# Ekeland Variational Principle for Generalized Vector Equilibrium Problems with Equivalences and Applications

Authors: De-ning Qu; Cao-zong Cheng
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

The aim of this paper is to introduce Ekeland variational principle with variants for generalized vector equilibrium problems and to establish some existence results of solutions of generalized vector equilibrium problems with compact or noncompact domain as applications. Finally, some equivalent results of the established Ekeland variational principle are presented.


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## Body

## 1. Introduction

An Ekeland variational principle [1] (also [2, 3]) appeared first as an existence result of approximate minimizer for a lower semicontinuous and bounded below function on complete metric spaces. It subsequently developed an important tool of many subjects, such as in nonlinear analysis (e.g., [4]), optimization theory (e.g., [5–11]), game theory (e.g., [12]), dynamical systems (e.g., [13]), and others (e.g., [14, 15]). By reason of the fact that equilibrium problems contain many problems as their special cases, such as optimization problems, fixed-point problems, variational inequality problems, complementary problems, and Nash equilibrium problems (see [16]), a new direction of research on variational principle for equilibrium problems has arisen. The variational principle for equilibrium problems (e.g., [17]) and for vector equilibrium problems (e.g., [18–23]) and/or their applications or equivalent results were discussed. In terms of set-valued objective mappings, variational principle for vector optimization problems was first introduced by Chen and Huang [5] and was reported in many literatures (e.g., see [7, 9–11]) in the sequel. In 2009, Zeng and Li [24] discussed Ekeland variational principle for vector equilibrium problems with set-valued objective mappings (generalized vector equilibrium problems).One of the most important tools of obtaining variational principle for vector problems is the scalarization functions (see [5, 6, 21–24], for instance). While these scalarization functions always involve single-valued mappings. Motivated by the works mentioned above, we establish the Ekeland variational principle for generalized vector equilibrium problems by applying a nonlinear scalarization function involving set-valued mappings. It is worth noting that the generalized vector equilibrium problems considered in this paper are rather than those in [24].Letℝ, ℝ+ and ℕ be denoted by the sets of real numbers, nonnegative real numbers and positive integers, respectively, and let 𝒩(*) be the collection of open neighborhoods of *, where * is a point or a set. A subset D of a real topological vector space Y is called a cone if λx∈D for all x∈D and λ>0. Let D be a cone in Y and A⊂Y. D is called proper if D≠Y. A is said to be D-closed [25] if A+clD is closed; to be D-bounded [25] if, for each neighborhood U∈𝒩(0Y), there exists λ>0 such that A⊂λU+D.Throughout this paper, unless otherwise specified, letX be a Hausdorff topological space and Y a real Hausdorff topological vector space, let D⊂Y be a proper, closed, and convex cone with nonempty interior and e∈intD, and let f:X×X→2Y be a strict, set-valued mapping, fx(·)=f(x,·) for each x∈X and fy(·)=f(·,y) for each y∈X. A set-valued mapping is said to be strict if it has nonempty values. Consider the following generalized vector equilibrium problems:
(GVEP1)tofindx¯∈Xsuchthatf(x¯,y)∩(-D)=∅,∀y∈X,(GVEP2)tofindx¯∈Xsuchthatf(x¯,y)∩(-intD)=∅,∀y∈X.This paper is divided into five sections. In Section2, some preliminaries are provided. In Section 3, Ekeland variational principle with variants for (GVEP1) is argued in complete quasimetric spaces. In Section 4, some existence results of solutions of (GVEP2) with compact or noncompact domain are established as applications. Finally, in Section 5, some equivalent results of the established Ekeland variational principle are presented.


## 2. Preliminaries

LetX be a nonempty set. “⪯” is called a quasiorder on X if it is of(o1)
reflexivity:x⪯x, for all x∈X;(o2)
transitivity:x⪯y,y⪯z⇒x⪯z.Here(X,⪯) is called a quasiorder space. An element x∈X is said to be a maximal elementof a quasiorder space (X,⪯) if there is no element y∈X, other than x, such that x⪯y, in other words, x⪯y for some y∈X implies x=y.LetX and Y be topological spaces. A real-valued function g:X→ℝ is said to be upper semicontinuous onX if {u∈X:g(u)<λ} is open for each λ∈ℝ; to be lower semicontinuous onX, {u∈X:g(u)>λ} is open for each λ∈ℝ. The following conceptions of continuity for a set-valued mapping can be found in [4]. A set-valued mapping G:X→2Y is said to be upper semicontinuous atu0∈X if, for any N∈𝒩(G(u0)), there exists B∈𝒩(u0) such that G(u)⊂N for all u∈B; to be lower semicontinuous atu0∈X, if for any y0∈G(u0) and any N∈𝒩(y0), there exists B∈𝒩(u0) such that G(u)∩N≠∅ for all u∈B; to be upper semicontinuous (resp., lower semicontinuous) onX, if G is upper semicontinuous (resp., lower semicontinuous) at each u∈X; to be continuous atu0∈X (resp., on X), if G is both upper semicontinuous and lower semicontinuous at u0 (resp., on X); to be closed, if its graph Graph(G)={(u,y)∈X×Y:y∈G(u)} is closed in X×Y.Lemma 1 (see [26]).
