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We consider a generalized equilibrium problem involving DC functions. By using the properties of the epigraph of the conjugate functions, some sufficient and/or necessary conditions for the weak and strong duality results and optimality conditions for generalized equilibrium problems are provided.

Consider the following generalized equilibrium problem:

As mentioned in [

Duality for equilibrium problems was first studied in [

Inspired by the works mentioned above, we continue to study the generalized equilibrium problems. Our main aim in the present paper is to give some new regularity conditions which characterize the weak duality, the strong duality, and optimality conditions for (GEP). In general, we do not impose any topological assumption on

The paper is organized as follows. Section

The notation used in the present paper is standard (cf. [

The indicator function

The following lemma is known in [

Let

If

If either

Let

We say that

the weak duality holds (between

the strong duality holds (between

the stable weak duality (resp., the stable strong duality) holds if the weak duality (resp., the strong duality) holds between

If

Let

To consider the weak duality, the strong duality, and optimality conditions for problem

The family

the weak closure condition at 0 (

the closure condition at 0 (

the weak closure condition (

the closure condition (

the Moreau-Rockafellar formula (MRF) at

(MRF) if it satisfies (MRF) at each point in

If

The following proposition describes the relationship between the

The family

Let

Under the assumption that

The following implication holds:

Suppose that the

Furthermore, suppose that (

To study the weak duality and the strong duality, we need the following lemma.

Let

(i) By the definition of the conjugate function, one has

(ii) Let

Conversely, suppose that

Our first theorem of this section shows that the

(i) The weak duality holds if and only if the family

(ii) The stable weak duality holds if and only if the family

As assertion (ii) is a global version of assertion (i). Hence, by Proposition

Conversely, suppose that the family

(i) The strong duality holds if and only if the family

(ii) The stable strong duality holds if and only if the family

As before, it is sufficient to prove assertion (i). Suppose that the strong duality holds. Let

Conversely, suppose that the

Let

Since

Furthermore, assume that

In the case when (

Recall the optimization problem

Let

(i) Suppose that for each

(ii) Suppose that for each

(i) Since for each

(ii) Since for each

The following theorems establish the relationships between the solutions of

(a) Obviously,

(b) Let

Let

By Theorem

Let

Let

Suppose that

Conversely, suppose that

Theorem

Below we will give a upper estimate for the Fréchet subdifferential of the function

Let

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

D. H. Fang was supported in part by the National Natural Science Foundation of China (Grant no. 11101186) and supported in part by the Scientific Research Fund of Hunan Provincial Education Department (Grant no. 13B095).