuity of pi(·,yi), the set {xi∈Xi:pi(xi,yi)∈-Ci∖{θ}} is open in Xi, and hence Ai-(yi)={xi∈Xi:qi(xi,yi)∈-Ci∖{θ}}×∏j≠iXj is open in X. For each x∈X, Ai(x) is convex from the Ci-convexity of pi(xi,·) for all i∈I. By (ii), there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y)∩Aj(y). Therefore all the conditions of Theorem 3.1 are satisfied. Applying Theorem 3.1, there exists v=(vi)i∈I∈X such that: (1)
qi(ui,vi)∉Ci∖{θ},(2)
pi(vi,xi)∉-Ci∖{θ} for all xi∈Ti(v).Theorem 3.4.
LetI, Xi, X, Ci, E, θ, Ti, and pi be the same as in Theorem 3.3. Let u=(ui)i∈I∈X with pi(ui,ui)=θ for all i∈I. For each i∈I, suppose that the following conditions are satisfied:(i)
for eachy∈X, coTi(y)⊆{xi∈Xi:pi(ui,xi)∉Ci∖{θ}} and Ti-(zi) is open for all zi∈Xi;(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y) such that pj(yj,zj)∈-Cj∖{θ}.
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has (a)
pi(ui,vi)∉Ci∖{θ},(b)
pi(vi,xi)∉-Ci∖{θ} for all xi∈Ti(v)∪{zi∈Xi:pi(ui,zi)∈Ci∖{θ}}.Proof.
For eachi∈I, let qi:Xi×Xi→E be defined by qi=pi. By Theorem 3.3, there exists v=(vi)i∈I∈X such that for each i∈I, we have (1)
pi(ui,vi)∉Ci∖{θ},(2)
pi(vi,xi)∉-Ci∖{θ} for all xi∈Ti(v).
We claim that for eachi∈I, pi(vi,xi)∉-Ci∖{θ} for all xi∈{zi∈Xi:pi(ui,zi)∈Ci∖{θ}}. For each xi∈{zi∈Xi:pi(ui,zi)∈Ci∖{θ}}, we have pi(ui,xi)∈Ci∖{θ}. If pi(vi,xi)∈-Ci∖{θ}, then, by (VQD2) (in Definition 2.6(b)), we obtain
(3.8)pi(ui,vi)∈pi(ui,xi)-pi(vi,xi)+Ci⊆Ci∖{θ},
which contradict with (1). Hence pi(vi,xi)∉-Ci∖{θ} for all xi∈Ti(v)∪{zi∈Xi:pi(ui,zi)∈Ci∖{θ}}, and the proof is completed.Remark 3.5.
(i) LetX be a nonempty Hausdorff t.v.s., and let C≠{θ} be a nonempty pointed convex cone in a t.v.s. E with zero vector θ. If a map p:X×X→E satisfies that for each x∈X, p(x,·) is C-quasiconvex, then for each u∈X with p(u,u)=θ, Wu:={x∈X:p(u,x)∉C∖{θ}} is a nonempty convex subset in X.
(ii)  In Theorem3.3, if for each i∈I, Wi:={xi∈Xi:pi(ui,xi)∉Ci∖{θ}} is convex and Ti(x)=Wi for all x∈X, then conclusion (b) can be replaced with conclusion (b)′, where
(b)′
pi(vi,xi)∉-Ci∖{θ}, for allxi∈Xi.
Since the sum of twoCi-vectorial quasi-distances is also a Ci-vectorial quasi-distance, the following results related with system of VSEVP for vectorial quasi-distances in a Hausdorff t.v.s. are immediate from Theorem 3.3.Theorem 3.6.
LetI, Xi, X, Ci, E, θ, Ti, and pi be the same as in Theorem 3.3. Let u=(ui)i∈I∈X with pi(ui,ui)=θ for all i∈I. For each i∈I, let fi:Xi→E be a Ci-v.l.s.c. and Ci-convex function and suppose that the following conditions are satisfied:(i)
for eachy∈X, coTi(y)⊆{xi∈Xi:fi(xi)-fi(ui)∉Ci∖{θ}} and Ti-(zi) is open for all zi∈Xi;(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y) such that pj(yj,zj)+fj(zj)-fj(yj)∈-Cj∖{θ}.
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has (a)
fi(vi)-fi(ui)∉Ci∖{θ},(b)
pi(vi,xi)+fi(xi)-fi(vi)∉-Ci∖{θ} for all xi∈Ti(v).In Theorem3.6, if E=(-∞,∞] and Ci=[0,∞) for all i∈I, then we have the following result.Corollary 3.7.
