Vectorial Ekeland Variational Principles and Inclusion Problems in Cone Quasi-Uniform Spaces

Abstract

Some new vectorial Ekeland variational principles in cone quasi-uniform spaces are proved. Some new equivalent principles, vectorial quasivariational inclusion principle, vectorial quasi-optimization principle, vectorial quasiequilibrium principle are obtained. Also, several other important principles in nonlinear analysis are extended to cone quasi-uniform spaces. The results of this paper extend, generalize, and improve the corresponding results for Ekeland's variational principles of the directed vectorial perturbation type and other generalizations of Ekeland's variational principles in the setting of 𝐹-type topological space and quasi-metric spaces in the literatures. Even in usual real metric spaces, some of our results are new.

1. Introduction

Ekeland’s variational principle [1] is a forceful tool in nonlinear analysis, control theory, global analysis, and many others. In the last two decades, it has been further studied, extended, and applied to many fields in mathematics (see, e.g., [2–17, 19–21, 26, 28, 29] and the references therein). Especially, we want to emphasize that many generalizations of Ekeland’s variational principle to vector-valued functions have been recently obtained in [2–5, 8–10, 14–17] and the references therein. For example, in [2], the author proved some vectorial Ekeland’s variational principles for vector-valued functions defined on quasi-metric spaces. In [3, 5, 14], the authors proved some vectorial Ekeland’s variational principles for vector-valued functions defined on metric spaces. In [8], the authors proved some vector Ekeland’s variational principle in a 𝐹-type topological space. However, all of these results are essentially Ekeland’s principle of the directed vectorial perturbation type. But, the study for the case of nondirected vectorial perturbation type has just been started in [15]. On the other hand, variational inclusion problem is a very important problem and it contains many important problems such as complementarity problems, minimax inequalities, equilibrium problems, saddle point problems, optimization theory, bilevel problems, mathematical programs with equilibrium constraints, variational inequalities, and fixed point problems (see, e.g., [18–21] and the references therein).

In this paper, we first prove some nondirected vectorial perturbation type Ekeland’s variational principles in cone quasi-uniform spaces, then by using our new principles, we introduce some variational inclusion principles and prove the equivalence among these results and other equivalents of our principle.

The paper is organized as follows: in Section 2, some properties of cone, some new equivalent characterizations of the pseudo-nuclear cone, and the definition of cone quasi-uniform spaces are given. In Section 3, we prove some new Ekeland’s variational principles for both non-directed vectorial perturbation type and directed vectorial perturbation type in the setting of general topological vector spaces for vector-valued functions defined on complete cone quasi-uniform spaces. Some of new equivalent principles, vectorial quasi-variational inclusion principle, vectorial quasi-optimization principle, and vectorial quasi-equilibrium principle are introduced and proved, which have a wide practical background in quasi-variational inclusion problems, quasi-optimization problems, and quasi-equilibrium problems (see, e.g., [21]). In addition to these new equivalent principles, generalized Caristi-Kirk type coincidence point theorem for multivalued maps defined on cone quasi-uniform spaces, nonconvex maximal element theorem for the family of multivalued maps defined on cone quasi-uniform spaces, generalized vectorial Takahashi nonconvex minimization theorem, and Oettli-Théra type theorem defined on cone quasi-uniform spaces are also presented. The results of this paper not only give some new Ekeland’s principles of the non-directed vectorial perturbation type and some new equivalent principles, but also extend and generalize or improve many corresponding results for Ekeland’s variational principles of the directed vectorial perturbation type and other generalizations of Ekeland’s variational principles in the setting of 𝐹-type topological space and quasi-metric spaces in the literatures [2, 3, 5, 8, 14, 15]. Even in usual real metric spaces, some of our results are new.

2. Preliminaries

Let 𝐸(𝜏) be a topological vector space and 𝑉_𝜏(𝜃) (resp., 𝑉_𝜎(𝜃)) denote the 𝜃-neighborhood base with respect to the topology 𝜏 (resp., to the weak topology 𝜎). A subset 𝐾⊆𝐸 is called a closed convex cone if 𝐾 is closed, 𝐾+𝐾⊆𝐾 and 𝜆𝐾⊆𝐾 for all 𝜆∈[0,+∞). A quasiorder (i.e., a reflexive and transitive relation) on 𝐸 can be defined by 𝐾, that is, 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝐾. We write 𝑥<𝑦 whenever 𝑥≤𝑦 and 𝑦≠𝑥. The quasi-order ≤ is called a partial order if it is antisymmetric. Let 𝐸^∗ be the topological dual space of 𝐸 (i.e., 𝐸^∗ is the set of all continuous linear functions on 𝐸) and let 𝐾^∗ be the dual cone of 𝐾, that is, 𝐾^∗={𝑓∈𝐸^∗∶𝑓(𝑥)≥0,forall𝑥∈𝐾}.

We recall that a subset 𝐵⊂𝐸 is said to be 𝐾-saturated if 𝐵=[𝐵], where [𝐵] is defined by[𝐵]=(𝐵+𝐾)∩(𝐵−𝐾)=∪𝑥,𝑦∶𝑥∈𝐵,𝑦∈𝐵, (2.1) where [𝑥,𝑦]={𝑧∈𝐸∶𝑥≤𝑧≤𝑦}=(𝑥+𝐾)∩(𝑦−𝐾). The cone 𝐾 is said to be normal if there is a 𝜃-neighborhood base for 𝜏 consisting of 𝐾-saturated sets.

The following Lemma 2.1 is an elementary result (refer to [22–24]) characterizing the concept of normality that will be used in our proof.

Lemma 2.1. Let 𝐸(𝜏) be a topological vector space and 𝐾⊆𝐸 be a cone. Then the following propositions are equivalent.(1)𝐾 is a normal cone.(2)There exists a 𝜃-neighborhood base 𝑉_𝜏(𝜃) for 𝜏 consisting of sets 𝑉 for which 𝜃≤𝑥≤𝑦∈𝑉 implies 𝑥∈𝑉.(3)For any two nets {𝑥_𝑖}_𝑖∈𝐼 and {𝑦_𝑖}_𝑖∈𝐼 in 𝐸, if 𝜃≤𝑥_𝑖≤𝑦_𝑖 for all 𝑖∈𝐼 and {𝑦_𝑖}_𝑖∈𝐼 converges to zero for 𝜏, then {𝑥_𝑖}_𝑖∈𝐼 converges to zero for 𝜏.(4)For any 𝜏-neighborhood 𝑉 of zero, there exists a 𝜏-neighborhood 𝑊 of zero such that 𝜃≤𝑥≤𝑦∈𝑊 implies 𝑥∈𝑉.

The following notion of a pseudo-nuclear cone is a generalization of nuclear cone which has many applications in optimizations, fixed point theory, and the best approximation theory, and so forth.

Definition 2.2 (see [25]). Let 𝐸(𝜏) be a topological vector space (not necessarily locally convex) and 𝐾⊆𝐸 be a closed convex cone (not necessary pointed). If 𝑓∈𝐸^∗, then we denote 𝑓_𝜋={𝑥∈𝐸∶𝑓(𝑥)≤1}. A set of the kind 𝑓_𝜋∩𝐾 is called a 𝜃-pseudoslice of 𝐾, where 𝑓∈𝐸^∗, the dual space of 𝐸. We say that 𝐾 is pseudo-nuclear if each 𝜃-neighborhood in 𝐸 contains a 𝜃-pseudoslice of 𝐾.

To discuss the properties of pseudo-nuclear cone, we need the following existence theorem of a quasi-norms family which can determine the topology of 𝐸(𝜏).

