Existence Theorems of 𝜺-Cone Saddle Points for Vector-Valued Mappings

Abstract

A new existence result of -vector equilibrium problem is first obtained. Then, by using the existence theorem of -vector equilibrium problem, a weakly -cone saddle point theorem is also obtained for vector-valued mappings.

1. Introduction

Saddle point problems are important in the areas of optimization theory and game theory. As for optimization theory, the main motivation of studying saddle point has been their connection with characterized solutions to minimax dual problems. Also, as for game theory, the main motivation has been the determination of two-person zero-sum games based on the minimax principle.

In recent years, based on the development of vector optimization, a great deal of papers have been devoted to the study of cone saddle points problems for vector-valued mappings and set-valued mappings, such as [1–8]. Nieuwenhuis [5] introduced the notion of cone saddle points for vector-valued functions in finite-dimensional spaces and obtained a cone saddle point theorem for general vector-valued mappings. Gong [2] established a strong cone saddle point theorem of vector-valued functions. Li et al. [4] obtained an existence theorem of lexicographic saddle point for vector-valued mappings. Bigi et al. [1] obtained a cone saddle point theorem by using an existence theorem of a vector equilibrium problem. Zhang et al. [9] established a general cone loose saddle point for set-valued mappings. Zhang et al. [8] obtained a minimax theorem and an existence theorem of cone saddle points for set-valued mappings by using Fan-Browder fixed point theorem. Some other types of existence results can be found in [3, 10–18].

On the other hand, in some situations, it may not be possible to find an exact solution for an optimization problem, or such an exact solution simply does not exist, for example, if the feasible set is not compact. Thus, it is meaningful to look for an approximate solution instead. There are also many papers to investigate the approximate solution problem, such as [19–21]. Kimura et al. [20] obtained several existence results for -vector equilibrium problem and the lower semicontinuity of the solution mapping of -vector equilibrium problem. Anh and Khanh [19] have considered two kinds of solution sets to parametric generalized -vector quasiequilibrium problems and established the sufficient conditions for the Hausdorff semicontinuity (or Berge semicontinuity) of these solution mappings. X. B. Li and S. J. Li [21] established some semicontinuity results on -vector equilibrium problem.

The aim of this paper is to characterize the -cone saddle point of vector-valued mappings. For this purpose, we first establish an existence theorem for -vector equilibrium problem. Then, by this existence result, we obtain an existence theorem for -cone saddle point of vector-valued mappings.

2. Preliminaries

Let be a real Hausdorff topological vector space and let be a real local convex Hausdorff topological vector space. Assume that is a pointed closed convex cone in with nonempty interior . Let be the topological dual space of . Denote the dual cone of by : Note that from Lemma 3.21 in [22] we have

Definition 1 (see [7, 23]). Let be a vector-valued mapping. is said to be -upper semicontinuous on if and only if, for each and any , there exists an open neighborhood of such that is said to be -lower semicontinuous on if and only if is -upper semicontinuous on .

Lemma 2 (see [17]). Let be a vector-valued mapping and . If is -lower semicontinuous, then is lower semicontinuous.

Definition 3 (see [24]). Let and be nonempty subsets of and be a vector-valued mapping.(i)is said to be -concavelike in its first variable on if and only if, for all and , there exists such that (ii) is said to be -convexlike in its second variable on if and only if, for all and , there exists such that (iii) is said to be -concavelike-convexlike on if and only if is -concavelike in its first variable and -convexlike in its second variable.

Definition 4. Let be a nonempty subset and .(i)A point is said to be a weak  -minimal point of if and only if and denotes the set of all weak -minimal points of .(ii)A point is said to be a weak  -maximal point of if and only if and denotes the set of all weak  -maximal points of .

Definition 5. Let be a vector-valued mapping and . A point is said to be a weak  --saddle point of on if

3. Existence of -Vector Equilibrium Problem

In this section, we deal with the following -vector equilibrium problem (for short VAEP). Find such that where is a vector-valued mapping, is a nonempty subset of and .

