THE EXISTENCE OF SOLUTIONS AND WELL-POSEDNESS FOR BILEVEL MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

Abstract

In this paper, a new class of bilevel mixed equilibrium problems (for short, (BMEP)) is introduced and investigated in reflexive Banach space and some topological properties of solution sets for the lower level mixed equilibrium problem and the problem (BMEP) are established without coercivity. Subsequently, we construct a new iterative algorithm which can directly compute some solutions of the problem (BMEP). Some strong convergence theorems of the sequence generated by the proposed algorithm are also presented. Finally, the well-posedness and generalized well-posedness for the problem (BMEP) are introduced by an $\epsilon$-bilevel mixed equilibrium problem. Also, we explore the sufficient and necessary conditions for (generalized) well-posedness of the problem (BMEP) and show that, under some suitable conditions, the well-posedness and generalized well-posedness of (BMEP) are equivalent to the uniqueness and existence of its solutions, respectively. These results are new and improve some recent results in this field.