Lemma 2.3 (see [26]). Let 𝐸(𝜏) be a topological vector space and 𝑈(𝜏) be a balanced 𝜃-neighborhood base for the topology 𝜏. Then there exist a directed set 𝐼 and a quasi-norms family {‖⋅‖_𝛼∶𝛼∈𝐼} such that(a)for any 𝑘∈𝑅, 𝛼∈𝐼 and 𝑥∈𝐸, ‖𝑘𝑥‖_𝛼=|𝑘|‖𝑥‖_𝛼;(b)for any 𝛼∈𝐼, there exists 𝜇∈𝐼 such that 𝛼≤𝜇 and ‖𝑥+𝑦‖_𝛼≤‖𝑥‖_𝜇+‖𝑦‖_𝜇 for any 𝑥,𝑦∈𝐸;(c)for any 𝛼,𝜇∈𝐼, if 𝛼≤𝜇, then ‖𝑥‖_𝛼≤‖𝑥‖_𝜇 for any 𝑥∈𝐸;(d){‖⋅‖_𝛼∶𝛼∈𝐼} and 𝑈(𝜏)determine the same topology 𝜏;(e)if 𝜏 is a 𝑇₀ topology (i.e., for any 𝑥≠𝜃, there exists a neighborhood of zero 𝑈∈𝑈(𝜏) such that 𝑥∉𝑈), then𝑥=𝜃⟺‖𝑥‖_𝛼=0,∀𝛼∈𝐼. (2.2)

The following lemma gives several equivalent conditions of pseudo-nuclearity.

Lemma 2.4. Let 𝐸(𝜏) be a topological vector space and 𝐾⊆𝐸 a cone. Then the following assertions are equivalent.(1)𝐾 is a pseudo-nuclear cone.(2)For all 𝑉∈𝑉_𝜏(𝜃), there exists 𝑊∈𝑉_𝜎(𝜃) such that 𝜃≤𝑥≤𝑦∈𝑊 implies 𝑥∈𝑉.(3)For all 𝑉∈𝑉_𝜏(𝜃), there exists 𝑓∈𝐾^∗ such that {𝑥∈𝐾∶𝑓(𝑥)≤1}⊂𝑉.(4)If {‖⋅‖_𝛼∶𝛼∈𝐼} is the family of quasi-norms which determines the topology of 𝐸(𝜏), then, for any 𝛼∈𝐼, there exists 𝑓_𝛼∈𝐾^∗ such that‖𝑥‖_𝛼≤𝑓_𝛼(𝑥),∀𝑥∈𝐾. (2.3)

Proof. It follows from Theorem  3.8 in [24] that (1) ⇒ (2).
(2) ⇒ (3) Since each neighborhood of zero with respect to the weak topology 𝜎 is also a neighborhood of zero with respect to the topology 𝜏, for any 𝑉∈𝑉_𝜎(𝜃), we have 𝑉∈𝑉_𝜏(𝜃). From (2), we know that there exists 𝑊∈𝑉_𝜎(𝜃) such that 𝜃≤𝑥≤𝑦∈𝑊 implies 𝑥∈𝑉. This and (4) of Lemma 2.1 imply that 𝐾 is a normal cone with respect to the weak topology 𝜎. Since the weak topology 𝜎 is defined by semi-norms family {|𝑓|∶𝑓∈𝐸^∗},   𝐸(𝜎) is a locally convex topological vector space. Therefore, 𝐸^∗=𝐾^∗−𝐾^∗ with respect to the weak topology 𝜎. From (2), it follows that, for any 𝑉∈𝑉_𝜏(𝜃), there exists 𝑊∈𝑉_𝜎(𝜃) such that 𝜃≤𝑥≤𝑦∈𝑊 implies 𝑥∈𝑉. Thus there exist 𝑓₁,…,𝑓_𝑚 and 𝜀>0 such that {𝑥∈𝐸∶|𝑓_𝑖(𝑥)|<𝜀,𝑖=1,…,𝑚}⊂𝑊. Also, there exist 𝑔_𝑖,ℎ_𝑖∈𝐾^∗ such that 𝑓_𝑖=𝑔_𝑖−ℎ_𝑖 for each 𝑖=1,…,𝑚. Let 𝑓=(2/𝜀)∑^𝑚_𝑖=1(𝑔_𝑖+ℎ_𝑖). Then 𝑓∈𝐾^∗ and 𝑥∈𝐾∶𝑓(𝑥)≤1=𝑥∈𝐾∶^𝑚_𝑖=1𝑔_𝑖+ℎ_𝑖(𝑥)≤𝜀2⊂𝑥∈𝐸∶𝑓_𝑖(𝑥)<𝜀,𝑖=1,…,𝑚⊂𝑊. (2.4) Thus {𝑥∈𝐾∶𝑓(𝑥)≤1}⊂𝑉, that is, (3) holds.
(3) ⇒ (4) Suppose that {‖⋅‖_𝛼∶𝛼∈𝐼} is the family of quasi-norms which determines the topology of 𝐸(𝜏). Then, for any 𝛼∈𝐼,   {𝑥∈𝐸∶‖𝑥‖_𝛼<1}∈𝑉_𝜏(𝜃). It follows from (3) that, for any 𝛼∈𝐼, there exists 𝑓_𝛼∈𝐾^∗ such that {𝑥∈𝐾∶𝑓_𝛼(𝑥)≤1}⊂{𝑥∈𝐸∶‖𝑥‖_𝛼<1}. If 𝑥∈𝐾 and 𝑓_𝛼(𝑥)>0, then ‖𝑥/𝑓_𝛼(𝑥)‖_𝛼<1, that is, ‖𝑥‖_𝛼<𝑓_𝛼(𝑥). If 𝑥∈𝐾 and 𝑓_𝛼(𝑥)=0, then, for any 𝑙∈𝑅⁺,  𝑓_𝛼(𝑙𝑥)=0, which implies that ‖𝑙𝑥‖_𝛼=𝑙‖𝑥‖_𝛼<1. Since 𝑙 is arbitrary, we know that ‖𝑥‖_𝛼=0 and so ‖𝑥‖_𝛼≤𝑓_𝛼(𝑥) for all 𝑥∈𝐾, that is, (4) is true.
(4) ⇒ (1) By Lemma 2.3(d), {‖⋅‖_𝛼∶𝛼∈𝐼} and 𝑈(𝜏) determine the same topology 𝜏. Then, for any 𝑉∈𝑉_𝜏(𝜃), there exist 𝛼₁,𝛼₂,…,𝛼_𝑚∈𝐼 and positive numbers 𝛿₁,𝛿₂,…,𝛿_𝑚 such that ⋂^𝑚_𝑖=1{𝑥∈𝐸∶‖𝑥‖_𝛼𝑖<𝛿_𝑖}⊂𝑉. By (4), there exists 𝑓_𝛼𝑖∈𝐾^∗(𝑖=1,2,…,𝑚) such that ‖𝑥‖_𝛼𝑖≤𝑓_𝛼𝑖(𝑥) for all 𝑥∈𝐾. Let 𝛿=min{𝛿₁,𝛿₂,…,𝛿_𝑚} and 𝑓(𝑥)=(2/𝛿)∑^𝑚_𝑖=1𝑓_𝛼𝑖(𝑥). Then 𝑓∈𝐸^∗ and 𝑥∈𝐸∶𝑓(𝑥)≤1∩𝐾=𝑥∈𝐾∶^𝑚_𝑖=1𝑓_𝛼𝑖(𝑥)≤𝛿2⊂^𝑚_𝑖=1𝑥∈𝐾∶𝑓_𝛼𝑖(𝑥⊂)<𝛿_𝑚𝑖=1𝑥∈𝐸∶‖𝑥‖_𝛼𝑖<𝛿_𝑖⊂𝑉. (2.5) This shows that (1) holds. This completes the proof.

To discuss Ekeland’s principles of the nondirected vectorial perturbation type, by enlightening of the work in [27], we define the following cone quasi-uniform space, which is an extension of the cone uniform space in [27]. About the discussion and applications of cone uniform space, one can refer to [15, 27–29].