LetX and Y be Hausdorff topological spaces and S:X→2Y a strict set-valued mapping. If X is compact and S is upper semicontinuous with compact values, then S(X)=∪{S(x):x∈X} is compact.Definition 2.
LetX be a nonempty set. A function d:X×X→ℝ+ is said to be a quasimetric onX if (d1)
d(x,y)=0 if and only if x=y;(d2)
d(x,z)≤d(x,y)+d(y,z),∀x,y,z∈X.X equipped a quasimetric d is said to be a quasimetric space, denoted by (X,d). Furthermore, if d also satisfies(d3)
d(x,y)=d(y,x),∀x,y∈X,then(X,d) is called a metric space, where d is a metric on X.Let(X,d) be a quasimetric space. xk→x means d(xk,x)→0. {xk}⊂X is called a Cauchy sequence on (X,d) if, for any ɛ>0, there exists K∈ℕ such that d(xk,xl)<ɛ,∀l>k>K.  (X,d) is said to be complete if, for each Cauchy sequence {xk}⊂X, there exists x∈X such that xk→x as k→+∞.Definition 3 (see [27]).
Let(X,d) be a quasimetric space. A function w:X×X→ℝ+ is said to be a W-distance on (X,d) if (w1)
w(x,z)≤w(x,y)+w(y,z),∀x,y,z∈X;(w2)
for each fixedx∈X, y↦w(x,y) is lower semicontinuous;(w3)
for anyϵ>0, there exists δ>0 such that w(x,y)≤δ and w(x,z)≤δ imply that d(y,z)≤ϵ.TheW-distance includes metric and quasimetric as its special cases. But the converse fails to be true. Moreover, W-distance is not necessary to be symmetric. Some examples and properties of a W-distance in metric spaces are provided by Kada et al. [27].Lemma 4 (see [27]).
Letw be a W-distance on a quasimetric space (X,d), let {xk},{yk}⊂X and {αk},{βk}⊂ℝ+ with αk,βk→0 as k→+∞, and let x,y,z∈X. Then the following assertions are true:(i)
ifw(xk,y)≤αk and w(xk,z)≤βk,∀k∈ℕ, then y=z. In particular, if w(x,y)=0 and w(x,z)=0, then y=z;(ii)
ifw(xk,xl)≤αk,∀l>k, then {xk} is a Cauchy sequence;(iii)
ift>0, then tw is also a W-distance on (X,d).LetX be a topological space and Y be a topological vector space in the rest of this section.Since each compact subset inY is both D-closed and D-bounded by Definition 3.1 and Proposition 3.1 in [25], the following general nonlinear scalarization function is well defined in view of Lemma 3.1 in [28].Definition 5.
LetG:X→2Y be a strict and compact-valued mapping. A generalized nonlinear scalarization function ζG:X→ℝ of G is defined by
(1)ζG(u)=min{λ∈ℝ:G(u)∩(λe-D)≠∅},∀u∈X.According to Proposition 3.1 in [28], we have the following.Lemma 6.
LetG:X→2Y be a strict and compact-valued mapping. The following assertions are true for each λ∈ℝ and u∈X:(i)
ζG(u)<λ⇔G(u)∩(λe-intD)≠∅.(ii)
ζG(u)≤λ⇔G(u)∩(λe-D)≠∅.Lemma 7.
LetG:X→2Y be a strict and compact-valued mapping.(i)
IfG is lower semicontinuous on X, then ζG is upper semicontinuous on X;(ii)
IfG is upper semicontinuous on X, then ζG is lower semicontinuous on X.Proof.
This proof is completed by lettingG(x)=0,C(x)=D, and E(x)={e} for all x∈X in Corollary 3.1 in [28] since upper (resp., lower) semicontinuity implies the D-upper (resp., lower) semicontinuity (see [29]).Lemma 8.
If(a1)
f is compact valued;(a2)
f(x,y)+f(y,z)⊂f(x,z)+D,∀x,y,z∈X,then
(2)ζfx(z)≤ζfx(y)+ζfy(z),∀x,y,z∈X.Proof.
