Nguyen Hong Quan

Generic Stability and Essential Components of Generalized KKM Points and Applications

We propose the definition of T-KKM points and consider generic stability of T-KKM mappings and essential components of sets of T-KKM points. As applications, using a unified approach, we derive from these results the existence of essential components of solution sets to various optimization-related problems. We do this in two steps. First, we deduce the corresponding results for variational inclusions, which are new. Then we obtain, as consequences, the existence of essential components of solutions to other problems, which are new or include recent ones in the literature.

Intersection theorems, coincidence theorems and maximal-element theorems in GFC-spaces

We propose a definition of GFC-spaces to encompass G-convex spaces, FC-spaces and many recent existing spaces with generalized convexity structures. Intersection, coincidence and maximal-element theorems are then established under relaxed assumptions in GFC-spaces. These results contain, as true particular cases, a number of counterparts which were recently developed in the literature.

Continuous Selections, Collectively Fixed Points and Weak Knaster–Kuratowski–Mazurkiewicz Mappings in Optimization

We prove theorems on continuous selections, collectively fixed points, collectively coincidence points, weak Knaster–Kuratowski–Mazurkiewicz mappings and provide their applications in various optimization-related problems. Each of our theorems is demonstrated by using its preceding assertions. The results contain and improve a number of existing ones in the recent literature. They are shown to be also more advantageous in applications in optimization.