If , , and if   is a solution of VAEP, then is a solution of -vector optimization of , where is a vector-valued mapping.

Denote the -solution set of (VAEP) by

Lemma 6 (see [20]). Let be a nonempty subset of . Suppose that is a vector-valued mapping and the following conditions are satisfied:(i) is a compact set;(ii);(iii) is -lower semicontinuous on .Then, for each , .

Next, we give a sufficient condition for the condition (ii) in Lemma 6.

Lemma 7. Let be a nonempty subset of . Suppose that is a vector-valued mapping with for all and the following conditions are satisfied:(i) is a compact set;(ii) is -concavelike-convexlike on ;(iii)for each , is -lower semicontinuous on .Then, there exists such that

Proof. For any and , we define a multifunction by
First, by assumptions, we must have In fact, if there exists such that , for all , then Particularly, taking , we have , which contradicts the assumption about .
Then, by Lemma 2, is a closed set, for each . By (11), for any , we have Since is compact, there exists a finite point set in such that Namely, for each , there exists such that Now, we consider the set Obviously, by the condition (ii), is a convex set. By (15), we have the fact that .
By the separation theorem of convex sets, there exists such that Since , we can get   and  , for all . By the definition of , for each , By (18), . Then, by (17) and (19), By (20), . Thus, by the condition (ii), for each , there exists such that By the assumption about and , there exists such that This completes the proof.

By Lemmas 6 and 7, we can get the following result.

Theorem 8. Let be a nonempty subset of . Suppose that is a vector-valued mapping with for all and the following conditions are satisfied:(i) is a compact set;(ii) is -concavelike-convexlike on ;(iii) is -lower semicontinuous on .Then, for each , .

Remark 9. Note that the condition (i) does not require the fact that is a convex set. So Theorem 8 is different from Theorem 3.2 in [20]. The following example explains this case.

Example 10. Let , , and , Obviously, is a compact set. However, is not a convex set. So, Theorem 3.2 in [20] is not applicable. By the definition of , is -concavelike-convexlike on and -lower semicontinuous on . Thus, all conditions of Theorem 8 hold. Indeed, for each , Namely, .

4. Existence of -Cone Saddle Points

Lemma 11. Let be a nonempty subset of and . Let and let be a vector-valued mapping with , where , , , and . If there exists such that then is a weak  --saddle point of on .

Proof. By assumptions, we have Then, By (27), taking , which implies . Then, by (27), taking , which implies . Thus, is a weak  --saddle point of on . This completes the proof.

Theorem 12. Let and be nonempty sets and . Suppose that is a vector-valued mapping and the following conditions are satisfied:(i) and are compact sets;(ii) is -concavelike-convexlike on ;(iii) is -upper semicontinuous on ;(iv) is -lower semicontinuous on .Then, has a weak  --saddle point on .

Proof. Let and be a vector-valued mappings by Next, we show that all assumptions of Theorem 8 are satisfied by .
Clearly, by the condition (i), is compact. Then, by the condition (ii), we have the fact that, for each and , there exists such that and, for each and , there exists By (31) and (32), for each and , there exists such that Namely, is -concavelike-convexlike on .
Now, we show that is -lower semicontinuous on . By the condition (iii), for each and , there exists an open neighborhood of and of such that and, for each and , there exists an open neighborhood of and of such that By (34) and (35), we have the fact that, for any , Namely, is -lower semicontinuous on . Therefore, by Lemma 11, has a weak  --saddle point on . This completes the proof.

Remark 13. The conditions (iii) and (iv) of Theorem 12 do not imply that is continuous (see [23]).

Conflict of Interests

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

Acknowledgments

The author would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This paper is dedicated to Professor Miodrag Mateljevi’c on the occasion of his 65th birthday.