Research On Some Nonlinear Problems In Fuzzy Metric Spaces

In this paper,some new concepts are introduced.Some problems,including the existence and uniqueness for the fixed point of nonlinear operator are investigated in fuzzy metric spaces and generalized fuzzy metric spaces.Utilizing the iterative method in fuzzy metric spaces,a series of new results are given.Moreover,some important theorems are proved and generalized.In chapter one,we introduce the backgrounds of fuzzy metric spaces,current situation about fixed point theory and some notions in fuzzy metric spaces.In chapter two,by improving the characteristic function and using fuzzy ?-contraction,we investigate the existence and uniqueness of fixed point in fuzzy metric spaces deeply.In chapter three,we introduce ?-partial ordered in fuzzy metric spaces,and extend the dimension of fuzzy metric spaces.After giving the notion of generalizing mixed monotone operators,we obtain some new coincidence point and common fixed point theorem.In chapter four,we construct a new metric to estimate the distance of two closed complete bounded sets by using concept of net.Then we introduce the notion of fuzzy net metric spaces and discuss the existence and uniqueness of fixed point in fuzzy net metric spaces after generalizing the notion of fuzzy contraction mappings.In chapter five,some definitions about ?-fuzzy graph are given in metric spaces.And some iterative scheme results for G-contractive and G-nonexpansive maps on graphs are presented.A fuzzy fixed point theorem in metric spaces are obtained by using ?-fuzzy graph.