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

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.

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].

Let ℝ, 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(0_𝑌), there exists 𝜆>0 such that 𝐴⊂𝜆𝑈+𝐷.

Throughout this paper, unless otherwise specified, let 𝑋 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→2^𝑌 be a strict, set-valued mapping, 𝑓_𝑥(⋅)=𝑓(𝑥,⋅) for each 𝑥∈𝑋 and 𝑓^𝑦(⋅)=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: tofind𝑥∈𝑋suchthat𝑓∩()()𝑥,𝑦−𝐷=∅,∀𝑦∈𝑋,GVEP1tofind𝑥∈𝑋suchthat𝑓∩()()𝑥,𝑦−int𝐷=∅,∀𝑦∈𝑋.GVEP2

This paper is divided into five sections. In Section 2, 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

Let 𝑋 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 element of 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.

Let 𝑋 and 𝑌 be topological spaces. A real-valued function 𝑔∶𝑋→ℝ is said to be upper semicontinuous on 𝑋 if {𝑢∈𝑋∶𝑔(𝑢)<𝜆} is open for each 𝜆∈ℝ; to be lower semicontinuous on 𝑋, {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→2^𝑌 is said to be upper semicontinuous at 𝑢₀∈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 at 𝑢₀∈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) on 𝑋, if 𝐺 is upper semicontinuous (resp., lower semicontinuous) at each 𝑢∈𝑋; to be continuous at 𝑢₀∈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]). Let 𝑋 and 𝑌 be Hausdorff topological spaces and 𝑆∶𝑋→2^𝑌 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. Let 𝑋 be a nonempty set. A function 𝑑∶𝑋×𝑋→ℝ₊ is said to be a quasimetric on 𝑋 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. 𝑥_𝑘→𝑥 means 𝑑(𝑥_𝑘,𝑥)→0. {𝑥_𝑘}⊂𝑋 is called a Cauchy sequence on (𝑋,𝑑) if, for any 𝜀>0, there exists 𝐾∈ℕ such that 𝑑(𝑥_𝑘,𝑥_𝑙)<𝜀,∀𝑙>𝑘>𝐾.  (𝑋,𝑑) is said to be complete if, for each Cauchy sequence {𝑥_𝑘}⊂𝑋, there exists 𝑥∈𝑋 such that 𝑥_𝑘→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 fixed 𝑥∈𝑋, 𝑦↦𝑤(𝑥,𝑦) is lower semicontinuous;(w3) for any 𝜖>0, there exists 𝛿>0 such that 𝑤(𝑥,𝑦)≤𝛿 and 𝑤(𝑥,𝑧)≤𝛿 imply that 𝑑(𝑦,𝑧)≤𝜖.

The 𝑊-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]). Let 𝑤 be a 𝑊-distance on a quasimetric space (𝑋,𝑑), let {𝑥_𝑘},{𝑦_𝑘}⊂𝑋 and {𝛼_𝑘},{𝛽_𝑘}⊂ℝ₊ with 𝛼_𝑘,𝛽_𝑘→0 as 𝑘→+∞, and let 𝑥,𝑦,𝑧∈𝑋. Then the following assertions are true:(i)if 𝑤(𝑥_𝑘,𝑦)≤𝛼_𝑘 and 𝑤(𝑥_𝑘,𝑧)≤𝛽_𝑘,∀𝑘∈ℕ, then 𝑦=𝑧. In particular, if 𝑤(𝑥,𝑦)=0 and 𝑤(𝑥,𝑧)=0, then 𝑦=𝑧;(ii)if 𝑤(𝑥_𝑘,𝑥_𝑙)≤𝛼_𝑘,∀𝑙>𝑘, then {𝑥_𝑘} is a Cauchy sequence;(iii)if 𝑡>0, then 𝑡𝑤 is also a 𝑊-distance on (𝑋,𝑑).

Let 𝑋 be a topological space and 𝑌 be a topological vector space in the rest of this section.

