Research On Well-posedness And Algorithms Of Vector Equilibrium Problems

In this paper,we mainly study the well-posedness and algorithm of vector equilibrium problem.Firstly,in the setting of real Banach spaces,we use the conditions of cosmically upper continuity and Hausdorff upper semi-continuity for variable domination structures to establish the well-posedness of two types of bilevel vector equilibrium problems with variable control structures.Secondly,we combine with the tecnniques of generalized projection and inertial to give an inertial extragradient method for finding a common solution of vector equilibrium problem and fixed point problem,and obtain convergence results.These results obtained in this paper extend and develop some recent works in the literature.The full text is divided into four chapters as follows:In the first chapter,we mainly introduce the research background and development process,and describe the main research content and the significance of equilibrium problem.And the structure of this paper is briefly described.In the second half of this chapter,we give some concepts,lemmas and known conclusions to be used in this paper.In the second chapter,we introduce and study well-posedness for two types of bilevel vector equilibrium problems with variable domination structures.By virtue of the conditions of cosmically upper continuity or Hausdorff upper semi-continuity for variable domination structures respectively,in the setting of real Banach spaces and under the conditions of proper continuity of equilibrium mappings,some sufficient and necessary conditions of Levitin-Polyak well-posedness and Levitin-Polyak well-posedness in the generalized sense for strong bilevel vector equilibrium problem and weak bilevel vector equilibrium problem with variable control structure are obtained respectively.In the third chapter,we mainly consider an algorithm for finding a common solution of vector equilibrium problem and fixed point problem.In a uniformly convex uniformly smooth real Banach space,by means of generalized projection and inertia iterative technique,an inertial extragradient method for solving common solution is presented.Then,by using the dual cone and its quasi-interior properties,the feasibility of the algorithm is analyzed under suitable conditions of continuity,conic convexity and monotonicity.Further,some convergence theorems are obtained.In the last chapter,we summarize the main research results and put forward a outlook.