Monotonicity Of Set-valued Mappings And Applications To The Existence Of Solutions Of Variational Inequalities

Equilibrium problems provide a mathematical framework to what includes optimization,variational inequalities,fixed-point and saddle point problems,and noncooperative games as particular case.Being a original model of equilibrium problems,the classical variational inequality problems are widely used at least in physics,engineering,economics and many other fields.With the expansion of the scientific research’s category,more and more researchers extend classical variational inequalities to set-valued variational inequalities.Furthermore,the methods of set-valued variational inequality problems are countless.Similaring to the mathematical programming problem in the constraint set of convexity,monotonicity of set-valued mappings play a vital role in solving set-valued variational inequalities.This dissertation mainly studies the monotonicity of set-valued mappings,constructs bifunctions and set-valued bifunctions and applies their properties to characterize the monotonicity of set-valued mappings.We afterwards apply these results to the existence of solutions of set-valued variational inequality problems.In this dissertation,the main contents are as follows:In the first chapter,we introduce the research background and the situation at home and abroad,and the main work of this dissertation.In the second chapter,we recall some related concepts and conclusions which are used in this dissertation.In the third chapter,we give some specific examples to verify the implication relations of the six monotonicity of set-valued mappings,bifunctions and set-valued bifunctions.In the fourth chapter,we construct bifunctions and set-valued bifunctions to characterize the monotonicity of set-valued mappings,and prove in detail some equivalent conditions between the monotonicity of bifunctions and set-valued bifunctions and that of set-valued mappings.In the fifth chapter,we obtain the existence of set-valued variational inequalities by using the equivalence relation of pseudomonotonicity of bifunctions and pseudomonotonicity of set-valued mappings.