Since each compact subset in 𝑌 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. Let 𝐺∶𝑋→2^𝑌 be a strict and compact-valued mapping. A generalized nonlinear scalarization function 𝜁_𝐺∶𝑋→ℝ of 𝐺 is defined by 𝜁_𝐺(𝑢)=min{𝜆∈ℝ∶𝐺(𝑢)∩(𝜆𝑒−𝐷)≠∅},∀𝑢∈𝑋.(1)

According to Proposition 3.1 in [28], we have the following.

Lemma 6. Let 𝐺∶𝑋→2^𝑌 be a strict and compact-valued mapping. The following assertions are true for each 𝜆∈ℝ and 𝑢∈𝑋:(i)𝜁_𝐺(𝑢)<𝜆⇔𝐺(𝑢)∩(𝜆𝑒−int𝐷)≠∅. (ii)𝜁_𝐺(𝑢)≤𝜆⇔𝐺(𝑢)∩(𝜆𝑒−𝐷)≠∅.

Lemma 7. Let 𝐺∶𝑋→2^𝑌 be a strict and compact-valued mapping.(i)If 𝐺 is lower semicontinuous on 𝑋, then 𝜁_𝐺 is upper semicontinuous on 𝑋;(ii)If 𝐺 is upper semicontinuous on 𝑋, then 𝜁_𝐺 is lower semicontinuous on 𝑋.

Proof. This proof is completed by letting 𝐺(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 𝜁_𝑓𝑥(𝑧)≤𝜁_𝑓𝑥𝑦+𝜁_𝑓𝑦(𝑧),∀𝑥,𝑦,𝑧∈𝑋.(2)

Proof. For each 𝑥,𝑦,𝑧∈𝑋, it follows from Definition 5 that 𝑢∈𝑓(𝑥,𝑦)∩(𝜁_𝑓𝑥(𝑦)𝑒−𝐷) and 𝑣∈𝑓(𝑦,𝑧)∩(𝜁_𝑓𝑦(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 𝑤∈𝜁_𝑓𝑥𝑦𝑒−𝐷+𝜁_𝑓𝑦(𝑧)𝜁𝑒−𝐷−𝐷⊂_𝑓𝑥𝑦+𝜁_𝑓𝑦(𝑧)𝑒−𝐷.(3) Therefore, 𝑓(𝑥,𝑧)∩((𝜁_𝑓𝑥(𝑦)+𝜁_𝑓𝑦(𝑧))𝑒−𝐷)≠∅ by Lemma 6(ii) and 𝜁_𝑓𝑥(𝑧)≤𝜁_𝑓𝑥(𝑦)+𝜁_𝑓𝑦(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 𝑆𝑦⊂𝑆(𝑥),∀𝑦∈𝑆(𝑥),∀𝑥∈𝑋,(4) where 𝑆∶𝑋→2^𝑋 is defined by 𝑆(𝑥)=𝑦∈𝑋∶𝑦≠𝑥,𝑓𝑥,𝑦∩−𝑤𝑥,𝑦𝑒−𝐷≠∅,∀𝑥∈𝑋.(5)(ii) Besides (a1)–(a3), if(a4) for each 𝑥∈𝑋,𝑓_𝑥(⋅) is upper semicontinuous on 𝑋;(a5) for each 𝑥∈𝑋,𝑓_𝑥(𝑋)∩(𝑏−𝐷)=∅ for some 𝑏∈𝑌,
then, for each 𝑥∈𝑋 with 𝑆(𝑥)≠∅, there exists 𝑥∈𝑆(𝑥) such that 𝑆(𝑥)=∅.

