Research On Some Problems Of Nonlinear Differential Equations

The present paper employs the cone theory, Krasnoselskii fixed point the-orem and the method of lower and upper solutions and so on, to investigate the existence and uniqueness of positive solutions to several kinds of singular semi-positone boundary value problems and integral boundary value problems. Meanwhile, we construct several new fixed point theorems for mixed monotone operators. By deep study, we obtain some new interesting results under weaker conditions.The thesis is divided into four chapters.In ChapterⅠ, we mainly introduce the background of nonlinear functional analysis and some concepts.In ChapterⅡ, without assuming operators to be continuous or compact, we study mixed monotone operators and give several new existence and uniqueness theorems.ChapterⅢfocuses on the study of integral boundary value problems. In§3.2, we establish necessary and sufficient conditions for the existence and unique-ness of C1[0, 1] positive solutions and C[0,1] positive solutions for a class of inte-gral boundary value problems. Moreover, we give the continuous dependence on a parameter of C1[0, 1] positive solutions. In§3.3, we only require that f(t,u) is decreasing on u and establish a sufficient condition for the existence of positive solutions for the second order integral boundary value problem. In§3.4, we in-vestigate boundary value problems of the perturbed equation - u"+k2u = f(t, u) which are rarely investigated. Nonlinearity f(t, u) is increasing in u. We obtain the existence and uniqueness of C1[0, 1] positive solutions in some set D. In§3.5, we also focus on the perturbed equation - u" + k2u = f(t, u). Here nonlinearity f(t, u) is decreasing in u. f may be singular at t = 0, 1 and u = 0. We obtain the existence and uniqueness of both C[0,1] and C1[0,1] positive solutions.In ChapterⅣ, we discuss two kinds of singular semi-positone boundary value problems. In§4.2, by using fixed point index theory on a cone and the method of transformation, we investigate the singular semi-positone sturm-liouville boundary value problem and gets a new conclusion of its CP1[0,1] positive solution. In§4.3, we deal with the existence of at least one positive solution and a unique positive solution for the semi-positone sturm-liouville boundary value problem, by using an effective operator and the fixed point theorems in cone, especially Krasnoselskii fixed point theorem.