For eachx,y,z∈X, it follows from Definition 5 that u∈f(x,y)∩(ζfx(y)e-D) and v∈f(y,z)∩(ζfy(z)e-D) can be chosen. Then u+v∈f(x,z)+D by (a2). Select w∈f(x,z) to satisfy w∈u+v-D. This deduces that
(3)w∈(ζfx(y)e-D)+(ζfy(z)e-D)-D⊂(ζfx(y)+ζfy(z))e-D.
Therefore, f(x,z)∩((ζfx(y)+ζfy(z))e-D)≠∅ by Lemma 6(ii) and ζfx(z)≤ζfx(y)+ζfy(z).


## 3. Ekeland Variational Principle for(GVEP1)

From now on, unless otherwise specified, suppose that(X,d) is a Hausdorff complete quasimetric space and that w is a W-distance on (X,d).Lemma 9.
(i) If (a1)-(a2) and(a3)
f(x,x)∩(-intD)=∅,∀x∈X
hold, then(4)S(y)⊂S(x),∀y∈S(x),∀x∈X,
where S:X→2X is defined by
(5)S(x)={y∈X:y≠x,f(x,y)∩(-w(x,y)e-D)≠∅},∀x∈X.(ii) Besides (a1)–(a3), if(a4)
for eachx∈X,fx(·) is upper semicontinuous on X;(a5)
for eachx∈X,fx(X)∩(b-D)=∅ for some b∈Y,
then, for eachx^∈X with S(x^)≠∅, there exists x¯∈S(x^) such that S(x¯)=∅.Proof.
Obviously, by (a1) and Lemma6(ii),
(6)S(x)={y∈X:y≠x,ζfx(y)+w(x,y)≤0}.
(i) For each x∈X and y∈S(x), if S(y)=∅, then the conclusion holds trivially. Otherwise, for each z∈S(y),
(7)ζfx(y)+w(x,y)≤0,ζfy(z)+w(y,z)≤0,
and so
(8)ζfx(z)+w(x,z)≤ζfx(y)+ζfy(z)+w(x,y)+w(y,z)≤0
according to Lemma 8. Moreover, by w(x,y)≥0 and w(y,z)≥0,
(9)ζfx(y)≤0,ζfy(z)≤0.
Now claim that z≠x. Otherwise, ζfx(z)≥0 by (a3) and Lemma 6(i). Thus,
(10)w(x,y)=0,
since
(11)w(x,y)≤ζfx(z)+w(x,y)≤ζfx(y)+w(x,y)+ζfy(z)≤0
by (2) and (9). Similarly, w(y,z)=0, and so w(x,z)=0 by (w1). This, together with w(x,y)=0, implies that y=z by Lemma 4(i), which contradicts with z∈S(y). Thus z∈S(x), and so S(y)⊂S(x),∀y∈S(x).
(ii) Forb∈Y given in (a5), b∈re+D for some r∈ℝ by Lemma 2.1 in [30], which leads to re-D⊂b-D. This, together with (a5), implies that fx(X)∩(re-D)=∅. It follows form Lemma 4(ii) that
(12)v(x)=inf{ζfx(y):y∈S(x)}>-∞,∀x∈X.
Then, for each x0=x^∈X,
(13)v(x0)=inf{ζfx0(y):y∈S(x0)}>-∞.
For each k∈ℕ, let xk-1 satisfy that v(xk-1)>-∞ and take xk∈S(xk-1) such that
(14)ζfxk-1(xk)≤v(xk-1)+12k.
If S(xk)=∅ for some k∈ℕ, this conclusion holds by letting x¯=xk. Now consider S(xk)≠∅,∀k∈ℕ. The rest proof is divided into four steps as follows.
(a) Show thatxk→x¯ as k→+∞ for some x¯∈X. Indeed, for each k∈ℕ, by (2),
(15)v(xk)=inf{ζfxk(y):y∈S(xk)}≥inf{ζfxk-1(y)-ζfxk-1(xk):y∈S(xk-1)}=v(xk-1)-ζfxk-1(xk)≥-12k,
which implies that
(16)w(xk,y)≤-ζfxk(y)≤-v(xk)≤12k,∀y∈S(xk),
and so
(17)w(xk,xl)≤∑i=kl-1‍w(xi,xi+1)≤∑i=kl-1‍12i≤12k-1,∀l>k.
Therefore, {xk} is a Cauchy sequence by Lemma 4(ii), and so there exists x¯∈X such that xk→x¯ as k→+∞ by the completeness of (X,d). In addition, according to (w2)
(18)w(xk,x¯)≤liminfl→+∞w(xk,xl)≤12k-1,∀k∈ℕ.