Proof. Obviously, by (a1) and Lemma 6(ii), 𝑆(𝑥)=𝑦∈𝑋∶𝑦≠𝑥,𝜁_𝑓𝑥𝑦+𝑤𝑥,𝑦≤0.(6) (i) For each 𝑥∈𝑋 and 𝑦∈𝑆(𝑥), if 𝑆(𝑦)=∅, then the conclusion holds trivially. Otherwise, for each 𝑧∈𝑆(𝑦), 𝜁_𝑓𝑥𝑦+𝑤𝑥,𝑦≤0,𝜁_𝑓𝑦(𝑧)+𝑤𝑦,𝑧≤0,(7) and so 𝜁_𝑓𝑥(𝑧)+𝑤(𝑥,𝑧)≤𝜁_𝑓𝑥𝑦+𝜁_𝑓𝑦(𝑧)+𝑤𝑥,𝑦+𝑤𝑦,𝑧≤0(8) according to Lemma 8. Moreover, by 𝑤(𝑥,𝑦)≥0 and 𝑤(𝑦,𝑧)≥0, 𝜁_𝑓𝑥𝑦≤0,𝜁_𝑓𝑦(𝑧)≤0.(9) Now claim that 𝑧≠𝑥. Otherwise, 𝜁_𝑓𝑥(𝑧)≥0 by (a3) and Lemma 6(i). Thus, 𝑤𝑥,𝑦=0,(10) since 𝑤𝑥,𝑦≤𝜁_𝑓𝑥(𝑧)+𝑤𝑥,𝑦≤𝜁_𝑓𝑥𝑦+𝑤𝑥,𝑦+𝜁_𝑓𝑦(𝑧)≤0(11) 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) For 𝑏∈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 𝑓_𝑥(𝑋)∩(𝑟𝑒−𝐷)=∅. It follows form Lemma 4(ii) that 𝑣(𝑥)=inf𝜁_𝑓𝑥𝑦∶𝑦∈𝑆(𝑥)>−∞,∀𝑥∈𝑋.(12) Then, for each 𝑥₀=𝑥∈𝑋, 𝑣𝑥₀=inf𝜁_𝑓𝑥0𝑦∶𝑦∈𝑆𝑥₀>−∞.(13) For each 𝑘∈ℕ, let 𝑥_𝑘−1 satisfy that 𝑣(𝑥_𝑘−1)>−∞ and take 𝑥_𝑘∈𝑆(𝑥_𝑘−1) such that 𝜁_{𝑓𝑥𝑘−1}𝑥_𝑘≤𝑣𝑥_𝑘−1+12^𝑘.(14) If 𝑆(𝑥_𝑘)=∅ for some 𝑘∈ℕ, this conclusion holds by letting 𝑥=𝑥_𝑘. Now consider 𝑆(𝑥_𝑘)≠∅,∀𝑘∈ℕ. The rest proof is divided into four steps as follows.
(a) Show that 𝑥_𝑘→𝑥 as 𝑘→+∞ for some 𝑥∈𝑋. Indeed, for each 𝑘∈ℕ, by (2), 𝑣𝑥_𝑘=inf𝜁_𝑓𝑥𝑘𝑦∶𝑦∈𝑆𝑥_𝑘≥inf𝜁_{𝑓𝑥𝑘−1}𝑦−𝜁_{𝑓𝑥𝑘−1}𝑥_𝑘∶𝑦∈𝑆𝑥_𝑘−1=𝑣𝑥_𝑘−1−𝜁_{𝑓𝑥𝑘−1}𝑥_𝑘1≥−2^𝑘,(15) which implies that 𝑤𝑥_𝑘,𝑦≤−𝜁_𝑓𝑥𝑘𝑦≤−𝑣𝑥_𝑘≤12^𝑘,∀𝑦∈𝑆𝑥_𝑘,(16) and so 𝑤𝑥_𝑘,𝑥_𝑙≤^𝑙−1_𝑖=𝑘𝑤𝑥_𝑖,𝑥_𝑖+1≤^𝑙−1_𝑖=𝑘12^𝑖≤12^𝑘−1,∀𝑙>𝑘.(17) Therefore, {𝑥_𝑘} is a Cauchy sequence by Lemma 4(ii), and so there exists 𝑥∈𝑋 such that 𝑥_𝑘→𝑥 as 𝑘→+∞ by the completeness of (𝑋,𝑑). In addition, according to (w2) 𝑤𝑥_𝑘,𝑥≤liminf_𝑙→+∞𝑤𝑥_𝑘,𝑥_𝑙≤12^𝑘−1,∀𝑘∈ℕ.