Equilibrium Theory

Since the existence of equilibrium in an abstract economy with compact strategy set in Rn was proved in a seminal paper of Debreu, many famous scholars have studied it, and there have been many generalization of Debreu's theorem .At same time , the development of equilibrium theory stimulates the study of many fundamental mathematical fields such as variational inequality theory and the fixed point theory . In the dissertation, equilibrium and several related topics have been systematically investigated, a series of results are established.The article is organized as follows .Chapterl is devoted to introducing the study background and some results obtained by the author.Chapter 2 mainly investigates the continuous selection theorem , variational inequality ,the fixed point theory. At first, the existence of extended form of approach to continuous selection for any set valued mapping without any continuity restriction in para-compact metric space is proved ;by the topologically separated mappings , the approximate selection theorem of sub-lower-semi-continuous mapping is established, furthermore , continuous selection problem in H-space is studied. Next, with W-correspondence, an improved variational inequality is obtained ; by the H-KKM mapping ,Ky Fan's minimax inequality is generalized to H-space . At last, with H-convexity instead of the linear topological structure, a new version of Browder fixed point theorem is established.Chapter 3 deals with set-valued mapping vector variational inequality and minimax problems. Firstly, the generalization of Fan-Ha section theorem and a general vector variational inequality without convexity assumption and minimax theorem of vector-valued function are obtained. Then, the quasi-montone vector variational inequality problem is concerned. Cx-quasi-monotone operator is defined in topological vector space, inner point of a closed convex set K is introduced, the relation between inner point and relative algebraic interior point is given, an existence result for quasi-monotone vector variational inequality is obtained.In chapter 4 , with S-KKM mapping , a new Ky Fan's minimax inequality is proved , the locally convex H-space with uniform structure is introduced , s-HKKM mapping is defined as the generalization of KKM mapping, with the H-convexity of Hausdorff locally convex H-space , a new fixed point theorem for the compact andclosed mapping belonging to the class s-HKKM(X,X,X) is established ;with local intersection property of muti-function ,K-H-quasi-convexity of mapping , connectedness of set and condensing correspondences, some new vector quasi-variational inequalities are established , respectively .In order to study vector-valued optimization problem, in chapter 5, a class of vector -valued function, that is, uniformly same-order set-valued function is introduced , which includes the separated functions as its proper subset; without hypothesis of convexity, new minimax theorem and saddle point theorem for uniformly same-order set-valued function are established. Next, by employing Ky Fan's lemma and H-KKM mapping , several existence results for generalized vector equilibrium problem established. At last, as the extension of (SVEP), a system of set-valued vector equilibrium problem(GSVEP) is introduced , with the quasi-convexity of set-valued function replacing the C-quasi-convexity of single-valued function, with the weakened continuity , the existence of a solution for (GSVEP) is proved .Chapter 6 discusses the equalibria of abstract economy . At first , an existence theorem of equilibrium for functional form of abstract economy is proved , then the Q e -class is introduced and an existence theorem of maximal element for a new type of preference correspondence which are Q0-majorized is given ; some existence theorems of equilibrium for abstract economy and qualitative game in which the constraint or preference correspondence are Q0-majorized are obtained in locally convex topological vector space. At last, the HM-class is introduced , some existence t...