(b) Show that x¯∈⋂k=0+∞S(xk). In fact, for each k∈ℕ,
(19)ζfxk(xl)+w(xk,xl)≤∑i=kl-1‍(ζfxi(xi+1)+w(xi,xi+1))≤0
by (2). Since, for each x∈X,  ζfx(·) is lower semicontinuous by (a1), (a4), and Lemma 7(ii),
(20)ζfxk(x¯)+w(xk,x¯)≤0
by letting l→+∞ in (19). Hence,
(21)ζfxk(x¯)≤0,∀k∈ℕ.
Now it is enough to prove that xk≠x¯,∀k∈ℕ by (20). Indeed, if xk0=x¯ for some k0∈ℕ, then ζfxk0(xk0)=ζfxk0(x¯)≥0 in view of (a3) and Lemma 6(i). By applying (21) and adopting the same argument of the proof of (10),
(22)w(xk0,xk0+1)=0.
Thus,
(23)ζfxk0(xk0+1)=ζfxk0(xk0+1)+w(xk0,xk0+1)≤0.
It follows form 0≤ζfxk0(x¯)≤ζfxk0(xk0+1)+ζfxk0+1(x¯) that ζfxk0+1(x¯)≥-ζfxk0(xk0+1)≥0. Hence, w(xk0+1,xk0+2)=0 by (21) and the similar argument of (10), which, together with w(xk0,xk0+2)=0, implies xk0+1=xk0+2 in virtue of (22) and Lemma 4(i). This contradicts with xk0+2∈S(xk0+1). Thus, x¯∈S(xk),∀k∈ℕ. Clearly, x¯∈S(x0).
(c) Show that⋂k=1+∞S(xk)={x¯}. Indeed, for any x~∈⋂k=1+∞S(xk), x~≠xk, and ζfxk(x~)+w(xk,x~)≤0,∀k∈ℕ. As a result,
(24)w(xk,x~)≤-ζfxk(x~)≤-v(xk)≤12k,∀k∈ℕ,
which, together with (18), implies that x~=x¯ by Lemma 4(i).
(d) Show thatx¯∈S(x^) and S(x¯)=∅. As a matter of fact, x¯∈S(x^) by (b). If S(x¯)≠∅, then S(x¯)={x¯} in view of (c). This leads to a contradiction by the definition of S.Theorem 10.
DefineA:X→2X as
(25)A(x)={y∈X:x=yorf(x,y)∩(-ɛw(x,y)e-D)≠∅},∀x∈X.
If (a1)–(a5) hold, then, for any ɛ>0 and for any x^∈X, there exists x¯∈X such that (i)
A(x¯)={x¯};(ii)
f(x¯,y)∩(-ɛw(x¯,y)e-D)=∅,∀y∈X with y≠x¯;(iii)
f(x^,x¯)∩(-ɛw(x^,x¯)e-D)≠∅ if, further, S(x^)≠∅, where S:X→2X defined as
(26)S(x)={y∈X:x≠y,f(x,y)∩(-ɛw(x,y)e-D)≠∅},∀x∈X.Proof.
Sincewɛ(x,y)=ɛw(x,y) is another W-distance by Lemma 4(iii), without loss of generality, we set ɛ=1. Then
(27)A(x)={y∈X:x=yorf(x,y)∩(-w(x,y)e-D)≠∅},∀x∈X,
and S is identical to (5). Take x¯=x^ if S(x^)=∅. Otherwise, there exists x¯∈S(x^) such that S(x¯)=∅ by Lemma 9(ii). All in all, A(x¯)={x¯}. The conclusion (i) is true. Also, y≠x¯ is equivalent to y∉A(x¯), and so the conclusion (ii) holds by Lemma 6(ii). If, further, S(x^)≠∅, (i) implies that
(28)f(x^,x¯)∩(-ɛw(x^,x¯)e-D)≠∅,
by Lemma 6(ii).Corollary 11.
If (a5) is replaced by(a6)
(X,d) is compactin Theorem 10, then the conclusions still hold.Proof.
It is easy to see thatfx(X) is compact for each x∈X by (a1), (a4), (a6), and Lemma 1. The rest proof is divided into three steps.
(a) For anyλ,μ∈ℝ with λ<μ,
(29)λe-D⊂μe-(μ-λ)e-D⊂μe-intD-D⊂μe-intD.
(b) For any u∈Y, there exists λu∈ℝ such that u∉λue-D. In fact, if u∈λe-D,∀λ∈ℝ, then u∈λe-intD,∀λ∈ℝ by (a), which implies
(30){λe-u:λ∈ℝ}={-λe-u:λ∈ℝ}⊂intD.
