Research On Some Problems Of Operator Theory In The PN-Space

In 1942,K.Menger introduced the concept of probabilistic metric space. In his base thought, the distance between two elements of the space was considered to be random. So it was presented by a distribution function and not by a nonncgativc real numbers. Obviously, it further conformed to reality. Moreover, general metric space can be looked on as the special case of probabilistic metric space. In the whole tliesis we investigate some problems of operator theory in the PN-space. This thesis is divided into three sections.In chapter one, we introduce the background, present status of operator theory and the preparatory knowledge in the PN-space.In chapter two, we study the solutions of operator equations Ax = x in the PN-space, and obtain some new results. Moreover, we create a new space, L-M-PN-space, on the base of Menger PN-space, in which the existence of thesolutions of nonlinear operator equations Ax = μx (μ ≥ 1) is discussed.In chapter three, based on the topological degree theory of A-proper mappings, we build and discuss the topological degree of monotone mappings in the PN-space,and establish some property of it.