Research On Some Nonlinear Problems In Probabilistic Metric Spaces

In probabilistic metric spaces, the distance between two elements is measured bya distribution function and an ordinary metric space can be viewed as a special caseof a probabilistic metric space. Consequently, the research of nonlinear operators inPM-spaces is of great significance. In this thesis, some problems for several kinds ofnonlinear operators in PM-spaces are studied. It is divided into the following foursections.In chapter one, the backgrounds and current situation of operator theory in PM-spaces are introduced and the preliminaries of PM-spaces are given.In chapter two, the existence and uniqueness theorems for a class of set-valuedand single-valued nonlinear operator equations sequence with (Φ,Δ)-typeprobabilistic contractor sequence and (φ,Δ)-type probabilistic contractor couplesequence in Menger PN-spaces are presented and proved. Utilizing these theorems,some fixed point theorems are obtained.In chapter three, the concepts of compatible mappings and weakly compatiblemappings in multi-valued case in Menger PM-spaces are introduced. Based on this,some fixed point theorems and coincidence point theorems under hybrid contractionsin Menger PM-spaces are established. Also, a common fixed point theorem forweakly compatible mappings in single-valued case is proved. Finally, someapplications in ordinary metric spaces or fuzzy metric spaces are given.In chapter four, the topological degree of 1-set-contractive operators inPN-spaces are newly defined, and some fixed point theorems are obtained.Meanwhile, using the concept and properties of accretive operators in PN-spaces, theexistence of the solution of 1-set-contractive perturbation equations with accretiveoperators are discussed.