Non-cooperative Game Strategy For Stability And Abstract Convex Structure Of A Number Of Issues

Recently, essential components of solution set is an important aspect of stability for nonlinear problems. It plays a crucial role in the study of stability of solutions including optimal solutions, Nash equilibrium, fixed points, etc. In the chapter one, a concept of continuously strongly essential sets of Nash equilibrium points is introduced, and the existence and the connectence of continuously strongly essential set are proved. That is, we show that every n-person noncooperative game has at least one continuously strongly essential component of the set of Nash equilibrium points.Then the relationg among some definitions of strategic stability is discussed.The convexity of space plays a very important role in fixed point theory and continuous selection theory. In the second chapter, the further properties of abstract convexity structures are introduced . A new fixed point theorem and the approximate selection property for upper semi-continuous set-valued mappings in abstract convexity space is established. At the same time, the Generalizations of Ky Fan's Lemma,Ky Fan's variational inequalities theorem,minimax inequalities theorem and Schauder's fixed point theorem in abstract convexity space are given. As application the existence of Nash equilibrium points for n-person noncooperative game and n-person noncooperative generalized game is proved.Let M be the collection of all KKM mappings T in abstract convexity space and denote byF(T) the set of all KKM points of T. We prove that there exists a dense residual subset Q of M such that for each T∈Q; T is stable and there exists at least one essential component of F(T) for each T∈M. And the KKM technique is presented, using thistechnique a new alternative theorem and a new coincidence theorem in abstract convexity space are established.