Since Y=∪{λe-D:λ∈ℝ} by Lemma 2.1 in [30], for each v∈Y, there exists d0∈intD and μ∈ℝ such that -v=μe-d0. Then v=-μe+d0=(-μe-u)+d0+u∈u+intD. Thus Y⊂u+intD and a contradiction with properness if D arises.
(c) For somer∈ℝ, fx(X)∩(re-D)=∅ and (a5) holds. Indeed, for each u∈fx(X), there exists λu∈ℝ such that u∉λue-D, and so Nu∩(λue-D)=∅ for some Nu∈𝒩(u). By the completeness of fx(X), there exists u1,…,un∈fx(X) such that fx(X)=∪{Nui:1≤i≤n}. Taking r=min{λui:1≤i≤n}, we have fx(X)∩(re-D)=∅.Apparently, we have the following by applying Theorem10.Corollary 12.
If(a7)
f(x,x)⊂-∂D,∀x∈X, where ∂D is the topological bound of Dinstead of (a3), hold in Theorem 10, then the conclusions are true as well.Remark 13.
Under all the hypotheses in Theorem10, Corollary 11, or Corollary 12, for each ɛ>0, there exists xɛ∈X such that
(31)f(xɛ,y)∩(-ɛw(xɛ,y)e-intD)=∅,∀y∈X.In fact, it is obvious thatf(xɛ,y)∩(-ɛw(xɛ,y)e-intD)=∅,∀y∈X with y≠xɛ. For y=xɛ, if
(32)f(xɛ,xɛ)∩(-ɛw(xɛ,xɛ)e-intD)≠∅,
then f(xɛ,xɛ)∩(-intD)≠∅, which contradicts with (a3) or (a7).Remark 14.
(a) The condition:(a8)
there existsb~∈Y such that fx(X)∩(b~-intD)=∅,often appeared in the results on Ekeland variational principle for (generalized) vector equilibrium problems, such as Theorem 2.1 in [22]. Actually, (a8) is equivalent to (a5). As the case stands, it is apparent that (a5) implies (a8). On the other hand, taking b∈b~-intD, then b-D⊂b~-intD-D⊂b~-intD, which guarantees (a5).
(b) The condition:(a9)
for eachx∈X, 0Y∈f(x,x) or f(x,x)={0Y},instead of (a3) or (a7), is always required to prove variational principle for equilibrium problems in many literatures, such as in [20–24]. But this condition is unnecessary in Theorem 10. See the following examples.Example 15.
LetX=[0,1],Y=ℝ2, and D={(x,y):x∈ℝ,y≥0}, and let d(x,y)=|x-y|,∀x,y∈X. Define f:X×X→2Y as
(33)f(x,y)={(x,|y-x|),(-x,|y-x|)},∀x,y∈X.
By computing simply, (a1)–(a5) hold. But (a9) is false since 0Y∉f(x,x) unless x=0.Example 16.
LetX=ℝ,Y=ℝ2, and D={(x,y):x∈ℝ,y≥0}, and let d(x,y)=|x-y|,∀x,y∈X, and f:X×X→2Y define as f(x,y)=[1,2]×{0},∀x,y∈X. It is clear that (a1)-(a2), (a4)-(a5), and (a7) are satisfied, but (a9) does not hold.
It is effortless to obtain the following variant of vectorial form of Ekeland variational principle by Theorem10.Theorem 17.
Letɛ,λ>0 be given. If, besides (a1)–(a5),
(
a
10
)
for some x^∈X with S(x^)≠∅,
(34)f(x^,y)∩(-ɛe-D)=∅,∀y∈X,
where S is defined as (26), then there exists x¯∈X such that (i)
f(x¯,x^)⊂D if, further, f(x,x)⊂D,∀x∈X;(ii)
w(x^,x¯)≤λ;(iii)
f(x¯,y)∩(-(ɛ/λ)w(x¯,y)e-D)=∅,∀y∈X with y≠x¯.Proof.
Letx¯∈X be the point provided by Theorem 10 with ɛ/λ instead of ɛ. Then (iii) is obtained by the conclusion (ii) in Theorem 10. In addition, in view of the conclusion (i) in Theorem 10,
(35)ζfx^(x¯)+ɛλw(x^,x¯)≤0.
It follows from (34)-(35), (a10), and Lemma 6(ii) that -ɛ≤ζfx^(x¯)≤-(ɛ/λ)w(x^,x¯) and so the conclusion (ii) holds. Moreover, according to the conclusion (iii) in Theorem 10, there exists u~∈f(x^,x¯) such that
(36)-u~∈ɛλw(x^,x¯)+D.
Taking x=z=x¯ and y=x^ in (a2), we have
(37)f(x¯,x^)+f(x^,x¯)⊂f(x¯,x¯)+D⊂D.