Definition 2.5. Let 𝑋 be a nonempty set and (𝐷,≺) be a directed set. Let 𝐸 be a topological vector space and 𝐾⊆𝐸 be a cone. Let the family 𝒫=𝑑_𝜆∶𝑋×𝑋⟶𝐸∶𝜆∈𝐷 (2.6) satisfy the following conditions:(d1) for any 𝜆∈𝐷,   𝑑_𝜆(𝑥,𝑦)∈𝐾 for all 𝑥,𝑦∈𝑋;(d2)for any 𝜆∈𝐷,    𝑑_𝜆(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦;(d3)for any 𝜆∈𝐷,   there exists 𝜇∈𝐷 with 𝜆≺𝜇 such that𝑑_𝜆𝑥,𝑦≤𝑑_𝜇(𝑥,𝑧)+𝑑_𝜇𝑧,𝑦,∀𝑥,𝑦,𝑧∈𝑋. (2.7) Then 𝒫 is called a family of cone quasi-metrics on 𝑋 and (𝑋,𝑃) is called a cone quasi-uniform space.

Definition 2.6. Let {𝑥_𝑛} be a sequence in a cone quasi-uniform space (𝑋,𝑃).(1)The sequence {𝑥_𝑛} is said to be convergent to a point 𝑥∈𝑋 if, for any 𝜆∈𝐷 and 𝜃-neighborhood 𝑈 in 𝐸, there exists a positive integer 𝑁 such that 𝑑_𝜆(𝑥,𝑥_𝑛)∈𝑈 for any 𝑛≥𝑁, that is, lim_𝑛→∞𝑑_𝜆(𝑥,𝑥_𝑛)=𝜃, which is denoted by lim_𝑛→∞𝑥_𝑛=𝑥.(2)The sequence {𝑥_𝑛} is called a Cauchy sequence in 𝑋 if, for any 𝜆∈𝐷 and 𝜃-neighborhood 𝑈 in 𝐸, there exists a positive integer 𝑁 such that 𝑑_𝜆(𝑥_𝑛,𝑥_𝑚)∈𝑈 for any 𝑚>𝑛≥𝑁, that is, lim_{𝑚,𝑛→∞}𝑑_𝜆(𝑥_𝑛,𝑥_𝑚)=𝜃.(3)If every Cauchy sequence is convergent in (𝑋,𝑃), then (𝑋,𝑃) is called a sequentially complete cone quasi-uniform space.(4)For any net {𝑥_𝛼}_𝛼∈𝐼 in a cone quasi-uniform space (𝑋,𝑃), the convergence of the net and the Cauchy net can be defined similarly. If every Cauchy net is convergent in (𝑋,𝑃), then (𝑋,𝑃) is called a complete cone quasi-uniform space.

Remark 2.7. If we take 𝐸=𝑅, then the cone quasi-uniform space is a generalization of 𝐹-type topological spaces [30]. If 𝑃={𝑑∶𝑋×𝑋→𝐸}, then (𝑋,P) reduces to a cone quasi-metric space (𝑋,𝑑). The notions on convergence and completeness in a cone quasi-uniform space are complicated because of the lack of symmetry of the cone quasi-metric. Similarly, we can define the other kinds of convergence and completeness in a cone quasi-uniform space. For more details, the reader can refer to same discussions in the case of quasi-metric spaces [31].

Recall that a cone 𝐾 has a base [8] or is well based [11] if there exists a convex subset 𝐵 with 𝜃∉𝐵 such that 𝐾=∪_𝑟>0𝑟𝐵.

Lemma 2.8 (see [8]). Let 𝐸(𝜏) be a topological vector space and 𝐾⊂𝐸 be a closed convex cone. Then the cone 𝐾 has a base if and only if 𝐾^∗∘=𝑓∈𝐸^∗∶𝑓(𝑥)>0,∀𝑥∈𝐾⧵{𝜃}≠∅. (2.8)

Definition 2.9 (see [16]). Let (𝑋,≤) be a quasiordered set.(1)𝑋 is said to be totally ordered upperseparable (resp., totally ordered lower-separable) if, for any totally ordered nonempty subset 𝑀 of 𝑋, there exists an increasing sequence (resp., decreasing sequence) {𝑥_𝑛}⊂𝑀 such that, for any 𝑥∈𝑀, there exists 𝑥_𝑛0∈{𝑥_𝑛} satisfying 𝑥≤𝑥_𝑛0 (resp., 𝑥_𝑛0≤𝑥).(2)𝑋 is said to be totally ordered separable if 𝑋 is both totally ordered upperseparable and totally ordered lower-separable.

Lemma 2.10 (see [16]). Let (𝑋,≤) be a partially ordered set such that every increasing sequence has an upper bound, let (𝑌,≤_𝑌) be a totally ordered upper-separable (resp., totally ordered lower-separable) partially ordered set, and let Φ∶𝑋→𝑌 be an increasing (resp., decreasing) mapping. Then, for any 𝑥∈𝑋, there exists 𝑣≥𝑥 such that Φ(𝑦)=Φ(𝑣) for all 𝑦≥𝑣. Moreover, if Φ is strictly monotone, then 𝑣 is a maximal element.

3. Vectorial Ekeland’s Variational Principle

Let 𝐸(𝜏) be a topological vector space, let 𝐾 be a closed convex cone in 𝐸, let (𝑋,𝑃) be a complete cone quasi-uniform space, and let 𝜑∶𝐸→(0,+∞) be a nondecreasing function, that is, 𝑥≤𝑦 implies that 𝜑(𝑥)≤𝜑(𝑦). For a function 𝐹∶𝑋×𝑋→𝐸, we first give the following conditions:(H1)𝐹(𝑥,𝑥)=𝜃 for all 𝑥∈𝑋;(H2)𝐹(𝑥,𝑦)+𝐹(𝑦,𝑧)∈𝐹(𝑥,𝑧)+𝐾 for all 𝑥,𝑦,𝑧∈𝑋;(H3) for any fixed 𝑥,𝑥₀∈𝑋, the mapping 𝐹(𝑥,⋅)∶𝑋→𝐸 is bound from below and𝑦∈𝑋∶𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑦+𝑑_𝜆𝑥,𝑥𝑦≤𝜃,∀𝜆∈𝐷. (3.1)

Is a closed subset in 𝑋.

Lemma 3.1. Let 𝐸(𝜏) be a topological vector space with a 𝑇₀ topology 𝜏 and 𝐾⊂𝐸 be a closed convex pseudo-nuclear cone or a closed point convex cone (i.e.,  𝐾∩(−𝐾)={𝜃}). Let (𝑋,𝑃) be a complete cone quasi-uniform space and 𝐹∶𝑋×𝑋→𝐸 be a function satisfying (H1) and (H2). For any fixed 𝑥₀∈𝑋, one defines a relation ≼ on 𝑋 as follows: 𝑥≼𝑥^′⟺𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑥^′+𝑑_𝜆𝑥,𝑥^′≤𝜃,∀𝜆∈𝐷. (3.2) Then (𝑋,≼)is a partially ordered set.

