A new existence result of

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 [

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 [

The aim of this paper is to characterize the

Let

Let

Let

Let

Let

A point

A point

Let

In this section, we deal with the following

If

Denote the

Let

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

Let

for each

For any

First, by assumptions, we must have

Then, by Lemma

By the separation theorem of convex sets, there exists

By Lemmas

Let

Note that the condition (i) does not require the fact that

Let

Let

By assumptions, we have

Let

Let

Clearly, by the condition (i),

Now, we show that

The conditions (iii) and (iv) of Theorem

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

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.