Then
(38)f(x¯,x^)⊂-u~+D⊂ɛλw(x^,x¯)+D+D⊂D.
Thereby, (i) holds.Remark 18.
(a) If the solution set of(GVEP2) or (GVEP1) is nonempty, then there must exist a point x^∈X satisfying (34).
(b) If, further,D is a point cone (that is, D∩(-D)={0Y}) in Theorems 17, then the condition (a10) can be weakened as follows: for some x^∈X with S(x^)≠∅,
(39)f(x^,x)∩(-ɛe-D∖{0Y})=∅,∀x∈X,
where S is defined as (5).
In fact, we only prove (ii) by the proof of Theorem17. In view of Theorem 10(iii),
(40)f(x^,x¯)∩(-ɛλw(x^,x¯)e-D)≠∅.
Then there exist y0∈f(x^,x¯) and d0∈D such that y0=-(ɛ/λ)w(x^,x¯)e-d0. By (39),
(41)-ɛλw(x^,x¯)e-d0∉-ɛe-D∖{0Y},
or equivalently,
(42)ɛλ(w(x^,x¯)-λ)e+d0∉D∖{0Y}.
If w(x^,x¯)>λ, then (ɛ/λ)(w(x^,x¯)-λ)e+d0∈D and so (ɛ/λ)(w(x^,x¯)-λ)e+d0=0Y, which implies that d0=-(ɛ/λ)(w(x^,x¯)-λ)e∈-intD. Thereupon, d0=0Y by the pointedness of D, which contradicts with d0∈-intD.


## 4. Applications of Vectorial Form of Ekelend Variational Principle for(GVEP1)

In this section, some existence results of solutions of(GVEP2) are established as applications of Ekelend variational principle for (GVEP1).Theorem 19.
LetX admit a topology τ (possibly different from the initial topology 𝒯 induced by d). Besides (a1)-(a2), (a4)-(a5) and (a7) in the sense of 𝒯, if the following conditions hold:(b1)
X is τ-compact; namely, X is compact with respect to τ;(b2)
for eachx∈X, xk→τx as k→+∞ implies the boundedness of {d(xk,x):k∈ℕ};(b3)
fy is τ-lower semicontinuous on X for each y∈X,then (GVEP2) has a solution.Proof.
Ifw=d and ɛ=1/k in (31), then there exists xk∈X such that
(43)f(xk,y)∩(-1kd(xk,y)e-intD)=∅,∀y∈X.
Thus ζfy(xk)≥-(1/k)d(xk,y) by Lemma 6(i). Without loss of generality, let xk→τx by (b1). Since ζfy(·) is τ-upper semicontinuous for each y∈X by (b3) and Lemma 7(i),
(44)ζfy(x¯)≥limsupk→+∞ζfy(xk)≥limsupk→+∞-1kd(xk,y)=0,∀y∈X,
due to (b2). This deduces that f(x¯,y)∩(-intD)=∅,∀y∈X by Lemma 6(i); that is, (GVEP2) has a solution.Remark 20.
It is worth noting that the condition (b2) cannot be guaranteed by (b1). For instance, letX=ℝ and d(x,y)=|x-y|,∀x,y∈X. For the trivial topology τ={∅,X}, X is τ-compact. For xk=k, it is clear that xk→τ0, but {d(xk,0):k∈ℕ} is unbounded.Corollary 21.
If (a1)-(a2), (a4), and (a6)-(a7) and(a11)
fy is lower semicontinuous on X for each y∈X,hold, then (GVEP2) has a solution.Proof.
Clearly, (a6) implies (a5). Takingτ=𝒯 in Theorem 19, (b1) and (b3) hold by (a6) and (a11), respectively, and (b2) holds trivially since d(xk,x¯)→0 when xk→x¯. Thus this conclusion is true by Theorem 19.When the existence of solutions of equilibrium problems on a noncompact domain is discussed, some sufficient assumptions, such as coercivity condition [31] and condition C1(x0) [20], and so forth, must be required to be substitute for compact domain. Motivated by these ideas, we obtain the following existence results of solutions of (GVEP2) by applying Theorem 19.Theorem 22.
Let (a1)-(a2), (a4)-(a5), and (a7) hold in the sense of𝒯 induced by d, and let X admit a topological τ (possibly different from the initial topology 𝒯 induced by d). If (b3) is satisfied and the following hold:(b4)
each𝒯-closed bounded ball on X is τ-compact,(b5)
for given pointx^∈X, there exists a τ-compact set C⊂X satisfying that, for each x∈C and {xk}⊂C, xk→τx implies the boundedness of {d(xk,x):k∈ℕ} and for each x∈X∖C,
(45)d(y,x^)<d(x,x^),f(x,y)∩(-D)≠∅for somey∈X,(b6)
for eachx∈X, L(x)={y∈X:f(x,y)∩(-D)≠∅} is τ-closed,(b7)
for eachy∈X, d(·,y) is τ-lower semicontinuous,then(GVEP2) has a solution.Proof.