Proof. It is clear that 𝑥≼𝑥. If 𝑥≼𝑦 and 𝑦≼𝑥, then we have 𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑦+𝑑_𝜆𝜑𝐹𝑥𝑥,𝑦≤𝜃,∀𝜆∈𝐷,₀,𝑦𝐹𝑦,𝑥+𝑑_𝜆𝑦,𝑥≤𝜃,∀𝜆∈𝐷. (3.3) This shows that 𝐹(𝑥,𝑦)≤𝜃 and 𝐹(𝑦,𝑥)≤𝜃. It follows from the condition (H2) that 𝐹𝑥₀,𝑥≤𝐹𝑥₀,𝑦+𝐹𝑦,𝑥≤𝐹𝑥₀,𝐹𝑥,𝑦₀,𝑦≤𝐹𝑥₀,𝑥+𝐹𝑥,𝑦≤𝐹𝑥₀,𝑥. (3.4) It follows from Lemma 2.3 that there exists a family {‖⋅‖_𝛼∶𝛼∈𝐼} of quasi-norms which defines the topology 𝜏 of 𝐸. If 𝐾 is pseudo-nuclear, by Lemma 2.4, it follows that, for any 𝛼∈𝐼, there exists 𝑓_𝛼∈𝐾^∗ with the property: ‖𝑥‖_𝛼≤𝑓_𝛼(𝑥),∀𝑥∈𝐾. (3.5) For any 𝛼∈𝐼, we know that 𝑓_𝛼 is an increasing function on 𝐸. Then it follows from (3.4) that 𝜑(𝐹(𝑥₀,𝑥))=𝜑(𝐹(𝑥₀,𝑦)), 𝑓_𝛼(𝐹(𝑥₀,𝑥))=𝑓_𝛼(𝐹(𝑥₀,𝑦)) and then 𝑓_𝛼(𝐹(𝑦,𝑥))=𝑓_𝛼(𝐹(𝑥,𝑦))=0. Thus, by (3.3), it follows that 𝑓_𝛼(𝑑_𝜆(𝑥,𝑦))=𝑓_𝛼(𝑑_𝜆(𝑦,𝑥))=0, for any 𝛼∈𝐼 and 𝜆∈𝐷. It follows from (3.5) that 𝑑_𝜆(𝑥,𝑦)=𝜃 for all 𝜆∈𝐷, that is, 𝑥=𝑦. If 𝐾 is a closed convex cone, then the ordering relation ≤ is a partial ordering. It follows from (3.3) and (3.4) that 𝐹(𝑥₀,𝑥)=𝐹(𝑥₀,𝑦) and 𝐹(𝑦,𝑥)=𝐹(𝑥,𝑦)=𝜃. Again by (3.3), it follows that 𝑑_𝜆(𝑥,𝑦)=𝜃 for all 𝜆∈𝐷, that is, 𝑥=𝑦. If 𝑥≼𝑦 and 𝑦≼𝑧, then we have 𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑦+𝑑_𝜆𝜑𝐹𝑥𝑥,𝑦≤𝜃,∀𝜆∈𝐷,₀,𝑦𝐹𝑦,𝑧+𝑑_𝜆𝑦,𝑧≤𝜃,∀𝜆∈𝐷, (3.6) which imply that 𝐹(𝑥,𝑦)≤𝜃 and 𝐹(𝑦,𝑧)≤𝜃 and so 𝐹𝑥₀,𝑦≤𝐹𝑥₀,𝑥+𝐹𝑥,𝑦≤𝐹𝑥₀,𝑥. (3.7) By (d3) of Definition 2.5, it follows that, for any 𝜆∈𝐷, there exists 𝜇∈𝐷 with 𝜆≺𝜇 such that 𝑑_𝜆(𝑥,𝑧)≤𝑑_𝜇(𝑥,𝑦)+𝑑_𝜇(𝑦,𝑧) and so 𝜑𝐹𝑥₀,𝑥𝐹(𝑥,𝑧)+𝑑_𝜆()𝐹𝑥𝑥,𝑧≤𝜑₀,𝑥𝐹𝑥,𝑦+𝐹𝑦,𝑧+𝑑_𝜇𝑥,𝑦+𝑑_𝜇𝐹𝑥𝑦,𝑧≤𝜑₀,𝑥𝐹𝑥,𝑦+𝜑𝐹𝑥₀,𝑦𝐹𝑦,𝑧+𝑑_𝜇𝑥,𝑦+𝑑_𝜇𝑦,𝑧≤𝜃. (3.8) This shows that 𝑥≼𝑧. Therefore, ≼ is a partial order on 𝑋. This completes the proof.

First, we give a general Ekeland’s principle of the nondirected vectorial perturbation type as follows:

Theorem 3.2. Let 𝐸(𝜏) be a topological vector space with a 𝑇₀ topology 𝜏 and 𝐾⊂𝐸 be a closed convex cone. Let (𝑋,𝑃) be a complete cone quasi-uniform space and let 𝐹∶𝑋×𝑋→𝐸 be a function satisfying the conditions (𝐻1), (𝐻2), and (𝐻3). If the cone 𝐾 is pseudo-nuclear, then, for any 𝑥₀∈𝑋, there exists 𝑣∈𝑋 such that(1)𝜑(𝜃)𝐹(𝑥₀,𝑣)+𝑑_𝜆(𝑥₀,𝑣)∈−𝐾 for all 𝜆∈𝐷;(2)for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists 𝜆∈𝐷 such that𝜑𝐹𝑥₀,𝑣𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)∉−𝐾. (3.9)

Proof. For any fixed 𝑥₀∈𝑋, if ⪯ is defined by (3.2), then Lemma 3.1 shows that (𝑋,⪯) is a partially ordered set.
First, we prove that any increasing net {𝑥_𝛾}_𝛾∈Λ is a Cauchy net. It follows from Lemma 2.3 that there exists a family {‖⋅‖_𝛼∶𝛼∈𝐼} of quasi-norms which defines the topology 𝜏 of 𝐸. Since 𝐾 is pseudo-nuclear, by Lemma 2.4, it follows that, for any 𝛼∈𝐼, there exists 𝑓_𝛼∈𝐾^∗ with the property: ‖𝑥‖_𝛼≤𝑓_𝛼(𝑥),∀𝑥∈𝐾. (3.10) From the proof of Lemma 3.1, we can show that {𝐹(𝑥₀,𝑥_𝛾)}_𝛾∈Λ is a decreasing net and, for any 𝜈≥𝛾, 𝑑_𝜆𝑥_𝛾,𝑥_𝜈+𝜑𝐹𝑥₀,𝑥_𝛾𝐹𝑥_𝛾,𝑥_𝜈≤𝜃,∀𝜆∈𝐷, (3.11) and so 𝑑_𝜆𝑥_𝛾,𝑥_𝜈≤−𝜑𝐹𝑥₀,𝑥_𝛾𝐹𝑥_𝛾,𝑥_𝜈,∀𝜆∈𝐷. (3.12) It follows from 𝐹(𝑥₀,𝑥_𝜈)≤𝐹(𝑥₀,𝑥_𝛾)+𝐹(𝑥_𝛾,𝑥_𝜈) that −𝐹𝑥_𝛾,𝑥_𝜈≤𝐹𝑥₀,𝑥_𝛾−𝐹𝑥₀,𝑥_𝜈. (3.13) Thus we have 𝑑_𝜆𝑥_𝛾,𝑥_𝜈≤𝜑𝐹𝑥₀,𝑥_𝛾𝐹𝑥₀,𝑥_𝛾−𝐹𝑥₀,𝑥_𝜈,∀𝜆∈𝐷, (3.14) and so, for any 𝛼∈𝐼, 𝑓_𝛼𝑑_𝜆𝑥_𝛾,𝑥_𝜈≤𝜑𝐹𝑥₀,𝑥_𝛾𝑓_𝛼𝐹𝑥₀,𝑥_𝛾−𝑓_𝛼𝐹𝑥₀,𝑥_𝜈,∀𝜆∈𝐷. (3.15) It follows from the condition (H3) that {𝐹(𝑥₀,𝑥_𝛾)} is bounded from below. This and (3.15) imply that lim_𝛾,𝜈𝑓_𝛼𝑑_𝜆𝑥_𝛾,𝑥_𝜈=0,∀𝜆∈𝐷. (3.16) It follows from (3.10) that, for any 𝛼∈𝐼, lim_𝛾,𝜈𝑑_𝜆𝑥_𝛾,𝑥_𝜈𝛼=0,∀𝜆∈𝐷. (3.17) This shows that {𝑥_𝛾}_𝛾∈Λ is a Cauchy net in 𝑋. The completeness implies that {𝑥_𝛾}_𝛾∈Λ converges to some 𝑣∈𝑋. It follows from 𝑥_𝛾≼𝑥_𝜈 for any 𝜈≥𝛾 that 𝑑_𝜆𝑥_𝛾,𝑥_𝜈+𝜑𝐹𝑥₀,𝑥_𝛾𝐹𝑥_𝛾,𝑥_𝜈≤𝜃,∀𝜆∈𝐷. (3.18) By (H3), we get that 𝑑_𝜆𝑥_𝛾,𝑣+𝜑𝐹𝑥₀,𝑥_𝛾𝐹𝑥_𝛾,𝑣≤𝜃,∀𝜆∈𝐷. (3.19) Thus 𝑥_𝛾≼𝑣, that is, 𝑣 is an upper bound of the net {𝑥_𝛾}_𝛾∈Λ. Assume that 𝐴⊂{𝑥∈𝑋∶𝑥≽𝑥₀} is a totally ordered set. Then 𝐴 is also a directed set. If we represent 𝐴 by {𝑥_𝑥}_𝑥∈𝐴, where 𝑥_𝑥=𝑥, then {𝑥_𝑥}_𝑥∈𝐴 is an increasing net. By the results just proved above, we can know that 𝐴 has an upper bound. By the well-known Zorn’s lemma, the set {𝑥∈𝑋∶𝑥≽𝑥₀} has a maximal element 𝑣≽𝑥₀, that is, 𝜑𝐹𝑥₀,𝑥₀𝐹𝑥₀,𝑣+𝑑_𝜆𝑥₀,𝑣≤𝜃,∀𝜆∈𝐷. (3.20) Therefore, it follows from 𝐹(𝑥₀,𝑥₀)=𝜃 that the conclusion (1) holds. For any 𝑥∈𝑋 with 𝑥≠𝑣, if 𝜑(𝐹(𝑥₀,𝑣))𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)≤𝜃 for all 𝜆∈𝐷, then 𝑣≼𝑥. Since 𝑣 is a maximal element, we have that 𝑣=𝑥, which is a contradiction. Therefore, conclusion (2) holds. This completes the proof.