(18) (b) Show that 𝑥∈⋂^+∞_𝑘=0𝑆(𝑥_𝑘). In fact, for each 𝑘∈ℕ, 𝜁_𝑓𝑥𝑘𝑥_𝑙+𝑤𝑥_𝑘,𝑥_𝑙≤^𝑙−1_𝑖=𝑘𝜁_𝑓𝑥𝑖𝑥_𝑖+1+𝑤𝑥_𝑖,𝑥_𝑖+1≤0(19) by (2). Since, for each 𝑥∈𝑋,  𝜁_𝑓𝑥(⋅) is lower semicontinuous by (a1), (a4), and Lemma 7(ii), 𝜁_𝑓𝑥𝑘(𝑥)+𝑤𝑥_𝑘,𝑥≤0(20) by letting 𝑙→+∞ in (19). Hence, 𝜁_𝑓𝑥𝑘(𝑥)≤0,∀𝑘∈ℕ.(21) Now it is enough to prove that 𝑥_𝑘≠𝑥,∀𝑘∈ℕ by (20). Indeed, if 𝑥_𝑘0=𝑥 for some 𝑘₀∈ℕ, then 𝜁_{𝑓𝑥𝑘0}(𝑥_𝑘0)=𝜁_{𝑓𝑥𝑘0}(𝑥)≥0 in view of (a3) and Lemma 6(i). By applying (21) and adopting the same argument of the proof of (10), 𝑤𝑥_𝑘0,𝑥_𝑘0+1=0.(22) Thus, 𝜁_{𝑓𝑥𝑘0}𝑥_𝑘0+1=𝜁_{𝑓𝑥𝑘0}𝑥_𝑘0+1+𝑤𝑥_𝑘0,𝑥_𝑘0+1≤0.(23) It follows form 0≤𝜁_{𝑓𝑥𝑘0}(𝑥)≤𝜁_{𝑓𝑥𝑘0}(𝑥_𝑘0+1)+𝜁_{𝑓𝑥𝑘0+1}(𝑥) that 𝜁_{𝑓𝑥𝑘0+1}(𝑥)≥−𝜁_{𝑓𝑥𝑘0}(𝑥_𝑘0+1)≥0. Hence, 𝑤(𝑥_𝑘0+1,𝑥_𝑘0+2)=0 by (21) and the similar argument of (10), which, together with 𝑤(𝑥_𝑘0,𝑥_𝑘0+2)=0, implies 𝑥_𝑘0+1=𝑥_𝑘0+2 in virtue of (22) and Lemma 4(i). This contradicts with 𝑥_𝑘0+2∈𝑆(𝑥_𝑘0+1). Thus, 𝑥∈𝑆(𝑥_𝑘),∀𝑘∈ℕ. Clearly, 𝑥∈𝑆(𝑥₀).
(c) Show that ⋂^+∞_𝑘=1𝑆(𝑥_𝑘)={𝑥}. Indeed, for any 𝑥∈⋂^+∞_𝑘=1𝑆(𝑥_𝑘), 𝑥≠𝑥_𝑘, and 𝜁_𝑓𝑥𝑘(𝑥)+𝑤(𝑥_𝑘,𝑥)≤0,∀𝑘∈ℕ. As a result, 𝑤𝑥_𝑘,𝑥≤−𝜁_𝑓𝑥𝑘(𝑥)≤−𝑣𝑥_𝑘≤12^𝑘,∀𝑘∈ℕ,(24) which, together with (18), implies that 𝑥=𝑥 by Lemma 4(i).
(d) Show that 𝑥∈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. Define 𝐴∶𝑋→2^𝑋 as 𝐴(𝑥)=𝑦∈𝑋∶𝑥=𝑦𝑜𝑟𝑓𝑥,𝑦∩−𝜀𝑤𝑥,𝑦𝑒−𝐷≠∅,∀𝑥∈𝑋.(25) If (a1)–(a5) hold, then, for any 𝜀>0 and for any 𝑥∈𝑋, there exists 𝑥∈𝑋 such that (i)𝐴(𝑥)={𝑥};(ii)𝑓(𝑥,𝑦)∩(−𝜀𝑤(𝑥,𝑦)𝑒−𝐷)=∅,∀𝑦∈𝑋 with 𝑦≠𝑥;(iii)𝑓(𝑥,𝑥)∩(−𝜀𝑤(𝑥,𝑥)𝑒−𝐷)≠∅ if, further, 𝑆(𝑥)≠∅, where 𝑆∶𝑋→2^𝑋 defined as 𝑆(𝑥)=𝑦∈𝑋∶𝑥≠𝑦,𝑓𝑥,𝑦∩−𝜀𝑤𝑥,𝑦𝑒−𝐷≠∅,∀𝑥∈𝑋.(26)