DefineA:X→2Y as
(46)A(x)={y∈X:d(y,x^)≤d(x,x^),ζfx(y)≤0}.
We see the following. (a)
For eachx∈X, A(x)≠∅ since x∈A(x) by (a7).(b)
For eachx∈X, y∈A(x) implies that A(y)⊂A(x) by the similar argument of the proof in Lemma 9.(c)
A(x) is τ-compact for each x∈X by (b4) and (b6).
By regarding theτ-compact set C as X in Theorem 19, there exists x¯C∈C such that
(47)f(x¯C,y)∩(-intD)=∅,∀y∈C.
Now argue the conclusion by contradiction. Assume that there exists y¯∈X∖C such that
(48)f(x¯C,y¯)∩(-intD)≠∅.
Let
(49)α=min{d(y,x^):y∈A(y¯)}.
Now assert that(50)A(y¯)∩C=∅.
In fact, if y∈A(y¯)∩C, then f(y¯,y)∩(-D)≠∅, which, together with (48), deduces that f(x¯C,y)∩(-intD)≠∅ in view of (a2) and Lemmas 6 and 8. This contradicts with (47).
The minimum in (49) is attained since A(y¯) is nonempty τ-compact by (a) and (c), and d(·,x^) is τ-lower semicontinuous. Take z0∈A(y¯) such that d(z0,x^)=α. Since z0∉C by (50), z1 can be chosen to satisfy that d(z1,x^)<d(z0,x^)=α and f(z0,z1)∩(-D)≠∅ by (b5). Accordingly, the assertion z1∈A(z0)⊂A(y¯) is absurd by the definition of α. Thereby, f(x¯C,y)∩(-intD)=∅,∀y∈X, or in other words, x¯C is a solution of (GVEP2).Theorem 23.
Suppose that(X,∥·∥) is a real reflexive Banach space. Besides (a1)-(a2), (a4)-(a5) and (a7), if the following conditions hold:(b8)
for eachy∈X, fy is weakly lower semicontinuous,(b9)
there existsr>0 such that for each x∈X∖Xr, y∈X can be selected such that ∥y∥<∥x∥ and f(x,y)∩(-D)≠∅, where Xr={x∈X:∥x∥≤r},then (GVEP2) has a solution.Proof.
Denote𝒯 and τ by the strong topology and the weak topology on X, respectively, and define A:X→2X as
(51)A(x)={y∈X:∥y∥≤∥x∥,ζfx(y)≤0},∀x∈X.
Letting x^=0X, C=Xr, and d(x,y)=∥x-y∥,∀x,y∈X, we see that A(x) is just equal to (46). (b3) and (b5) are satisfied by (b8) and (b9), respectively. Also, (b4) and (b7) hold thanks to weak compactness of each closed bounded ball and weak lower semicontinuity of ∥·∥ in a real reflexive Banach space, respectively. Besides, (b6) guarantees the weak compactness of A(x) in the proof of Theorem 22. While this property can be guaranteed by the assertions that X∥x∥ is weakly compact and {y∈X:ζfx(y)≤0} is closed by (a4) and Lemma 7(ii), this rest proof completes by the similar argument of the proof in Theorem 22.


## 5. Several Equivalent Results of Vectorial Form of Ekelend Variational Principle

In this section, we shall present several equivalent results of the established Ekelend variational principle. Now Define⪯ as follows:
(52)x⪯y⟺x=yorf(x,y)∩(-w(x,y)e-D)≠∅.
Then, under the conditions (a1)–(a3), ⪯ is a quasiorder on X by Lemma 8(i). Also, (X,⪯) has a maximal element x¯ under all assumptions of Theorem 10. Indeed, A(x)={y∈X:x⪯y}, where A(x) is defined as (27). Then there exists x¯∈X such that A(x¯)={x¯} by Theorem 10(i), which implies that x¯ is a maximal element of (X,⪯).Theorem 24.
Let (a1)–(a3) hold andX equip a quasiorder ⪯ defined as (52). Then the following assertions are equivalent.(i)
(Existence result of maximal element)(X,⪯) has a maximal element x¯.(ii)
(Ekelend variational principle for(GVEP1)) There exists x¯∈X such that
(53)f(x¯,y)∩(-w(x¯,y)e-D)=∅,∀y∈Xwithy≠x¯.(iii)
(Caristi-kirk fixe-point theorem) LetP:X→2X be a strict, set-valued mapping such that, for any x∈X,  x⪯y,∀y∈P(x). Then there exists x¯∈X such that P(x¯)={x¯}.Proof.
