Oscillation Of Delay Difference Equations And Boundary Value Problems Of Differential Equation

This thesis, focus on the oscillation of delay difference equations and the existence of multiple positive solutions for p-Laplacian boundary value problems(BVP), consists of three parts.In the preface, the author introduces the development of these topics briefly, and gives the main questions studied.Chapter one concentrates on the relations about the properties of the solutions between the higher-order nonlinear difference equationsand the equationwhere m is a positive integer, and Ï„(n) is an integer. The author analyses the effect on the properties of solutions caused by the advanced argument (T(n) > n) and the retarded argument (Ï„(n) < n). The odd-order and the even-order difference equations are studied respectively. Especially, the author establishes the equivalence of the oscillation of the above mentioned two difference equations when m is even, i. e, the deviating arguments have no effect upon the oscillation of the equations. These results extend the corresponding results in [22,23]. Meanwhile, some interesting conclusions are obtained when m is odd.Chapter two discusses the oscillation of a second order nonlinear differential equation. Sufficient conditions for the oscillation of all solutions of the equation are obtained by using the Riccati transformation and certain inequality.The author proves the existence of multiple positive solutions for p-Laplacian BVPs by applying some fixed point index theorems in cones in Chapter three. Our results can solve some BVPs which can not settled in previous papers. So the results in Chapter three make up and complement the former results in the literature on this topic.