If the cone 𝐾 has a base, we have the following result.

Theorem 3.3. Let 𝐸(𝜏) be a topological vector space with a 𝑇₀ topology 𝜏 and let 𝐾⊂𝐸 be a closed convex cone. Let (𝑋,𝑃) be a sequentially complete cone quasi-uniform space and let 𝐹∶𝑋×𝑋→𝐸 be a mapping satisfying the conditions (𝐻1), (𝐻2), and (𝐻3). If the cone 𝐾 has a base, then, for any 𝑥₀∈𝑋, there exists 𝑣∈𝑋 such that(1)𝜑(𝜃)𝐹(𝑥₀,𝑣)+𝑑_𝜆(𝑥₀,𝑣)∈−𝐾 for all 𝜆∈𝐷;(2)for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists 𝜆∈𝐷 such that𝜑𝐹𝑥₀,𝑣𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)∉−𝐾. (3.21)

Proof. Since cone 𝐾 has a base, it follows from Lemma 2.3 that (𝐾^∗)^∘≠∅. Let 𝜉∈(𝐾^∗)^∘. For any fixed 𝑥₀∈𝑋, assume that ⪯ is defined by (3.2). Similarly, as in the proofs of Lemma 3.1 and Theorem 3.2, we can prove that (𝑋,⪯) is a partially ordered set and any increasing sequence {𝑥_𝑛}_𝑛∈𝑁 has an upper bound. The only difference is that 𝑓_𝛼∈𝐾^∗ is replaced by 𝜉∈(𝐾^∗)^∘. Assume that 𝑥≼𝑦 and 𝑥≠𝑦. Then we have 𝑑_𝜆𝑥,𝑦≤𝜑𝐹𝑥₀,𝑥𝐹𝑥₀,𝑥−𝐹𝑥₀,𝑦,∀𝜆∈𝐷. (3.22)𝑥≠𝑦 implies that there exists 𝜆₀∈𝐷 such that 𝑑_𝜆0(𝑥,𝑦)≠𝜃. Then we have 0<𝜉𝑑_𝜆0𝑥,𝑦≤𝜑𝐹𝑥₀,𝑥𝜉𝐹𝑥₀,𝑥−𝜉𝐹𝑥₀,𝑦, (3.23) that is, 𝜉(𝐹(𝑥₀,𝑥))<𝜉(𝐹(𝑥₀,𝑦)). Let 𝜂(𝑥)=𝜉(𝐹(𝑥₀,𝑥)). Then 𝜂∶𝑋→𝑅 is strictly increasing. Assume that 𝐴⊂𝑋 is a totally ordered set. Then 𝜂(𝐴) is also a totally set and, for any 𝑥,𝑦∈𝐴,  𝑥≼𝑦 if and only if 𝜂(𝑥)≤𝜂(𝑦). It follows from Theorem  2.3 in [16] that the set 𝑅 of real numbers is totally ordered separable and so there exist an increasing sequence {𝑦_𝑛}⊂𝜂(𝐴) and a decreasing sequence {𝑧_𝑛}⊂𝜂(𝐴) such that, for any 𝑦∈𝜂(𝐴), there exist positive integers 𝑛₁ and 𝑛₂ satisfying 𝑧_𝑛1≤𝑦≤𝑦_𝑛2. (3.24) Let {𝑢_𝑛} and {𝑣_𝑛} be two sequences in 𝐴 such that 𝜂(𝑢_𝑛)=𝑦_𝑛 and 𝜂(𝑣_𝑛)=𝑧_𝑛. Since 𝜂∶𝑋→𝑅 is strictly increasing, we know that {𝑢_𝑛} is increasing and {𝑣_𝑛} is decreasing. For any 𝑥∈𝐴, by (3.24), we have 𝜂𝑣_𝑛1=𝑧_𝑛1≤𝜂(𝑥)≤𝑦_𝑛2=𝜂𝑢_𝑛2 (3.25) and so 𝑣_𝑛1≤𝑥≤𝑢_𝑛2. (3.26) Thus (𝑋,⪯) is totally ordered separable. By using Lemma 2.10, we can get that there exists a maximal element 𝑣≽𝑥₀, that is, 𝜑𝐹𝑥₀,𝑥₀𝐹𝑥₀,𝑣+𝑑_𝜆𝑥₀,𝑣≤𝜃,∀𝜆∈𝐷. (3.27) Then, in the same way as in the proof of Theorem 3.2, we can prove that conclusions (1) and (2) holds. This completes the proof.

Remark 3.4. From the proof of Theorems 3.2 and 3.3 we can see that if cone 𝐾 is a pointed convex cone and replace the pseudo-nuclearity of 𝐾 with the regularity of 𝐾 (i.e., any monotone and bounded net in 𝐾 is convergent), then the conclusions of Theorems 3.2 and 3.3 still hold.

Clearly, our nondirected vectorial perturbation type Ekeland’s principle contains the directed vectorial perturbation type Ekeland’s principle as a special case. From Theorem 3.2, we have the following Theorem 3.5, which improves the corresponding results in [2, 3, 5, 8, 14].

Theorem 3.5. Let (𝑋,{𝑑_𝜆}_𝜆∈𝐷) be a complete 𝐹-type topological space, let 𝐸(𝜏) be a Hausdorff topological vector space, and let 𝐾⊂𝐸 be a closed convex cone. Let 𝐹∶𝑋×𝑋→𝐸 be a function satisfying the conditions (𝐻1), (𝐻2), and (𝐻3). Let 𝑒_𝜆∈𝐾⧵{𝜃} for all 𝜆∈𝐷. If 𝐾 is pseudo-nuclear, or 𝐸(𝜏) is a locally convex space and 𝐾∩(−𝐾)={𝜃}, then, for any 𝑥₀∈𝑋, there exists 𝑣∈𝑋 such that(1)𝜑(𝜃)𝐹(𝑥₀,𝑣)+𝑑_𝜆(𝑥₀,𝑣)𝑒_𝜆∈−𝐾 for all 𝜆∈𝐷;(2)for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists 𝜆∈𝐷 such that𝜑𝐹𝑥₀,𝑣𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)𝑒_𝜆∉−𝐾. (3.28)

