Nonlinear Differential Equation Boundary Value Problems And Its Applications

Nonlinear functional analysis is an important branch of modern analysis mathematics, because it can explain all kinds of natural phenomena, more and more mathematicians are devoting their time to it. Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analytical mathematics. The present paper employs the cone theory, fixed point index theory, and Krasnoselskii fixed point theorem and so on, to investigate the existence of solutions to boundary value problem of several kinds of nonlinear differential equations. The results obtained are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into three sections according to contents.In chapter 1, by constructing a special cone and applying the fixed index theory in the cone, the existence of positive solutions for a class of singular m-point boundary value problemsis investigated, which improve on some known results.In Chapter 2, by constructing a special cone and applying the theorem on cone expansion and compression and a fixed point theorem on cone, we investigate the existence of positive solutions for some singular boundary value problem of p-Laplacian operator whereφ_p(x) = |x|^p-2x,p > 1.φ_q = (φ_p)^-1,1/p +1/q = 1. A sufficient conditions of boundary value problem (2.1.1) including one, two, three positive solutions are obtained.In Chapter 3, by means of the fixed point theorems of the cone expansion and compression of norm type, an explicit interval for A is derived such that for any A in this interval, the existence of at least one positive solution to singular boundary value problemis guaranteed. Our results extend and improve many known results including singular and non-singular cases.