LetI, Xi, X, and Ti be the same as in Theorem 3.3. For each i∈I, let fi:Xi→(-∞,∞] be a l.s.c. and convex function and let pi:Xi×Xi→(-∞,∞] be a quasi-distance. Let u=(ui)i∈I∈X with pi(ui,ui)=0 for all i∈I. Suppose that the following conditions are satisfied: (i)
for eachy∈X, coTi(y)⊆{xi∈Xi:fi(xi)≤fi(ui)} and Ti-(zi) is open for all zi∈Xi;(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y) such that pj(yj,zj)<fj(yj)-fj(zj).
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has (a)
fi(vi)≤fi(ui),(b)
pi(vi,xi)≥fi(vi)-fi(xi) for all xi∈Ti(v).Theorem 3.8.
LetI, Xi, X, Ci, E, θ, Ti, and pi be the same as in Theorem 3.3. Let u=(ui)i∈I∈X with pi(ui,ui)=θ for all i∈I. For each i∈I, let fi:Xi→E be a Ci-v.l.s.c. and Ci-convex function and suppose that the following conditions are satisfied: (i)
for eachy∈X, coTi(y)⊆{xi∈Xi:pi(ui,xi)+fi(xi)-fi(ui)∉Ci∖{θ}} and Ti-(zi) is open for all zi∈Xi;(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y) such that pj(yj,zj)+fj(zj)-fj(yj)∈-Cj∖{θ}.
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has (a)
pi(ui,vi)+fi(vi)-fi(ui)∉Ci∖{θ},(b)
pi(vi,xi)+fi(xi)-fi(vi)∉-Ci∖{θ} for all xi∈Ti(v)∪{zi∈Xi:pi(ui,zi)+fi(zi)-fi(ui)∈Ci∖{θ}}.Corollary 3.9.
LetI, Xi, X, and Ti be the same as in Theorem 3.3. For each i∈I, let fi:Xi→(-∞,∞] be a l.s.c. and convex function and pi:Xi×Xi→(-∞,∞] be a quasi-distance. Let u=(ui)i∈I∈X with pi(ui,ui)=0 for all i∈I. Suppose that the following conditions are satisfied: (i)
for eachy∈X, coTi(y)⊆{xi∈Xi:pi(ui,xi)≤fi(ui)-fi(xi)} and Ti-(zi) is open for all zi∈Xi;(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y) such that pj(yj,zj)<fj(yj)-fj(zj).
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has (a)
pi(ui,vi)≤fi(ui)-fi(vi),(b)
pi(vi,xi)≥fi(vi)-fi(xi) for all xi∈Ti(v)∪{zi∈Xi:pi(ui,zi)>fi(ui)-fi(zi)}.By using Theorem3.1 again, we have the following result.Theorem 3.10.
LetI, Xi, X, Ci, E, θ, u, pi, and qi be the same as in Theorem 3.3. Suppose that there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that qj(uj,zj)∈-Cj and pj(yj,zj)∈-Cj∖{θ}. Then there exists v=(vi)i∈I∈X such that for each i∈I, one has (a)
qi(ui,vi)∉Ci∖{θ},(b)
pi(vi,xi)∉-Ci∖{θ} for all xi∈{zi∈Xi:qi(ui,zi)∈-Ci}.Proof.
For eachi∈I, let Hi and multivalued maps Fi, Gi, and Ai be defined as in the proof of Theorem 3.3. For each i∈I, define Ti:X⊸Xi by
(3.9)Ti(x)=Li:={zi∈Xi:qi(ui,zi)∈-Ci},∀x∈X⟺Ti-(zi)={Xifzi∈Li,∅ifzi∈Xi∖Li.
Clearly, for each i∈I, Ti-(zi) is open in X for all zi∈Xi. By the Ci-convexity of qi(ui,·), Li is a nonempty convex subset of Xi for all i∈I. Since Li is convex in Xi and Li⊆Hi, we have coTi(x)⊆Hi for all i∈I. By our hypothesis, there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj∩Tj(y)∩Aj(y). Thus all the conditions of Theorem 3.1 are satisfied, and the conclusion follows from Theorem 3.1.Remark 3.11.
In Theorem3.10, the multivalued map Ti and the condition “for each y∈Y, coTi(y)⊆{xi∈Xi:qi(ui,xi)∉Ci∖{θ}}” are not assumed. So Theorems 3.10 and 3.3 are different.Theorem 3.12.
