| Nonlinear functional analysis is an important branch of mordern analysismathematics,more and more mathematicians are devoting their time to it which can explain many kinds of natural phenomena. Among them,the positivefixed point theory for nonlinear operators plays a very important role in nonlinear integral and differential equations, especially in the study of positive solutions of nonlinear integral equations in nuclear physics.In this paper,we discuss the existence of the differential equations with integralboundary value problems using methods of cone theory and Topological degree. And we get the new results. The thesis is divided into three sections according to contents.Chapter 1 is the introduction of this paper, which introduces development of nonlinear functional analysis and give some instruction of our problems.In Chapter 2, we consider the singular second-order integral boundary-valueproblems under some conditions concerning the first eigenvalues to the relevant linear operators. In particular, f is not necessary to be nonnegative.The existence results of nontrivial solutions are given by means of the topological degree theory. The results arc illustrated with an example.In Chapter 3, we study the existence of nonncgativc solutions of integralboundary value problems for second-order impulsive differential equations. Our analysis relies on combining a nonlinear alternative of Lerary-Schauder type with Krasnoselskii's fixed-point theorems in a cone. The results arc illustratedwith an example, and give some remarks in the end.The improvements of this paper will arise in two aspects. First, to our knowledge, the general boundary value condition ofhas not been studied, because there are some difficulties in the process of differentialequation converts into integral equations. Motivated by [1], we conqued this difficulty. Secondly, to the best of our knowledge, there is not much con- cerning for impulsive equation with the integral boundary value condition of... |
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