Proof. Since 𝑤_𝜀(𝑥,𝑦)=𝜀𝑤(𝑥,𝑦) is another 𝑊-distance by Lemma 4(iii), without loss of generality, we set 𝜀=1. Then 𝐴(𝑥)=𝑦∈𝑋∶𝑥=𝑦or𝑓𝑥,𝑦∩−𝑤𝑥,𝑦𝑒−𝐷≠∅,∀𝑥∈𝑋,(27) 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 𝑓(𝑥,𝑥)∩(−𝜀𝑤(𝑥,𝑥)𝑒−𝐷)≠∅,(28) 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 that 𝑓_𝑥(𝑋) is compact for each 𝑥∈𝑋 by (a1), (a4), (a6), and Lemma 1. The rest proof is divided into three steps.
(a) For any 𝜆,𝜇∈ℝ with 𝜆<𝜇, 𝜆𝑒−𝐷⊂𝜇𝑒−𝜇−𝜆𝑒−𝐷⊂𝜇𝑒−int𝐷−𝐷⊂𝜇𝑒−int𝐷.(29) (b) For any 𝑢∈𝑌, there exists 𝜆_𝑢∈ℝ such that 𝑢∉𝜆_𝑢𝑒−𝐷. In fact, if 𝑢∈𝜆𝑒−𝐷,∀𝜆∈ℝ, then 𝑢∈𝜆𝑒−int𝐷,∀𝜆∈ℝ by (a), which implies {𝜆𝑒−𝑢∶𝜆∈ℝ}={−𝜆𝑒−𝑢∶𝜆∈ℝ}⊂int𝐷.(30) 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 some 𝑟∈ℝ, 𝑓_𝑥(𝑋)∩(𝑟𝑒−𝐷)=∅ and (a5) holds. Indeed, for each 𝑢∈𝑓_𝑥(𝑋), there exists 𝜆_𝑢∈ℝ such that 𝑢∉𝜆_𝑢𝑒−𝐷, and so 𝑁_𝑢∩(𝜆_𝑢𝑒−𝐷)=∅ for some 𝑁_𝑢∈𝒩(𝑢). By the completeness of 𝑓_𝑥(𝑋), there exists 𝑢₁,…,𝑢_𝑛∈𝑓_𝑥(𝑋) such that 𝑓_𝑥(𝑋)=∪{𝑁_𝑢𝑖∶1≤𝑖≤𝑛}. Taking 𝑟=min{𝜆_𝑢𝑖∶1≤𝑖≤𝑛}, we have 𝑓_𝑥(𝑋)∩(𝑟𝑒−𝐷)=∅.

