The Existence And The Uniqueness Of Positive Solutions Of Two Classes Of Nonlinear Boundary Value Problems

In the first chapter,we introduce the development of nonlinear boundary value problem and nonlinear operator theory.In the second chapter,we describe the basic concepts and basic lemmas.In the first section of the third chapter.by using the fixed point theorems of generalized concave and convex operators and the properties of the Green functions,a class of fourth-order boundary value problems is studied as the follow: The existence and uniqueness of positive solutions for this boundary value problem is obtained under two different conditions.The results obtained generalize and com-pliment the related existing ones in references, which has theoretical significance and reference value on boundary value problems.In the second section of the third chapter, the purpose is to investigate the exis-tence and the uniqueness of symmetric positive solutions for the following fourth-order boundary value problem: By using the fixed point index method, we establish the existence of at least one or at least two symmetric positive solutions for the above boundary value problem. Further, by using a fixed point theorem of generalα-concave operators, we also present criteria which guarantee the existence and uniqueness of symmetric positive solutions for the above boundary value problem.The fourth chapter is concerned with the existence and uniqueness of positive solutions for a class of Neumann boundary value problems of second order impulsive differential equations as the follow:The result is obtained by using a fixed point theorem of generalized concave operators.