Research On Some Problems In Probabilistic Metric Analysis

It is well known that an ordinary metric space can be regarded as a special probabilistic metric space. Consequently, the research on nonlinear problems in probabilistic metric spaces is great significant. In this thesis, some problems for nonlinear operators and their applications in probabilistic metric spaces are studied. It is divided into four sections as following.In chapter one, the background and current situation of the nonlinear operator theory in probabilistic metric spaces are introduced, and the preliminaries of probabilistic metric spaces are given. The main problems studied in the thesis, the expected conclusions and the study and research significance of this thesis are introduced.In chapter two, by virtue of the Topological Degree in Menger PN-spaces the problems on the existence of the fixed point of nonlinear operators in the Z-P-S space are studied and some new fixed point theorems are obtained. Meanwhile, some important conclusions are generalized.In chapter three, the new concepts of the bifurcation point and asymptotic bifurcation point of compact continuous operators in Menger PN-spaces are introduced. Some sufficient conditions for the existence of the bifurcation point and asymptotic bifurcation point are obtained. And the problems on the intrinsic value and intrinsic element of compact continuous operators are studied in Menger PN-spaces. A series of theorems on the existence of the intrinsic value and intrinsic element are obtained. Meanwhile some results are generalized.In chapter four, the Leray-Schauder Topological Degree of compact continuous operators in probabilistic normed spaces is generalized. The new concept of the fixed point index of compact continuous operators in probabilistic normed spaces is introduced. Some new conclusions of the fixed point index in probabilistic normed spaces are obtained; meanwhile some of their properties are studied. As an application, the problems of the existence of the fixed point and the solution of a class of differential equations in PN-spaces are considered.