Proof. Let 𝑑_𝜆∶𝑋×𝑋→𝐸 be a mapping defined by 𝑑_𝜆(𝑥,𝑦)=𝑑_𝜆(𝑥,𝑦)𝑒_𝜆 and 𝑃=𝑑_𝜆∶𝑋×𝑋⟶𝐸∶𝜆∈𝐷. (3.29) It is clearly that, for any net {𝑥_𝛾}_𝛾∈Λ and 𝑥∈𝑋, lim_𝛾𝑑_𝜆𝑥_𝛾,𝑥=𝜃⟺lim_𝛾𝑑_𝜆𝑥_𝛾,𝑥=0,lim_𝛾,𝜈𝑑_𝜆𝑥_𝛾,𝑥_𝜈=𝜃⟺lim_𝛾,𝜈𝑑_𝜆𝑥_𝛾,𝑥_𝜈=0. (3.30) Since (𝑋,{𝑑_𝜆}_𝜆∈𝐷) is a complete 𝐹-type topological space, we have that (𝑋,𝑃) is a complete cone quasi-uniform space. If 𝐾 is pseudo-nuclear, then the conclusion of Theorem 3.5 can be obtained by using Theorem 3.2 for (𝑋,𝑃). If 𝐸(𝜏) is a locally convex space and 𝐾∩(−𝐾)={𝜃}, then, for each 𝑒_𝜆∈𝐾⧵{𝜃}, we have −𝑒_𝜆∉𝐾. By using well-known separation theorem of convex sets, there exist a continuous linear function 𝑓_𝜆∈𝐸^∗ and a number 𝑐, such that 𝑓_𝜆(−𝑒_𝜆)<𝑐, and 𝑓_𝜆(𝑥)>𝑐 for any 𝑥∈𝐾. This shows that 0=𝑓_𝜆(𝜃)>𝑐 and 𝑓_𝜆(−𝑒_𝜆)<0. If 𝑥∈𝐾, then 𝑛𝑥∈𝐾 for any 𝑛∈𝑁. So we have that 𝑓_𝜆(𝑥)>𝑐/𝑛. Then by letting 𝑛→∞, we get that 𝑓_𝜆(𝑥)≥0 for any 𝑥∈𝐾. Thus, for each 𝑒_𝜆∈𝐾⧵{𝜃}, there exists a continuous function 𝑓_𝜆∈𝐾^∗, such that 𝑓_𝜆(𝑒_𝜆)>0. For any fixed 𝑥₀∈𝑋, if ⪯ is defined by 𝑥≼𝑥^′⟺𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑥^′+𝑑_𝜆𝑥,𝑥^′≤𝜃,∀𝜆∈𝐷. (3.31) By Lemma 3.1, (𝑋,⪯) is a partially ordered set. Similarly as in the proofs of Theorem 3.2, we can prove that any increasing net {𝑥_𝛾}_𝛾∈Λ has an upper bound. The only difference is that 𝑓_𝛼∈𝐾^∗ is replaced by 𝑓_𝜆∈𝐾^∗. Then, in the same way as in the proof of Theorem 3.2, we can prove that the conclusion of Theorem 3.5 holds. This completes the proof.

In the following Theorem 3.6, we introduce some new principles, which are equivalent with our Ekeland principle.

Theorem 3.6. Let 𝐸(𝜏) be a topological vector space with a 𝑇₀ topology 𝜏,  𝐾 be a closed convex and pseudo-nuclear cone in 𝐸 and (𝑋,𝑃) be a complete cone quasi-uniform space. Let 𝐹∶𝑋×𝑋→𝐸 be a function satisfying the conditions (𝐻1)-(𝐻3). Then, for any 𝑥₀∈𝑋, the following conclusions hold and they are equivalent.
(1) (Vectorial Ekeland type variational principle in cone quasi-uniform spaces) There exists𝑣∈𝑆𝑥₀=𝑥∈𝑋∶𝜑(𝜃)𝐹𝑥₀,𝑥+𝑑_𝜆𝑥₀,𝑥∈−𝐾,∀𝜆∈𝐷 (3.32)such that, for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists 𝜆∈𝐷 satisfying the following:𝜑𝐹𝑥₀,𝑣𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)∉−𝐾. (3.33)(2) (Vectorial quasivariational inclusion principle in cone quasi-uniform spaces) Let 𝑍 be a vector space and 𝑃⊂𝑍 be a nonempty convex subset. Let 𝐺∶𝑋×𝑋→2^𝑍⧵∅ and 𝐻∶𝑋→2^𝑋⧵∅ be both multivalued mappings with nonempty values. Suppose that𝑑_𝜆𝑥,𝑦+𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑦≤𝜃,∀𝜆∈𝐷, (3.34)holds, if 𝑦∈𝐻(𝑥) and 𝜃∈𝐺(𝑥,𝑦). Then there exists 𝑣∈𝑆(𝑥₀) such that 𝜃∉𝐺(𝑣,𝑦) for any 𝑦∈𝐻(𝑣)⧵{𝑣}.(3) (Vectorial quasi-optimization principle in cone quasi-uniform spaces) Let 𝑍 be a vector space and 𝑃⊂𝑍 be a nonempty convex subset. Let 𝐵∶𝑋×𝑋→2^𝑍⧵∅ and 𝐻∶𝑋→2^𝑋⧵∅ be both multivalued mappings with nonempty values. Suppose that (3.34) holds if 𝑦∈𝐻(𝑥) and 𝐵(𝑥,𝑦)⊈𝑃. Then there exists 𝑣∈𝑆(𝑥₀) such that 𝐵(𝑣,𝑦)⊂𝑃 for any 𝑦∈𝐻(𝑣)⧵{𝑣}.(4) (Vectorial quasi-optimization principle in cone quasi-uniform spaces) Let 𝑍 be a vector space and 𝑃⊂𝑍 be a nonempty convex subset. Let 𝐵∶𝑋×𝑋→2^𝑍⧵∅ and 𝐻∶𝑋→2^𝑋⧵∅ be both multivalued maps with nonempty values. Suppose that (3.34) holds if 𝑦∈𝐻(𝑥) and 𝐵(𝑥,𝑦)∩[𝐵(𝑥,𝑥)−𝑃]≠∅. Then there exists 𝑣∈𝑆(𝑥₀) such that 𝐵(𝑣,𝑦)∩[𝐵(𝑣,𝑣)−𝑃]=∅ for any 𝑦∈𝐻(𝑣)⧵{𝑣}.(5)(Generalized Caristi-Kirk type coincidence point in cone quasi-uniform spaces) Let 𝐼 be an index set. For each 𝑖∈𝐼, let 𝑇_𝑖∶𝑋→2^𝑋 be a multivalued mapping, let 𝑀 be a nonempty subset of 𝑋, and let 𝑔∶𝑀→𝑋 be a surjective mapping. Suppose further that, if, for each 𝑥∈𝑀 with 𝑔(𝑥)∉⋂_𝑙∈𝐼𝑇_𝑙(𝑥), there exist 𝑙₀∈𝐼 and 𝑦∈𝑇_𝑙0(𝑥)⧵{𝑔(𝑥)} such that𝑑_𝜆𝑔(𝑥),𝑦+𝜑𝐹𝑥₀,𝑔(𝑥)𝐹𝑔(𝑥),𝑦≤𝜃,∀𝜆∈𝐷. (3.35)Then there exists a coincidence point 𝑢∈𝑀 of 𝑔 and {𝑇_𝑖}_𝑖∈𝐼, that is, 𝑔(𝑢)∈⋂_𝑙∈𝐼𝑇_𝑙(𝑢) such that 𝑔(𝑢)∈𝑆(𝑥₀).(6) (Nonconvex maximal element for a family of multivalued maps in cone quasi-uniform spaces) Let 𝐼 be any index set and for each 𝑖∈𝐼,   𝑇_𝑖∶𝑋→2^𝑋 be a multivalued mapping. Assume that, for each (𝑥,𝑖)∈𝑆(𝑥₀)×𝐼 with 𝑇_𝑖(𝑥)≠∅, there exists 𝑦=𝑦(𝑥,𝑖)∈𝑋 with 𝑦≠𝑥 such that𝑑_𝜆𝑥,𝑦+𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑦≤𝜃,∀𝜆∈𝐷. (3.36) Then there exists 𝑣∈𝑆(𝑥₀) such that 𝑇_𝑖(𝑣)=∅ for each 𝑖∈𝐼.(7) (Generalized Takahashi nonconvex minimization Theorem in cone quasi-uniform spaces) Suppose that, for each 𝑥∈𝑆(𝑥₀) with{𝑢∈𝑋∶𝐹(𝑥,𝑢)≱𝜃}≠∅, (3.37)there exists 𝑦=𝑦(𝑥)∈𝑋 with 𝑦≠𝑥 such that (3.34) holds. Then there exists 𝑣∈𝑆(𝑥₀) such that 𝐹(𝑣,𝑦)≥𝜃 for all 𝑦∈𝑋.(8) (Oettli-Théra type theorem in cone quasi-uniform spaces) Let 𝑀⊂𝑋 and suppose that, for any 𝑥∈𝑆(𝑥₀)⧵𝑀, there exists 𝑦≠𝑥 such that (3.34) holds. Then there exists 𝑣∈𝑆(𝑥₀)∩𝑀.

