Nonlinear Differential Equations And Their Applications

?Nonlinear functional analysis is an important branch of mathmatics, be-cause it can explain all kinds of natural phenomena, more and more mathe-maticians are devoting themselves to it. The boundary value problems (BVPs)for nonlinear differential equations arise in a variety of areas of applied math-ematics, physics and variational problems of control theory, it is at presentone of the most active fields in analyse mathematics. Among them, impulsivedifferential equations come from a lot of branches of applied mathematics andphysics, and they are very meaningful in both practical and theoretical aspects.The present paper employs the cone theory, fixed point theory, topological de-gree theory and upper-lower solutions method and so on, to investigate theexistence of positive solutions to some kinds of nonlinear impulsive differentialequations. The results obtained are either new or essentially generalize andimprove the previous relevant ones under weaker conditions.?The thesis is divided into three chapters according to contents:?In Chapter 1, motivated by [4, 8], we consider the the existence of minimaland maximal solutions, the unique positive solutions of periodic boundaryvalue problem for the second order impulsive differential equations in Bananchspaces,when the problem have upper and lower solutions in the reversed order, byusing of the order method and new comparison principles. we aiso obtain aniterative sequence approximating the unique solution of the problem betweenlower and upper solutions. The theorem improves and generalizes the resultsof [8] to the impulsive differential equations.?In Chapter 2, by using the Schauder fixed-point theorem, we study theexistence of positive solutions of infinite boundary value problem for the second order differential equations,The theorems generalizes the results of [11-15].?In Chapter 3, by using of fixed point theorem, we study the solutions ofboundary value problem for the second order impulsive differential equations,and obtain the sufficient conditions of the existence of three positive solutions.