Apparently, we have the following by applying Theorem 10.

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 Theorem 10, Corollary 11, or Corollary 12, for each 𝜀>0, there exists 𝑥_𝜀∈𝑋 such that 𝑓𝑥_𝜀,𝑦∩−𝜀𝑤𝑥_𝜀,𝑦𝑒−int𝐷=∅,∀𝑦∈𝑋.(31)

In fact, it is obvious that 𝑓(𝑥_𝜀,𝑦)∩(−𝜀𝑤(𝑥_𝜀,𝑦)𝑒−int𝐷)=∅,∀𝑦∈𝑋 with 𝑦≠𝑥_𝜀. For 𝑦=𝑥_𝜀, if 𝑓𝑥_𝜀,𝑥_𝜀∩−𝜀𝑤𝑥_𝜀,𝑥_𝜀𝑒−int𝐷≠∅,(32) then 𝑓(𝑥_𝜀,𝑥_𝜀)∩(−int𝐷)≠∅, which contradicts with (a3) or (a7).

Remark 14. (a) The condition: (a8) there exists 𝑏̃∈𝑌 such that 𝑓_𝑥(𝑋)∩(𝑏̃−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∈𝑏̃−int𝐷, then 𝑏−𝐷⊂𝑏̃−int𝐷−𝐷⊂𝑏̃−int𝐷, which guarantees (a5).
(b) The condition:(a9) for each 𝑥∈𝑋, 0_𝑌∈𝑓(𝑥,𝑥) or 𝑓(𝑥,𝑥)={0_𝑌},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. Let 𝑋=[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→2^𝑌 as 𝑓𝑥,𝑦=𝑥,𝑦−𝑥,−𝑥,𝑦−𝑥,∀𝑥,𝑦∈𝑋.(33) By computing simply, (a1)–(a5) hold. But (a9) is false since 0_𝑌∉𝑓(𝑥,𝑥) unless 𝑥=0.

Example 16. Let 𝑋=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→2^𝑌 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 Theorem 10.

Theorem 17. Let 𝜀,𝜆>0 be given. If, besides (a1)–(a5),
(a10) for some 𝑥∈𝑋 with 𝑆(𝑥)≠∅, 𝑓𝑥,𝑦∩(−𝜀𝑒−𝐷)=∅,∀𝑦∈𝑋,(34) 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)∩(−(𝜀/𝜆)𝑤(𝑥,𝑦)𝑒−𝐷)=∅,∀𝑦∈𝑋 with 𝑦≠𝑥.

Proof. Let 𝑥∈𝑋 be the point provided by Theorem 10 with 𝜀/𝜆 instead of 𝜀. Then (iii) is obtained by the conclusion (ii) in Theorem 10. In addition, in view of the conclusion (i) in Theorem 10, 𝜁_𝑓𝑥(𝑥)+𝜀𝜆𝑤(𝑥,𝑥)≤0.(35) It follows from (34)-(35), (a10), and Lemma 6(ii) that −𝜀≤𝜁_𝑓𝑥(𝑥)≤−(𝜀/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 −𝜀𝑢∈𝜆𝑤(𝑥,𝑥)+𝐷.(36) Taking 𝑥=𝑧=𝑥 and 𝑦=𝑥 in (a2), we have 𝑓(𝑥,𝑥)+𝑓(𝑥,𝑥)⊂𝑓(𝑥,𝑥)+𝐷⊂𝐷.(37) Then 𝑓(𝑥)𝜀𝑥,⊂−𝑢+𝐷⊂𝜆𝑤(𝑥,𝑥)+𝐷+𝐷⊂𝐷.(38) 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)={0_𝑌}) in Theorems 17, then the condition (a10) can be weakened as follows: for some 𝑥∈𝑋 with 𝑆(𝑥)≠∅, 𝑓(𝑥,𝑥)∩−𝜀𝑒−𝐷⧵0_𝑌=∅,∀𝑥∈𝑋,(39) where 𝑆 is defined as (5).
In fact, we only prove (ii) by the proof of Theorem 17. In view of Theorem 10(iii), 𝑓(𝑥,𝑥)∩−𝜀𝜆𝑤(𝑥,𝑥)𝑒−𝐷≠∅.(40) Then there exist 𝑦₀∈𝑓(𝑥,𝑥) and 𝑑₀∈𝐷 such that 𝑦₀=−(𝜀/𝜆)𝑤(𝑥,𝑥)𝑒−𝑑₀. By (39), −𝜀𝜆𝑤(𝑥,𝑥)𝑒−𝑑₀∉−𝜀𝑒−𝐷⧵0_𝑌,(41) or equivalently, 𝜀𝜆(𝑤(𝑥,𝑥)−𝜆)𝑒+𝑑₀∉𝐷⧵0_𝑌.(42) If 𝑤(𝑥,𝑥)>𝜆, then (𝜀/𝜆)(𝑤(𝑥,𝑥)−𝜆)𝑒+𝑑₀∈𝐷 and so (𝜀/𝜆)(𝑤(𝑥,𝑥)−𝜆)𝑒+𝑑₀=0_𝑌, which implies that 𝑑₀=−(𝜀/𝜆)(𝑤(𝑥,𝑥)−𝜆)𝑒∈−int𝐷. Thereupon, 𝑑₀=0_𝑌 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. Let 𝑋 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 each 𝑥∈𝑋, 𝑥→−_𝑘^𝜏𝑥 as 𝑘→+∞ implies the boundedness of {𝑑(𝑥_𝑘,𝑥)∶𝑘∈ℕ};(b3)𝑓^𝑦 is 𝜏-lower semicontinuous on 𝑋 for each 𝑦∈𝑋,then (GVEP2) has a solution.

