Theory Research On Two Classes Of Equilibrium Problems

In this dissertation, we study the theory aspects on one-level equilibrium problems and bilevel equilibrium problems. The research content of this disser-tation includes six parts as follows:In the first part, we consider a class of parametric generalized mixed equi-librium problems (PGMEP) in Hausdorff topological vector spaces, where the constraint set K and a set-valued mapping T are perturbed by different param-eters, and establish the nonemptiness and upper semicontinuity of the solution mapping S for (PGMEP) under some suitable conditions. By virtue of the gap function, sufficient conditions for the H-continuity and B-continuity of the solu-tion mapping S of (PGMEP) are also derived. Finally, examples are provided for illustrating the presented results.In the second part, a concept of weak f-property for set-valued mapping is introduced, and then under some suitable assumptions, which do not involve any information about the solution set, the lower semicontinuity of the solution mapping to the parametric set-valued quasi-equilibrium-like problems are derived by using a density result and scalarization method, where the constraint set K and a set-valued mapping H are perturbed by different parameters.In the third part, we study the Levitin-Polyak well-posedness by perturba-tions for a class of general systems of set-valued vector quasi-equilibrium problems (SSVQEP) in Hausdorff topological vector spaces. Existence of solution for the system of set-valued vector quasi-equilibrium problem with respect to a param-eter (PSSVQEP) is established. Some sufficient and necessary conditions for the Levitin-Polyak well-posedness by perturbations are derived by the method of continuous selection. We also explore the relationships among these Levitin-Polyak well-posedness by perturbations, the existence and uniqueness of solution to (SSVQEP). By virtue of the nonlinear scalarization technique, a parametric gap function g for (PSSVQEP) is introduced, which is distinct from that in [141]. The continuity of the parametric gap function g is proved. Finally, the relations between these Levitin-Polyak well-posedness by perturbations of (SSVQEP) and that of a corresponding minimization problem with functional constraints are also established under quite mild assumptions.In the fourth part, 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 prob-lem and the problem (BMEP) are established without eoercivity. Subsequently, we construct a new iterative algorithm which can directly compute some solutions of the problem (BMEP). Some strong convergence theorems of the sequence gen-erated by the proposed algorithm are also presented. Finally, the well-posedness and generalized well-posedness for the problem (BMEP) are introduced by an e-bilevel mixed equilibrium problem. Also, we explore the sufficient and nec-essary 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.In the fifth part, a class of bilevel invex equilibrium problems of Hartman-Stampaeehia type and Minty type (resp., in short,(HSBEP) and (MBEP)) are firstly introduced in finite Euclidean spaces. The relationships between (HSBEP) and (MBEP) are presented under some suitable conditions. By using fixed point technique, the nonemptiness and compactness of solution sets to (HSBEP) and (MBEP) are established under the invexity, respectively. As applications, we in-vestigate the existence of solution and the behavior of solution set, to the bilevel pseudoinonotone variational inequalities (see [2]) and the solvability of minimiza-tion problem with variational inequality constraint.In the sixth part, a class of bilevel variational inequalities (for short,(B-VI)) having hierarchical nesting structure is firstly introduced and investigated. The relationships among (BVI) and some existing bilevel problems are present-ed. Subsequently, the existence of solution and some characteristics of solution sets to (BVI) and the lower level variational inequality are discussed without eoercivity. By using the penalty method, we transform (BVI) into one-level vari-ational inequality, and establish the equivalence between (BVI) and the one-level variational inequality. A new iterative algorithm to compute the approximate solutions of (BVI) is also suggested and analyzed. The strong convergence of the iterative sequence generated by the proposed algorithm is derived under some mild conditions. Last but not the least, some relationships among (BVI), system of variational inequalities and vector variational inequalities are also given.