LetI, Xi, X, Ci, E, θ, and pi be the same as in Theorem 3.3. Let u=(ui)i∈I∈X with pi(ui,ui)=θ for all i∈I. Suppose that there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(uj,zj)∈-Cj and pj(yj,zj)∈-Cj∖{θ}. Then there exists v=(vi)i∈I∈X such that for each i∈I, one has (a)
pi(ui,vi)∉Ci∖{θ},(b)
pi(vi,xi)∉-Ci∖{θ} for all xi∈{zi∈Xi:pi(ui,zi)∈-Ci}∪{zi∈Xi:pi(ui,zi)∈Ci∖{θ}}.Proof.
For eachi∈I, let qi=pi. Applying Theorem 3.10 and following the same argument as in the proof of Theorem 3.4, one can prove the theorem.Theorem 3.13.
LetI, Xi, X, Ci, E, θ, and pi be the same as in Theorem 3.3. Let u=(ui)i∈I∈X with pi(ui,ui)=θ for all i∈I. For each i∈I, let fi:Xi→E be a Ci-v.l.s.c. and Ci-convex function. Suppose that there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that fj(zj)-fj(uj)∈-Cj and pj(yj,zj)+fj(zj)-fj(yj)∈-Cj∖{θ}. Then there exists v=(vi)i∈I∈X such that for each i∈I, one has (a)
fi(vi)-fi(ui)∉Ci∖{θ},(b)
pi(vi,xi)+fi(xi)-fi(vi)∉-Ci∖{θ} for all xi∈{zi∈Xi:fi(zi)-fi(ui)∈-Ci}.Corollary 3.14.
LetI, Xi, and X be the same as in Theorem 3.3. For each i∈I, let fi:Xi→(-∞,∞] be a l.s.c. and convex function and pi:Xi×Xi→(-∞,∞] be a quasi-distance. Let u=(ui)i∈I∈X with pi(ui,ui)=0 for all i∈I. Suppose that there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that fj(zj)≤fj(uj) and pj(yj,zj)<fj(yj)-fj(zj). Then there exists v=(vi)i∈I∈X such that for each i∈I, one has (a)
fi(vi)≤fi(ui),(b)
pi(vi,xi)≥fi(vi)-fi(xi) for all xi∈{zi∈Xi:fi(zi)≤fi(ui)}.Remark 3.15.
[19, Theorem  4.3] is a special case of Corollary 3.14.Theorem 3.16.
LetI, Xi, X, Ci, E, θ, and pi be the same as in Theorem 3.3. Let u=(ui)i∈I∈X with pi(ui,ui)=θ for all i∈I. For each i∈I, let fi:Xi→E be a Ci-v.l.s.c. and Ci-convex function, and suppose that there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(uj,zj)+fj(zj)-fj(uj)∈-Cj and pj(yj,zj)+fj(zj)-fj(yj)∈-Cj∖{θ}. Then for each u=(ui)i∈I∈X with pi(ui,ui)=θ for all i∈I, there exists v=(vi)i∈I∈X such that for each i∈I, one has (a)
pi(ui,vi)+fi(vi)-fi(ui)∉Ci∖{θ},(b)
pi(vi,xi)+fi(xi)-fi(vi)∉-Ci∖{θ} for all xi∈{zi∈Xi:pi(ui,zi)+fi(zi)-fi(ui)∈-Ci}∪{zi∈Xi:pi(ui,zi)+fi(zi)-fi(ui)∈Ci∖{θ}}.
In Theorem3.16, if E=(-∞,∞] and Ci=[0,∞) for all i∈I, then we have the following system of Lin and Du's variant of system of Ekeland's variational principle in t.v.s.Corollary 3.17.
LetI, Xi, and X be the same as in Theorem 3.3. For each i∈I, let fi:Xi→E be a l.s.c. and convex function and let pi:Xi×Xi→(-∞,∞] be a quasi-distance. Let u=(ui)i∈I∈X with pi(ui,ui)=0 for all i∈I. Suppose that there exist a nonempty compact subset K of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(uj,zj)≤fj(uj)-fj(zj) and pj(yj,zj)<fj(yj)-fj(zj). Then for each u=(ui)i∈I∈X with pi(ui,ui)=0 for all i∈I, there exists v=(vi)i∈I∈X such that for each i∈I, one has (a)
pi(ui,vi)≤fi(ui)-fi(vi),(b)
pi(vi,xi)≥fi(vi)-fi(xi) for all xi∈Xi.Remark 3.18.