Proof. If 𝑤=𝑑 and 𝜀=1/𝑘 in (31), then there exists 𝑥_𝑘∈𝑋 such that 𝑓𝑥_𝑘∩−1,𝑦𝑘𝑑𝑥_𝑘,𝑦𝑒−int𝐷=∅,∀𝑦∈𝑋.(43) Thus 𝜁_𝑓^𝑦(𝑥_𝑘)≥−(1/𝑘)𝑑(𝑥_𝑘,𝑦) by Lemma 6(i). Without loss of generality, let 𝑥→−_𝑘^𝜏𝑥 by (b1). Since 𝜁_𝑓^𝑦(⋅) is 𝜏-upper semicontinuous for each 𝑦∈𝑋 by (b3) and Lemma 7(i), 𝜁_𝑓^𝑦(𝑥)≥limsup_𝑘→+∞𝜁_𝑓^𝑦𝑥_𝑘≥limsup_𝑘→+∞−1𝑘𝑑𝑥_𝑘,𝑦=0,∀𝑦∈𝑋,(44) 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, let 𝑋=ℝ and 𝑑(𝑥,𝑦)=|𝑥−𝑦|,∀𝑥,𝑦∈𝑋. For the trivial topology 𝜏={∅,𝑋}, 𝑋 is 𝜏-compact. For 𝑥_𝑘=𝑘, it is clear that 𝑥→−_𝑘^𝜏0, but {𝑑(𝑥_𝑘,0)∶𝑘∈ℕ} is unbounded.

Corollary 21. If (a1)-(a2), (a4), and (a6)-(a7) and(a11)𝑓^𝑦 is lower semicontinuous on 𝑋 for each 𝑦∈𝑋,hold, then (GVEP2) has a solution.

Proof. Clearly, (a6) implies (a5). Taking 𝜏=𝒯 in Theorem 19, (b1) and (b3) hold by (a6) and (a11), respectively, and (b2) holds trivially since 𝑑(𝑥_𝑘,𝑥)→0 when 𝑥_𝑘→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 of 𝒯 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) each 𝒯-closed bounded ball on 𝑋 is 𝜏-compact,(b5) for given point 𝑥∈𝑋, there exists a 𝜏-compact set 𝐶⊂𝑋 satisfying that, for each 𝑥∈𝐶 and {𝑥_𝑘}⊂𝐶, 𝑥→−_𝑘^𝜏𝑥 implies the boundedness of {𝑑(𝑥_𝑘,𝑥)∶𝑘∈ℕ} and for each 𝑥∈𝑋⧵𝐶, 𝑑𝑦,𝑥<𝑑(𝑥,𝑥),𝑓𝑥,𝑦∩(−𝐷)≠∅(45) for some 𝑦∈𝑋,(b6) for each 𝑥∈𝑋, 𝐿(𝑥)={𝑦∈𝑋∶𝑓(𝑥,𝑦)∩(−𝐷)≠∅} is 𝜏-closed,(b7) for each 𝑦∈𝑋, 𝑑(⋅,𝑦) is 𝜏-lower semicontinuous, then (GVEP2) has a solution.

