α-well-posedness Of Systems Of Equilibrium Problems And Characterization Of The Solution Sets Of Mixed Variational Inequality Problems

Variational inequality is the important part of the nonlinear analysis. The theory ofvariational inequalities is widely applied to optimization, cybernetics, economic balance,etc. In the appropriate condition assumptions, the solution set of variational inequalityproblem is consistent with the solution set of optimization. In order to meet the demandsof solving practical problems, many generalizations to variational inequality have beenmade, among which the mixed variational inequalities is an important form.Equilibrium problems provide a unifying formulation of minimization problems,variational inequality problems, fixed point problems and complementarity problems,Nash equilibrium problems, which are applied to abstract economic. In addition,well-posedness is a classical notion in optimization problems, variational inequalitiestheory and equilibrium problems, it has had the profound influence on optimizationproblems, variational inequalities and equilibrium problems theory. The well-posednessof equilibrium problems is to study the represent of the solution,it considers theapproximating sequence of equilibrium problems converging to solution for equilibriumproblems.This paper is concerned with characterizations of the solution sets of mixedvariational inequality problems under the subdifferential and dual gap function, discussthe relationships of solution sets between the mixed variational inequalities problemsand the dual problems, introduces the concepts of α-well-posedness and generalizedα-well-posedness for a system of equilibrium problems, research related nature ofα-well-posedness and generalized α-well-posedness.In chapter1, we sum up academic research significance of mixed variationalinequality, characterizations of solution sets, equilibrium problems and well-posednessand current research situation at home and abroad.In chapter2, we introduce the concepts of α-well-posedness and generalizedα-well-posedness for system of equilibrium problems; derive metric characterization ofα-well-posedness by considering the diameter of α-approximating solution set and ch -aracterization of generalized α-well-posedness by considering the noncompactnessmeasure of α-approximating solution set. Under suitable conditions, we research thesufficient conditions of generalized α-well-posedness.In chapter3, we introduce knowledge in relate to mixed variational inequalityproblems, discuss the relationships of solution sets between the mixed variationalinequalities problems and the dual problems and then introduce the dual gap function inrelate to mixed variational inequality problems,thus the corresponding relationshipsbetween the mixed variational inequalities problems and the optimization problems areestablished. Finally, we characterize the solution sets of mixed variational inequalityproblems.In chapter4, we give a summary of this paper and put forward some problems forfurther study.