Research On Some Problems Of Nonlinear Operators In The PM-Space

In probabilistic metric spaces, the distance between two elements is measured by a distribution function and an ordinary metric space can be viewed as a special case of a probabilistic metric space. Consequently, the research of nonlinear operators in probabilistic metric spaces is of great significance. In this thesis, some problems for nonlinear operators and their applications in PM spaces are studied. It is divided into the following four sections.In chapter one, the backgrounds and current situation of operator theory in PM spaces are introduced and the preliminaries of PM spaces is given.In chapter two, utilizing the topological degree method, the existence of fixed points for compact continuous operators(1/μ)T (μ≥1) in PN spaces is studied and anapplication for a class of differential equations is given.In chapter three, some problems of the intrinsic value and the intrinsic element for nonlinear operator T in PN spaces are investigated. A series of sufficient conditions for which the compact continuous operator T has intrinsic values y larger than 1 and has intrinsic elements on dW corresponding with y are established, Finally, the existence of solutions for operator equations Tx = μx + p(μ≥1) is discussed.In chapter four, a partial order is introduced in PM spaces. By using the partial order method, the solvability for a class of nonlinear operator equations Lx=Ax in PM spaces is discussed and some new conclusions are obtained, which turns out to be the improvement and generalization of some important theorems.