Research On Some Problems In Random Nonlinear Analysis

Random topological degree theory is an indispensable part of random nonlinear functional analysis and is of great significance in the research of random nonlinear differential and integral equations. In this paper, by using the random topological degree theory and combining a new kind of functions which is more general than convex functions and concave functions, we investigate some computational problems of random topological degree. And the definitions of partially weakly increasing property of a pair of random mappings(F, G) with respect to g, partially weakly increasing property of(F, G), random g-mixed monotone property and random commutative property for a pair of random mappings F: Ω ×X ×X ×X â†'X and g: Ω ×X â†'X are introduced. The exstence of common coupled random coincidence points, common coupled random fixed points, tripled random coincidence points and tripled random fixed points under various contractive conditions in complete and separable partially ordered metric spaces are studied. Many new results are obtained, which generalize some results in the corresponding literatures. Finally, some examples are given to illustrate our main results. There are three chapters in this thesis.In chapter one, Mainly introduce the historical backgrounds and current situation of random fixed point theories in random nonlinear analysis and the document related knowledge.In chapter two, using(α-m)-convex and strictly(α-m)-concave functions which are more general than common functions, we investigate some computational problems of random topological degree.In chapter three, the definitions of partially weakly increasing property of a pair of random mappings(F, G) with respect to g, partially weakly increasing property of(F, G), random g-mixed monotone property and random commutative property for a pair of random mappings F: Ω ×X ×X ×X â†'X and g: Ω×X â†'X are introduced. And the exstence of common coupled random coincidence points, common coupled random fixed points, tripled random coincidence points and tripled random fixed points under various contractive conditions in complete and separable partially ordered metric spaces are studied.