(i)⇒(iii) Let x¯ be a maximal element on (X,⪯). If P(x¯)≠{x¯}, then there exists y~≠x¯ such that y~∈P(x¯) since P is strict. By the assumption (iii), x¯⪯y~, which is absurd. Thus P(x¯)={x¯}.
(iii)⇒(ii) Let P=A, where A(x) is defined as (27). Then for any x∈X, x⪯y,∀y∈P(x); that is, the assumption of (iii) is satisfied. Consequently, P(x¯)={x¯} for some x¯∈X and so (ii) holds.
(ii)⇒(i) Define A(x) as (27). Then A(x)={y∈X:x⪯y}, and there exists x¯∈X such that A(x¯)={x¯} by (ii). This implies that x¯ is a maximal element on (X,⪯).Theorem 25.
LetS:X→2X be defined as (5) and x^∈X be any given point with S(x^)≠∅. Then the results below are equivalent.(i)
(Ekelend variational principle for(GVEP1)) There exists x¯∈S(x^) such that (53) holds.(ii)
(Existence result of solutions of(GVEP1)) If, for any x∈S(x^),
(54)f(x,y)∩(-D)≠∅forsomey∈Ximplies that there existsy∈X with y≠x such that
(55)f(x,y)∩(-w(x,y)e-D)≠∅,then(GVEP1) has a solution x¯∈S(x^).(iii)
(Caristi-kirk type fixe-point Theorem) LetP:X→2X be a strict, set-valued mapping such that, for any x∈S(x^), there exists y∈P(x) satisfying (54). Then there exists x¯∈S(x^) such that x¯∈P(x¯).(iv)
(Oettli-Théra type Theorem) LetΦ⊂X. If, for every x∈S(x^)∖Φ, there exists y∈X with y≠x such that (54) holds, then S(x^)∩Φ≠∅.Proof.
(i)⇒(iv) Let x¯∈S(x^) satisfy (53). If x¯∉Φ, then there exists y∈X with y≠x such that f(x¯,y)∩(-w(x¯,y)e-D)≠∅ holds by the hypothesis of (iv). This is absurd by (53) and so x¯∈S(x^)∩Φ.
(iv)⇒(i) Let (iv) hold and define Φ={x∈X:S(x)=∅}. If x∉Φ, then S(x)≠∅; that is, there exists y∈X with y≠x such that (54) holds; that is, the assumption of (iv) is satisfied. Thus, S(x^)∩Φ≠∅. By taking x¯∈S(x^)∩Φ, x¯ satisfies (53); that is, (i) is true.
(ii)⇒(iv) This conclusion is argued by contradiction. Assume that S(x^)∩Φ=∅. If x∈S(x^), then x∉Φ and so there exists y∈X with y≠x such that (54) holds by the hypothesis of (iv). Thus the assumption (ii) is satisfied and so there exists x¯∈S(x^) such that
(56)f(x¯,y)∩(-D)=∅,∀y∈X,
which implies that
(57)f(x¯,y)∩(-w(x¯,y)e-D)=∅,∀y∈X
by w(x¯,y)≥0 and e∈intD. This contradicts with (54).
(iv)⇒(ii) Let Φ={x∈X:f(x,y)∩(-D)=∅,∀y∈X}. By the assumption of (ii), for any x∈S(x^)∖Φ, there exists y∈X with y≠x such that (54) holds. This is just the assumption of (iv). Choosing x¯∈S(x^)∩Φ, we have that x¯∈S(x^) and f(x¯,y)∩(-D)=∅,∀y∈X.
(iii)⇒(iv) Define P:X→2X as P(x)={y∈X:y≠x}. Argue this by contradiction. If S(x^)∩Φ=∅, then the assumption of (iii) holds by the assumption of (iv). Hence, P has a fixed point on X, which is absurd by the definition of P.
(iv)⇒(iii) Let Φ={x∈X:x∈P(x)}. The assumption of (iv) is deduced straight by the assumption of (iii). Resultingly, S(x^)∩Φ≠∅; that is, x¯∈P(x¯) for some x¯∈S(x^).Remark 26.
Suppose thatf is a single-valued mapping. Theorem 25 reduces Theorem 5.1 in [21] if f(x,x)=0Y,∀x∈X hold instead of (a3), and if, further, (a4) hold and fx is (e,D)-lower semicontinuous for each x∈X (namely, {y∈X:f(x,y)∈re-D} is closed for each r∈ℝ and each x∈X).



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*Source: 267585-2013-08-25.xml*