Corollary3.17 generalizes [19, Theorem  4.1].


## 4. A Vectorial Minimization Theorem and Equivalent Formulations of VSEVP

Using Theorem3.1 again, we also obtain an existence theorem of system of generalized vectorial equilibrium problem of the Stampacchia-type which can be regarded as a weak form of VSEVP for C-vectorial ℓ-function in a Hausdorff t.v.s.Theorem 4.1.
LetI be any index set. For each i∈I, let Xi be a nonempty Hausdorff t.v.s. and Ci≠{θ} be a nonempty pointed convex cone in a t.v.s. E with zero vector θ. Let X=∏i∈IXi. For each i∈I, let pi, qi:Xi×Xi→E be ℓCi-function. Suppose that (i)
ℒ={x=(xi)i∈I∈X:pi(xi,xi)=qi(xi,xi)=θforalli∈I}≠∅,(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(yj,zj)+qj(yj,zj)∈-Ci∖{θ}.
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has pi(vi,xi)+qi(vi,xi)∉-Ci∖{θ} for all xi∈Xi.Proof.
Letu=(ui)i∈I∈ℒ be given. For each i∈I, we define Fi:X×Xi⊸E, Gi:X×X×Xi⊸E and Ai:X⊸Xi by
(4.1)Fi(x,yi)=Ci∖{θ},∀(x,yi)∈X×Xi,G(t,x,yi)=pi(xi,yi)+qi(xi,yi)+Ci∖{θ},∀(t,x,yi)∈X×X×Xi,(4.2)Ai(x)={yi∈Xi:θ∈Gi(u,x,yi)},∀x=(xi)i∈I∈X,
respectively. Clearly,
(4.3)Hi:={xi∈Xi:θ∉Fi(u,xi)}=X,∀i∈I.
Using the same argument in the proof of Theorem 3.3, one can verify that all the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, there exists v=(vi)i∈I∈X such that for each i∈I, we have pi(vi,xi)+qi(vi,xi)∉-Ci∖{θ} for all xi∈Xi.An existence theorem of system of generalized vector equilibrium problem is immediate from Theorem4.1 if qi≡0 (the zero map) for all i∈I.Theorem 4.2.
LetI be any index set. For each i∈I, let Xi be a nonempty Hausdorff t.v.s., and let Ci≠{θ} be a nonempty pointed convex cone in a t.v.s. E with zero vector θ. Let X=∏i∈IXi. For each i∈I, let pi:Xi×Xi→E be a ℓCi-function. Suppose that (i)
ℋ={x=(xi)i∈I∈X:pi(xi,xi)=θfor alli∈I}≠∅;(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(yj,zj)∈-Ci∖{θ}.
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has pi(vi,xi)∉-Ci∖{θ} for all xi∈Xi.Remark 4.3.
(a) Notice that Theorems4.1 and 4.2 are indeed equivalent since the sum of two ℓCi-functions is also a ℓCi-function.
(b) In Theorem4.1, the maps pi and qi are only assumed to be ℓCi-functions which need not be Ci-vectorial quasi-distances. So Theorem 4.1 is different from any theorem in Section 3 and is not a special case of any theorem in Section 3.Theorem 4.4.
LetI, Xi, X, Ci, E, and θ be the same as in Theorem 4.1. For each i∈I, let pi:Xi×Xi→E be a ℓCi-function and let fi:Xi→E be a Ci-v.l.s.c. and Ci-convex function. Suppose that (i)
ℋ={x=(xi)i∈I∈X:pi(xi,xi)=θfor alli∈I}≠∅,(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(yj,zj)+fj(zi)-fj(yi)∈-Cj∖{θ}.
Then there existsv=(vi)i∈I∈X such that for each i∈I, one has pi(vi,xi)+fi(xi)-fi(vi)∉-Ci∖{θ} for all xi∈Xi.Remark 4.5.
[19, Theorem  4.2] is a special case of Theorem 4.4.
Applying Theorem4.4, we obtain the following vectorial minimization theorem.Theorem 4.6 (vectorial minimization theorem).
LetI, Xi, X, Ci, E, and θ be the same as in Theorem 4.1. For each i∈I, let pi:Xi×Xi→E be a ℓCi-function and let fi:Xi→E be a Ci-v.l.s.c. and Ci-convex function. Suppose that (i)
ℋ={x=(xi)i∈I∈X:pi(xi,xi)=θfor alli∈I}≠∅,(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(yj,zj)+fj(zi)-fj(yi)∈-Cj∖{θ},(iii)
fori∈I and xi∈Xi with fi(xi)∉MinCifi(Xi), there exists yi∈Xi with yi≠xi such that pi(xi,yi)+fi(yi)-fi(xi)∈-Ci∖{θ}.
