Study On Some Problems For Nonlinear Operators In PM Spaces

In this paper,some new concepts are introduced.Some problems,including solvability of nonlinear operator equation,common solvability of nonlinear operator equations,intrinsic value and intrinsic element for continuous compact operator,are investigated in probabilistic metric spaces.Utilizing the topological degree method and the iterative method in probabilistic metric spaces,some important theorems are improved and generalized.Moreover,a series of new results are gained.It is divided into the following four sections.In chapter one,the backgrounds,current situation and the preliminaries of PM spaces is given.In chapter two,utilizing the topological degree method,the existence of solution for nonlinear operator equations Tx=μx and Tx=μx+q in PM spaces is studied, and the restrict conditionμ≥1 is generalized toμ＞0.In chapter three,the concept of strong probabilistic contractor be introduced, utilizing the iterative method,some theorems for nonlinear operator equations in non-Archimedean Menger PN spaces are generalized to Menger PN spaces.In chapter four,some problems of the intrinsic value and the intrinsic element for nonlinear operator are investigated and several theorems of continuous compact operator for the intrinsic value and the intrinsic element are obtained in Menger PN spaces and Z-P-S spaces.