KKM Type Theorems And Its Applications In Equilibrium Problems

KKM theory is one of the important parts in nonlinear analysis. In most of knownKKM theorems and applications, the convexity assumptions play a crucial role whichstrictly restrict the applicable area of KKM principle. How to weaken convexity is theimportant subject of KKM principle. Since KKM theory has aroused continuous interestdue to its promising applications in many fields of applicable sciences, such as equi-librium problem, game theory, mechanics and physics, economics and finance, trans-portation and operations research, variational inequality and complementarity problem,optimization and control problem. For recent decades, the KKM theory and Its applica-tions have been extensively studied.In this thesis, some new KKM type theorems and vector equilibrium problems areinvestigated. The thesis is composed of two parts.Part I(2-3):KKM type theorems and Its applications in topological spacesChapter 1 mainly introduces the history of KKM theory.Chapter 2 introduces a new concept of generalized L-KKM mapping and estab-lishes some new generalized L-KKM type theorems without any convexity structure intopological space. As application, an existence theorem of equilibrium point for abstractgeneralized vector equilibrium problem is proved in topological space.Based on Chapter 2, Chapter 3 first establishes some new nonempty intersectiontheorems for generalized L-KKM mappings and proves some new fixed point theoremsfor set-valued mappings in topological spaces. As applications, an existence theorem foran equilibrium problem with lower and upper bounds and two existence theorems fora quasi-equilibrium problem with lower and upper bounds are obtained in topologicalspaces. Part II(4-7): KKM type theorems and some vector equilibrium problems in FC-space.A new coincidence theorem and some KKM type theorems for better admissibleset-valued mappings are established in Chapter 4. As applications, some new existencetheorems of solutions for several classes of generalized vector equilibrium problems areestablished in noncompact FC-space.Chapter 5 studies a class of generalized vector variational-type inequality problems(in short, GVVTIP) in FC-space, which includes the most of vector equilibrium prob-lems, vector variational inequality problems, generalized vector equilibrium problemsand generalized vector variational inequality problem as special cases. As applications,some new existence results for GVVTIP are established in noncompact FC-space.Chapter 6 obtains a new nonempty intersection theorem. As application, some newtheorems of equilibrium problems and quasi-equilibrium problems with lower and upperbounds are established.Chapter 7 introduces and studies a new system of generalized mixed vector quasi-equilibrium problems, which includes as special cases the system of generalized vectorvariational-like inequality problems, the system of generalized quasi-equilibrium prob-lems, and the system of generalized equilibrium problems. By using the maximal ele-ment theorem, some existence results of solutions for the system of generalized mixedvector quasi-equilibrium problems are given.