Proof. It follows from Theorem 3.2 that (1) holds.
(1) ⇒ (2) By (1), there exists 𝑣∈𝑆(𝑥₀) such that, for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists 𝜆∈𝐷 satisfying 𝜑𝐹𝑥₀,𝑣𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)∉−𝐾. (3.38)
If (2) does not hold for 𝑣, then there exists 𝑦∈𝐻(𝑣)⧵{𝑣} with 𝜃∈𝐺(𝑣,𝑦) such that 𝜑𝐹𝑥₀,𝑣𝐹𝑣,𝑦+𝑑_𝜆𝑣,𝑦≤𝜃, (3.39) which contradicts (3.38). Thus 𝜃∉𝐺(𝑣,𝑦) for any 𝑦∈𝐻(𝑣)⧵{𝑣}, that is, (2) holds.
(2) ⇒ (3) Let 𝐺(𝑥,𝑦)=𝐵(𝑥,𝑦)−(𝑍⧵𝑃). Then 𝜃∈𝐺(𝑥,𝑦) implies that 𝐵𝑥,𝑦∩(𝑍⧵𝑃)≠∅. (3.40) This shows that 𝐵(𝑥,𝑦)⊈𝑃 and so, if 𝑦∈𝐻(𝑥) and 𝜃∈𝐺(𝑥,𝑦), then the conditions of (3) implies that (3.34) holds. It follows from (2) that there exists 𝑣∈𝑆(𝑥₀) such that 𝜃∉𝐺(𝑣,𝑦) for any 𝑦∈𝐻(𝑣)⧵{𝑣}. This implies that 𝐵(𝑣,𝑦)∩(𝑍⧵𝑃)=∅ for any 𝑦∈𝐻(𝑣)⧵{𝑣}. Thus 𝐵(𝑣,𝑦)⊂𝑃 for any 𝑦∈𝐻(𝑣)⧵{𝑣}, that is, (3) holds.
(3) ⇒ (4) Let 𝑄=∩(𝐵())∩(𝐵())𝑥,𝑦−𝑃,if𝐵𝑥,𝑦𝑥,𝑥−𝑃≠∅,𝑃,if𝐵𝑥,𝑦𝑥,𝑥−𝑃=∅. (3.41) If 𝑦∈𝐻(𝑥) and 𝑄(𝑥,𝑦)⊈𝑃, then 𝐵(𝑥,𝑦)∩(𝐵(𝑥,𝑥)−𝑃)≠∅. By the condition of (4), we have that (3.34) holds. This shows that the condition of (3) holds for 𝑄(𝑥,𝑦). By the conclusion of (3), there exists 𝑣∈𝑆(𝑥₀) such that 𝑄(𝑣,𝑦)⊂𝑃 for any 𝑦∈𝐻(𝑣)⧵{𝑣}. Thus 𝐵(𝑣,𝑦)∩(𝐵(𝑣,𝑣)−𝑃)=∅, that is, (4) holds.
(4) ⇒ (1) Let 𝑍=𝑧=𝑧_{𝜆𝜆∈𝐷}∶𝑧_𝜆∈𝐸,∀𝜆∈𝐷,𝑃=𝑧_{𝜆𝜆∈𝐷}∈𝑍∶𝑧_𝜆∈𝐾,∀𝜆∈𝐷. (3.42)
If, for any 𝑥,𝑦∈𝑍 and any scalar 𝛼, we define 𝑥+𝑦=𝑥_𝜆+𝑦_{𝜆𝜆∈𝐷},𝛼⋅𝑥=𝛼𝑥_{𝜆𝜆∈𝐷}, (3.43) then 𝑍 is a vector space and 𝑃 is a cone in 𝑍. Let 𝐵∶𝑋×𝑋→2^𝑍⧵∅ be a multivalued mapping defined by 𝐵𝑥,𝑦=𝑑_𝜆𝑥,𝑦+𝜑𝐹𝑥₀,𝑥𝐹𝑥,𝑦_𝜆∈𝐷 (3.44) and 𝐻∶𝑋→2^𝑋⧵∅ be a multivalued mapping defined by 𝐻(𝑥)=𝑋 for any 𝑥∈𝑋. If 𝐵𝑥,𝑦∩[𝐵(𝑥,𝑥)−𝑃]≠∅ (3.45) (note that 𝐵(𝑥,𝑥)=0), then 𝐵(𝑥,𝑦)∩(−𝑃)≠∅. This shows that (3.34) holds. It follows from (4) that there exists 𝑣∈𝑆(𝑥₀) such that 𝐵(𝑣,𝑦)∩[𝐵(𝑣,𝑣)−𝑃]=∅ for any 𝑦∈𝐻(𝑣)⧵{𝑣}. Thus, for all 𝑦∈𝑋 with 𝑦≠𝑣, there exists 𝜆∈𝐷 satisfying the following: 𝜑𝐹𝑥₀,𝑣𝐹𝑣,𝑦+𝑑_𝜆𝑣,𝑦∉−𝐾. (3.46) That is, (1) holds.
(1) ⇒ (5) From (1), there exists 𝑣∈𝑆(𝑥₀) such that, for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists 𝜆∈𝐷 satisfying the following: 𝜑𝐹𝑥₀,𝑣𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)∉−𝐾. (3.47) Since 𝑔 is a surjective mapping, there exists 𝑢∈𝐷 such that 𝑔(𝑢)=𝑣. We claim that 𝑔(𝑢)∈⋂_𝑙∈𝐼𝑇_𝑙(𝑢). If 𝑔(𝑢)∉⋂_𝑙∈𝐼𝑇_𝑙(𝑢), by the hypotheses of (V), there exist 𝑙₀∈𝐼 and 𝑦∈𝑇_𝑙0(𝑢)⧵{𝑔(𝑢)} such that 𝑑_𝜆𝑔(𝑥),𝑦+𝜑𝐹𝑥₀,𝑔(𝑥)𝐹𝑔(𝑥),𝑦≤𝜃,∀𝜆∈𝐷, (3.48) which contradicts (3.47). Thus 𝑔(𝑢)∈⋂_𝑙∈𝐼𝑇_𝑙(𝑢). 𝑣∈𝑆(𝑥₀) implies that 𝑔(𝑢)=𝑣∈𝑆(𝑥₀).
(5) ⇒ (6) If the conclusion of (6) dose not hold, then, for any 𝑥∈𝑆(𝑥₀), there exists 𝑖∈𝐼 such that 𝑇_𝑖(𝑥)≠∅. By the hypotheses of (6), there exists 𝑦=𝑦(𝑥,𝑖)∈𝑋 with 𝑦≠𝑥 such that (3.34) holds. Let 𝐷=𝑋, 𝑔=𝐼_𝑑: (the identity mapping of 𝑋) and 𝐻_𝑖(𝑥)=𝑇_𝑖(𝑥)∪𝑦(𝑥,𝑖)⧵{𝑥},if𝑥∈𝑆𝑥₀and𝑇_𝑖(𝑥)𝑥≠∅,∅,if𝑥∈𝑆₀and𝑇_𝑖(𝑥){𝑥}𝑥=∅,,if𝑥∉𝑆₀. (3.49) Then the conditions of (5) are satisfied for 𝐻_𝑖, 𝐷=𝑋 and 𝑔=𝐼_𝑑. Thus, from (5), it follows that there exists 𝑣∈𝑆(𝑥₀) such that 𝑣∈⋂_𝑙∈𝐼𝐻_𝑙(𝑣), which contradicts the definition of 𝐻_𝑖. Therefore, there exists 𝑣∈𝑆(𝑥₀) such that 𝑇_𝑖(𝑣)=∅ for any 𝑖∈𝐼.
(6) ⇒ (7) Let 𝑇_𝑦(𝑥)=𝑦∩𝑢∈𝑋∶𝐹(𝑥,𝑢)_╱≧𝜃,if𝑥∈𝑆𝑥₀,∅,if𝑥∉𝑆𝑥₀. (3.50) From this, we know that, if 𝑇_𝑦(𝑥)≠∅, then 𝐹(𝑥,𝑦)_╱≧𝜃. By the hypotheses of (7), there exists 𝑧=𝑧(𝑥)∈𝑋 with 𝑧≠𝑥 such that 𝑑_𝜆(𝑥,𝑧)+𝜑𝐹𝑥₀,𝑥𝐹(𝑥,𝑧)≤𝜃,∀𝜆∈𝐷. (3.51) By using (6), there exists 𝑣∈𝑆(𝑥₀) such that 𝑇_𝑦(𝑣)=∅ for any 𝑦∈𝑋, that is, 𝐹(𝑣,𝑦)≥𝜃 for any 𝑦∈𝑋.
(7) ⇒ (8) Suppose that (7) and the hypothesis of (8) hold. Suppose that, for any 𝑥∈𝑆(𝑥₀),  𝑥∉𝑀. By the hypothesis of (8), there exists 𝑣≠𝑥 such that 𝑑_𝜆(𝑥,𝑣)+𝜑𝐹𝑥₀,𝑥𝐹(𝑥,𝑣)≤𝜃,∀𝜆∈𝐷. (3.52)Equation (3.52) shows that the conditions of (7) naturally hold. Then there exists 𝑣∈𝑆(𝑥₀) such that 𝐹(𝑣,𝑦)≥𝜃 for any 𝑦∈𝑋, which contradicts (3.52). Therefore, there exists 𝑣∈𝑆(𝑥₀)∩𝑀.
(8) ⇒ (1) Let Γ(𝑢)=𝑥∈𝑋∶𝑢≠𝑥,𝑑_𝜆(𝑢,𝑥)+𝜑𝐹𝑥₀𝐹(),{(𝑢)}.,𝑢𝑢,𝑥≤𝜃,∀𝜆∈𝐷𝑀=𝑢∈𝑋∶Γ=∅ (3.53) If 𝑢∉𝑀, then Γ(𝑢)≠∅, that is, there exists 𝑥∈Γ(𝑢) such that 𝑑_𝜆(𝑢,𝑥)+𝜑𝐹𝑥₀,𝑢𝐹(𝑢,𝑥)≤𝜃,∀𝜆∈𝐷. (3.54) This shows that the conditions of (8) hold and so there exists 𝑣∈𝑆(𝑥₀)∩𝑀.𝑣∈𝑀 shows that, for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists 𝜆∈𝐷 satisfying the following: 𝜑𝐹𝑥₀,𝑣𝐹(𝑣,𝑥)+𝑑_𝜆(𝑣,𝑥)∉𝐾. (3.55) This completes the proof.

