The Study Of The Essential Set Of Fixed Points Of Set-valued Mapping And Related Problems

Brouwer fixed point theorem and Kakutani fixed point theorem were successfully applied to the field of game theory, mathematical economics and so on.especially Nash used Brouwer fixed point theorem to establish the non cooperative game theory, Arrow and Debreu used Kakutani fixed point theorem to prove the existence of general equilibrium.The fixed point theorem attracted the attention of more widely.Game theory is a branch of operations research, non cooperative game is foundation and the core in the game theory,moreover,the Nash equilibrium is the most important core concept in the non cooperative game theory.Between Nash equilibrium and nonlinear problems' solution,such as set-valued mapping fixed point and variational inequalities' solution.they have some equivalent relations, In this thesis,it is from these equivalence, through the stability of set-valued mapping fixed point,Ky-Fan lemma and variational inequalities' solution,mainly by improving the perturbation of set-valued mapping and function,we introduced several strong perturbation to study the stability of solutions of these problems.In this thesis,we got the strong essential stability of set-valued mapping's fixed point under the strong ?-perturbation,Using this result, we got the strong essential set of Nash equilibrium,and the existence of essential component of Ky-Fan section problem and Ky-Fan inequality's solution.Moreove,we studied the solution's stability of non cooperative vector game.The first chaper:we introduced the basic knowledge,including the Upper and lower semi continuity of set valued mappings,Fan-Glicksberg fixed point theorem, Fort lemma, dense residual set, the essential fixed point, the essential set of fixed point,the essential component,Hausdorff distance and so on.In the second chapter:we studied the stability of the fixed point and the stability of Nash equilibrium strategy set in the case of strong ?-perturbation.The stability of the fixed point under the strong ?-perturbation is the strongest,The stable set of the corresponding Nash equilibrium strategy——the SBR- stable set has the stability under the mapping perturbation,the mapping and definition domain common perturbation.The corresponding strong?-perturbation of the SBR- stable set is defined by the Hausdorff distance,no any other conditions attach to distance,so this method can be used to study any finite noncooperative game.The third chapter:Using the strong ?-perturbation,we got the strong continuity and stability of Ky-Fan section problem,and we got the strong essential component of Ky-Fan section problem under the strong ?-perturbation and graph ?-perturbation.From this strong essential stability,we got the strong essential component of Ky-Fan inequalities solution,through the result of Ky-Fan point,we got the strong essential component of Nash equilibrium.Section 3.5 is aimed at the stability of Ky-Fan point and Nash equilibrium point,we considered the perturbation of the point where the inequality function or the payment function may satisfy the point where the Ky-Fan point or Nash equilibrium point condition is satisfied.Further expanded the stable perturbation range of Ky-Fan point set and Nash equilibrium point set.In the fourth chapter:we studied the stability of the Pareto-Nash equilibrium of the non cooperative vector game.The ideal-Nash equilibrium point, satisfactory-Nash equilibrium point and the layer Pareto-Nash equilibrium point of the vector game, are all valid Pareto points.They all have the continuity(upper semi continuity need condition), generic stability and essential component of Pareto-Nash equilibrium.