Research On Some Problems Of Existence Of Fixed Points For Nonlinear Mappings In Menger PM-Spaces

In this paper, we introduce many new concepts in Menger probabilistic metric spaces. Based on this, utilizing partial order method and iterative method, the existence and uniqueness of fixed points for nonlinear mappings in Menger probabilistic metric spaces are investigated and some new theorems are established.Meanwhile, many examples are presented as applications of our main results. The obtained results generalize and improve many fixed point theorems and coincidence point theorems in recent corresponding literatures. It is divided into the following three sections. The following chapter:In chapter one, we introduce the development situation, the latest cutting-edge and the related issues of the research in Menger PM-spaces of nonlinear functional analysis. Meantime, we also give some basic concepts in Menger PM-spaces.In chapter two, we introduce a new concept of generalized cyclic)( Lκh,j-weak contraction mappings and obtain some coincidence point results for such mappings in complete partially ordered Menger PM-spaces.In chapter three, we introduce the concepts of F-contraction and probabilistic -contraction mappings in generalized Menger probabilistic metric spaces, and prove some fixed point theorems for such two types of mappings in generalized Menger probabilistic metric spaces.Our results generalize and extend many results in existing literature.Some specific examples are given. Meantime, an application to the existence of solutions for a class of integral equations is presented by utilizing one of our main results.