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The existence of solutions and well-posedness for bilevel mixed equilibrium problems in Banach spaces. (English) Zbl 1280.49016

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

MSC:

49J40 Variational inequalities
49K40 Sensitivity, stability, well-posedness
49J27 Existence theories for problems in abstract spaces
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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