Then there existsx̂=(x̂i)i∈I∈X such that fi(x̂i)∈MinCifi(Xi) for all i∈I.Proof.
Applying Theorem4.4, there exists v=(vi)i∈I∈X such that for each i∈I, we have pi(vi,xi)+fi(xi)-fi(vi)∉-Ci∖{θ} for all xi∈Xi. We claim that fi(vi)∈MinCifi(Xi) for all i∈I. Suppose to the contrary that there exists i0∈I such that fi0(vi0)∉MinCi0fi0(Xi0). Then, by our assumption, there exists yi0=yi0(vi0)∈Xi0 with yi0≠vi0 such that pi0(vi0,yi0)+fi0(yi0)-fi0(vi0)∈-Ci0∖{θ}, which leads to a contradiction. Therefore fi(vi)∈MinCifi(Xi) for all i∈I.The following scalar minimization theorem follows from Theorem4.6 immediately.Corollary 4.7.
LetI, Xi, and X be the same as in Theorem 4.1. For each i∈I, let fi:Xi→(-∞,∞] be a l.s.c. and convex function and let pi:Xi×Xi→(-∞,∞] be a ℓ-function. Suppose that (i)
𝒟={x=(xi)i∈I∈X:pi(xi,xi)=0for alli∈I}≠∅,(ii)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset Mi of Xi for each i∈I such that for each y∈X∖K there exist j∈I and zj∈Mj such that pj(yj,zj)<fj(yi)-fj(zi),(iii)
for anyi∈I and xi∈Xi with fi(xi)>infzi∈Xifi(zi) there exists yi∈Xi with yi≠xi such that pi(xi,yi)<fi(xi)-fi(yi).
Then there existsx̂=(x̂i)i∈I∈X such that fi(x̂i)=infzi∈Xifi(zi) for all i∈I.Remark 4.8.
(a) [19, Theorem  5.5] is a special case of Corollary 4.7.
(b) Theorems4.4 and 4.6 are equivalent if they further add the condition  “for each i∈I, pi(xi,yi)∈Ci for all xi, yi∈Xi.” Indeed, it suffices to show that Theorem 4.6 implies Theorem 4.4. Suppose that for each x=(xi)i∈I∈X, there exists ix∈I such that pix(xix,yix)+fix(yix)-fix(xix)∈-Cix∖{θ} for some yix∈Xix with yix≠xix. Then, by Theorem 4.6, there exists v=(vi)i∈I∈X such that fi(vi)∈MinCifi(Xi) or
(4.4)fi(xi)-fi(vi)∉-Ci∖{θ},∀xi∈Xi,∀i∈I.
But from our assumption, there exists iv∈I such that
(4.5)piv(viv,wiv)+fiv(wiv)-fiv(viv)∈-Ci∖{θ}
for some wiv∈Xiv with wiv≠viv. It follows that
(4.6)fiv(wiv)-fiv(viv)∈-piv(viv,wiv)-Ci∖{θ}⊆-Ci∖{θ},
which leads to a contradiction.Now, we give some equivalent formulations of Theorem4.1 (when I is a singleton) as follows.Theorem 4.9.
LetX be a Hausdorff t.v.s., let C≠{θ} be a nonempty pointed convex cone in a t.v.s. E with zero vector θ, p, q:X×X→E be ℓC-functions. Suppose that (1)
ℛ={x∈X:p(x,x)=q(x,x)=0}≠∅,(2)
there exist a nonempty compact subsetK of X and a nonempty compact convex subset M of X such that for each y∈X∖K there exists z∈M such that p(y,z)+q(y,z)∈-C∖{θ}.
Then the following statements are equivalent.(i)
(Vectorial Variant of Ekeland’s Variational Principle). There existsv∈X such that p(v,x)+q(v,x)∉-C∖{θ} for all x∈X.(ii)
(Common Fixed Point Theorem for a Family of Multivalued Maps). LetΛ be an index set. For each i∈Λ, let Ti:X⊸X be a multivalued map with nonempty values such that for each (i,x)∈Λ×X with x∉Ti(x), there exists y=y(x,i)∈X with y≠x such that
(4.7)p(x,y)+q(x,y)∈-C∖{θ}.