Proof. Define 𝐴∶𝑋→2^𝑌 as 𝐴(𝑥)=𝑦∈𝑋∶𝑑𝑦,𝑥≤𝑑(𝑥,𝑥),𝜁_𝑓𝑥𝑦≤0.(46) We see the following. (a) For each 𝑥∈𝑋, 𝐴(𝑥)≠∅ since 𝑥∈𝐴(𝑥) by (a7). (b) For each 𝑥∈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 the 𝜏-compact set 𝐶 as 𝑋 in Theorem 19, there exists 𝑥_𝐶∈𝐶 such that 𝑓𝑥_𝐶,𝑦∩(−int𝐷)=∅,∀𝑦∈𝐶.(47) Now argue the conclusion by contradiction. Assume that there exists 𝑦∈𝑋⧵𝐶 such that 𝑓𝑥_𝐶,𝑦∩(−int𝐷)≠∅.(48) Let 𝛼=min𝑑𝑦,x∶𝑦∈𝐴𝑦.(49)
Now assert that 𝐴𝑦∩𝐶=∅.(50) 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 each 𝑦∈𝑋, 𝑓^𝑦 is weakly lower semicontinuous,(b9) there exists 𝑟>0 such that for each 𝑥∈𝑋⧵𝑋_𝑟, 𝑦∈𝑋 can be selected such that ‖𝑦‖<‖𝑥‖ and 𝑓(𝑥,𝑦)∩(−𝐷)≠∅, where 𝑋_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟},then (GVEP2) has a solution.

Proof. Denote 𝒯 and 𝜏 by the strong topology and the weak topology on 𝑋, respectively, and define 𝐴∶𝑋→2^𝑋 as 𝐴(𝑥)=𝑦∈𝑋∶𝑦≤‖𝑥‖,𝜁_𝑓𝑥𝑦≤0,∀𝑥∈𝑋.(51) Letting 𝑥=0_𝑋, 𝐶=𝑋_𝑟, 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∶𝜁_𝑓𝑥(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: 𝑥⪯𝑦⟺𝑥=𝑦or𝑓𝑥,𝑦∩−𝑤𝑥,𝑦𝑒−𝐷≠∅.(52) 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 and 𝑋 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 𝑓𝑥,𝑦∩−𝑤𝑥,𝑦𝑒−𝐷=∅,∀𝑦∈𝑋𝑤𝑖𝑡ℎ𝑦≠𝑥.(53)(iii) (Caristi-kirk fixe-point theorem) Let 𝑃∶𝑋→2^𝑋 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. Let 𝑆∶𝑋→2^𝑋 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), 𝑓𝑥,𝑦∩(−𝐷)≠∅𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑦∈𝑋(54) implies that there exists 𝑦∈𝑋 with 𝑦≠𝑥 such that 𝑓𝑥,𝑦∩−𝑤𝑥,𝑦𝑒−𝐷≠∅,(55) then (GVEP1) has a solution 𝑥∈𝑆(𝑥).(iii) (Caristi-kirk type fixe-point Theorem) Let 𝑃∶𝑋→2^𝑋 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) Let Φ⊂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 𝑓𝑥,𝑦∩(−𝐷)=∅,∀𝑦∈𝑋,(56) which implies that 𝑓𝑥,𝑦∩−𝑤𝑥,𝑦𝑒−𝐷=∅,∀𝑦∈𝑋(57) 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→2^𝑋 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 that 𝑓 is a single-valued mapping. Theorem 25 reduces Theorem 5.1 in [21] if 𝑓(𝑥,𝑥)=0_𝑌,∀𝑥∈𝑋 hold instead of (a3), and if, further, (a4) hold and 𝑓_𝑥 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).

Acknowledgments

This research was supported by the Doctoral Fund of Innovation of Beijing University of Technology; the Science and Technology Development Project of Siping (2011005).