Research On Nonlinear Operators Equations And Related Problems In The Partial Banach Space

Nonlinear operator fixed points and equations theory is an important part of the nonlinear function analysis theory, it plays an essential role in all kinds of the mathematical equations especially. People often discuss the problems which were attributed to some certain fixed points and operator equations when studying differential integral equations and value theory. The purpose of this thesis is to discuss the fixed points of nonlinear operators and the existence and uniqueness of the solutions about nonlinear operator equations. Moreover, the theory of cone and the techniques of partial order theory are used to discuss the above questions under certain conditions. This thesis includes four chapters:In chapter one, a brief introduction and an overview of background materials and current situation of my research are given. At the same time, we recall some elementary concepts for this thesis.In chapter two, firstly, by using the theory of cone and iterated method, the solveability theorems for ordered contractive operator equations Lx= Ax and Lx= A(x, y) are discussed in Banach space. New concept and method are introduced, and the existence theorems of solutions for those operator equations are given where L needn't to be continuous, We also construct iterative sequences and study their approximation. Secondly, fixed point theorems of mixed monotone operator are studied, the contraction mapping is extended to the form of function and operator where the operator needn't to be continuous, the corresponding conclusions are generalized.In chapter three, the concept of rτ<φconcave (-Ψ) convex operator is introduced and a class of mixed monotone operators are discussed. The existence and uniqueness for the fixed points of such a class of operators is obtained, so the fixed point theorems for mixed monotone operators such as concavity-Guo convexity, concavity-(-α) convexity, a concavity- convexity, a concavity- Guo convexity, e concavity- Guo convexity, e concavity-(-α) convexity, e concavity- convexity,α1 concavity-(-α2) convexity are obtianed. Finally, some of the present results are applied to the Hammerstein integral equations on unbounded regions.