In the following, we provide some examples to illustrate our results.

Example 3.7. Let 𝐸=𝐿^𝑝[0,1], 𝐾={𝑥∈𝐿^𝑝[0,1]∶𝑥(𝑡)≥0,a.e.𝑡∈[0,1]}, where 𝑝≥1. Then 𝐾 is a nuclear and regular cone if 𝑝=1 and a regular cone, but not a nuclear cone in 𝐿^𝑝[0,1] if 𝑝>1. However, for any 𝑝≥1, 𝐾 is not a solid cone (see, e.g., [32]). Let 𝑋=𝐿^𝑝[0,1] and define a mapping 𝑑∶𝑋×𝑋→𝐿^𝑝[0,1] by 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Then 𝑑 is a cone quasi-metric on 𝑋. Since 𝐿^𝑝[0,1] is complete, (𝑋,𝑑) is also a complete quasi-metric space. Define a mapping 𝐹∶𝑋×𝑋→𝐿^𝑝[0,1] by 𝐹(𝑥,𝑦)=sin𝑦−sin𝑥 for any 𝑥,𝑦∈𝑋. Then, for each 𝑥∈𝑋, 𝐹(𝑥,⋅)∶𝑋→𝐸 is bound from below and 𝑦∈𝑋∶𝐹𝑥,𝑦+𝑥−𝑦≤0 (3.56) is a closed subset in 𝑋. It follows from Theorem 3.2 and Remark 3.4 that, for any 𝑥₀∈𝑋, there exists 𝑣∈𝑋 such that(1)sin𝑣(𝑡)−sin𝑥₀(𝑡)+|𝑥₀(𝑡)−𝑣(𝑡)|≤0 for a.e. 𝑡∈[0,1];(2)for all 𝑥∈𝑋 with 𝑥≠𝑣, there exists a measuable set 𝐴⊂[0,1] with a positive measure such thatsin𝑥(𝑡)−sin𝑣(𝑡)+|𝑥(𝑡)−𝑣(𝑡)|>0,a.e.𝑡∈𝐴. (3.57)

In fact, 𝑣=𝑥₀ meets the demands.

Example 3.8. Let 𝐸=𝑙₁ endowed with the topology defined by the semi-norms {𝑝_𝑛}_𝑛∈𝑁, where 𝑝_𝑛((𝑥_𝑘)_𝑘∈𝑁)=∑^𝑛_𝑘=1|𝑥_𝑘|. The cone 𝐾=𝑥=𝑥_{𝑘𝑘∈𝑁}∈𝑙₁∶𝑥_𝑘≥0,∀𝑘∈𝑁 (3.58) is a nuclear cone, but not well based (see [33]). Let 𝑋=[0,1], define a mapping 𝑑∶𝑋×𝑋→𝑙₁ by 𝑑𝑥,𝑦=^𝑘𝑥−𝑦𝑘²_𝑘∈𝑁1,if𝑥<𝑦,𝑘²_𝑘∈𝑁,if𝑥>𝑦, (3.59) and a mapping 𝐹∶𝑋×𝑋→𝑙₁ by 𝐹=1𝑥,𝑦𝑘1sin−1𝑦+𝑘𝑘1sin𝑥+𝑘_𝑘∈𝑁. (3.60) Then (𝑋,𝑑) is a complete cone quasi-metric space, for each 𝑥∈𝑋, 𝐹(𝑥,⋅)∶𝑋→𝐸 is bound from below and 𝑦∈𝑋∶𝐹𝑥,𝑦+𝑑𝑥,𝑦≤0 (3.61) is a closed subset in 𝑋. It follows from Theorem 3.2 and Remark 3.4 that, for any 𝑥₀∈𝑋, there exists 𝑣∈𝑋 such that 1𝑘1sin−1𝑣+𝑘𝑘1sin𝑥₀+𝑘_𝑘∈𝑁+𝑑𝑥₀,𝑣∈−𝐾 (3.62) and, for all 𝑥∈𝑋 with 𝑥≠𝑣, 1𝑘1sin−1𝑣+𝑘𝑘1sin𝑥+𝑘_𝑘∈𝑁+𝑑(𝑣,𝑥)∉−𝐾. (3.63)

Remark 3.9. In above examples, the cone quasi-metric is not directed metric, the cone is not solid or not well based, thus, Example 3.7 and Example 3.8 show that our results are different from the results in [2–5, 8–10, 14–17].

Acknowledgments

First, the authors are very grateful to the referees for their careful reading of the paper, valuable comments and suggestions that enable us to revise and greatly improve this paper. Next, this paper was supported by the National Natural Science Foundation of China (11171286, 11126344) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.: 2011-0021821).