Then there exists x0∈X such that x0∈⋂i∈ΛTi(x0). That is, {Ti}i∈Λ has a common fixed point in X.(iii)
(Common Fixed Point Theorem for a Family of Single-Valued Maps). LetΛ be an index set. For each i∈Λ, suppose that Ti:X→X is a self-map satisfying
(4.8)p(x,Tix)+q(x,Tix)∈-C∖{θ}.
for all x≠Ti(x). Then there exists x0∈X such that Tix0=x0 for all i∈Λ;(iv)
(Maximal Element Theorem for a Family of Multivalued Maps). LetΛ be an index set. For each i∈Λ, let Ti:X⊸X be a multivalued map. Suppose that for each (i,x)∈Λ×X with Ti(x)≠∅, there exists y=y(x,i)∈X with y≠x such that(4.9)p(x,y)+q(x,y)∈-C∖{θ}.
Then there existsx0∈X such that Ti(x0)=∅ for all i∈Λ.Proof.
By Theorem4.1, conclusion (i) holds.
(a) “(i)⇔ (ii)”
(⇒) By (i), there exists v∈X such that p(v,x)+q(v,x)∉-C∖{θ} for all x∈X. We claim that v∈Ti(v) for all i∈I. If v∉Ti0v for some i0∈I, then, by hypothesis, there exists w(v,i0)∈X with w(v,i0)≠v such that
(4.10)p(v,w(v,i0))+q(v,w(v,i0))∈-C∖{θ},
which leads to a contradiction. Hence v∈⋂i∈ITi(v) and v is a common fixed point of {Ti}i∈I.
(⇐) Suppose that for each x∈X, there exists y∈X with y≠x such that p(x,y)+q(x,y)∈-C∖{θ}. Then for each x∈X, we can define a multivalued map T:X⊸X∖{∅} by
(4.11)T(x)={y∈X:p(x,y)+q(x,y)∈-C∖{θ}}.
Clearly, x∉T(x) for all x∈X. But, by (ii), T has a fixed point v in X, a contradiction. So (i) is true.(b) “(ii)⇔ (iii)”
(⇒) Suppose (ii) holds. Under the assumption of (iii), for each i∈I, let φi:X⊸X be defined by φi(x)={Ti(x)}. Then for each (i,x)∈I×X with x∉φi(x), we have x≠Ti(x). By hypothesis of (iii), p(x,Tix)+q(x,Tix)∈-C∖{θ}. Therefore, by (ii), there exists v∈X such that v∈⋂i∈Iφi(v) or Tiv=v for all i∈I, and hence (iii) is proved.
(⇐) Assume (iii) holds. Under the assumption of (ii), for each (i,x)∈I×X with x∉Ti(x), there exists y(x,i)∈X with y(x,i)≠x such that
(4.12)p(x,y(x,i))+q(x,y(x,i))∈-C∖{θ}.
Defineτi:X→X by
(4.13)τi(x)={xifx∈Ti(x),y(x,i)ifx∉Ti(x).
Hence τi is a self-map of X into X satisfying p(x,τi(x))+q(x,τi(x))∈-C∖{θ} for all x≠τi(x). By (iii), there exists v∈X such that v=τi(v)∈Ti(v) for all i∈I. This shows that (iii) implies (ii).(c) "(i)⇔ (iv)"
(⇒) Using (i), there exists v∈X such that p(v,x)+q(v,x)∉-C∖{θ} for all x∈X. We want to show that Ti(v)=∅ for all i∈I. Suppose to the contrary that there exists i0∈I such that Ti0(v)≠∅. By hypothesis of (iv), there exists w=w(v,i0)∈X with w≠v such that
(4.14)p(v,w)+q(v,w)∈-C∖{θ},
which is a contradiction. Therefore Ti(v)=∅ for all i∈I.
(⇐) Suppose that for each x∈X, there exists y∈X with y≠x such that p(x,y)+q(x,y)∈-C∖{θ}. For each x∈X, define a multivalued map T:X⊸X∖{∅} by
(4.15)T(x)={y∈X:p(x,y)+q(x,y)∈-C∖{θ}}.
Then T(x)≠∅ for all x∈X. But applying (iv), there exists x0∈X such that T(x0)=∅, a contradiction. Hence (i) holds.Remark 4.10.
Theorem4.9 improves and generalizes Theorems  4.2, 5.1, 5.2, 5.3, and 5.4 in [19].



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*Source: 247236-2011-09-29.xml*