Some Problems For Several Classes Of Nonlinear Operators In Banach Space

Nonlinear functional analysis is an important branch of mathematics. The fixed point theory for nonlinear operators, which originated from both the theoretical study and practical applications of many disciplines such as physics, biology and economics, has become one of the most active realm of research, and thus has theoretical as well as practical significance. In this paper, utilizing cone theory and partial ordering method as well as topological degree theory, the existence and uniqueness of fixed points as well as the solutions of operator equations for several classes of nonlinear operators are extensively studied. The thesis is divided into three chapters as following:In Chapter one, the historical background and current situation of the work related to this thesis are introduced and some preliminaries are given.In Chapter two, using fixed point index theory, the existence and uniqueness of the solutions for semi-closed1-set-cotractive operators are investigated, which are applied to study a class of second order2-point boundary value problem. It is worth pointing out that existence theorems for operator equations Ax=μx generalize many famous fixed point theorems to the case of semi-closed1-set-contractive operators.In chapter three, first of all, the existence and uniqueness of fixed points for a class of symmetric contractive binary operators is studied. Based on this, the existence and uniqueness theorem of the solutions for a class of binary operator equations is obtained. Secondly, making use of cone theory and Banach's contraction mapping principle, the existence and uniqueness of the solutions for a class of mixed non-monotone binary operator equations in ordered Banach spaces are discussed, and some new results are obtained.