Positive Solutions Of Boundary Value Problems For Nonlinear Differential Equations And Its Applications

Nonlinear functional analysis is a research field of mathematics which has pro-found theories and extensive applications. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general the-ories and methods to handle nonlinear problems. Its rich theory and advanced method have provided the effective theory tool for solving many kinds of nonlinear differential equations, nonlinear integral equations and some other types of equations, and han-dling many nonlinear problems in computational mathematics, cybernetics, optimized theory, dynamic system, economical mathematics and so on. At present, the contents of nonlinear functional analysis mainly have topology degree theory, critical point the-ory, partial order method, analysis method, monotone mapping theory and so on. In recent years nonlinear problems have received highly attention of the domestic and foreign mathematics and natural science field, so the research on nonlinear functional analysis and its applications is very important in both theory and applications.The boundary value problems of nonlinear differential equations are important subjects in the theory of differential equations. Owing to the important in both theory and in applications, boundary value problems for ordinary diffrential equations have been attracted many researchers, and a large number of results have been obtained. Under the impetus of functional analysis and practical problems, the development of the research on boundary value problems for nonlinear differential equations is rapid.The present paper employs the nonlinear functional analysis theory and method, such as cone theory, fixed point theory, fixed point index theory, Krasnosel'skii fixed point theorem, Global Continuation Theorem, monotone iterative technique and the method of lower and upper solutions, to investigate the existence, multiplicity, depen-denceon a parameter and monotony for positive solutions to several kinds of (singular) boundary value problems of nonlinear differential equations (system), including some periodic boundary value problems, high singularity of higher order differential equa-tions, nonlocal problems, semipositone problems, impulsive boundary value problems and singular boundary value problems with p-Laplacian operator systems. By deep study, we obtain some new interesting results.The thesis is divided into six chapters. In Chapter I, we mainly introduce the background of nonlinear functional analysis and some basic concepts and theorems. In Chapter II, the existence, multiplicity and nonexistence results for positive solutions are derived to second order periodic boundary value problems, and the uniqueness of solutions and the dependence of solutions on the parameter are also studied. In Chapter III, we discuss three kinds of singular higher order nonlocal boundary value problems. In§3.2, the existence of positive solutions is established for nth-order m-point singular boundary value problem depends on higher derivatives of unknown function. In§3.3, using the first eigenvalue corresponding to the relevant linear operator, we study the nonlinear nth-order singular boundary value problem with nonlocal condition which is given by Riemann-Stieltjes integral with a signed measure.§3.4 deals with multiple positive solutions to nth-order singular nonlocal boundary value problem in Banach space. Chapter IV focuses on the study of monotone positive solution for higher order boundary value problems.§4.1 establishes the existence of monotone positive solution for semipositone right focal boundary value problem with a sign-changing nonlinear term which may be unbounded from below. In§4.2, the existence of multiple monotone positive solutions for higher order integral boundary value problems is established. Chapter V deals with the positive solutions to a class of nonlinear impulsive Sturm-Liouville problem with integral boundary conditions. In Chapter VI, the existence, nonexistence and multiplicity of positive solutions for singular boundary value problems with p-Laplacian operator systems involving two parameters are investigated. By the fixed point index theory and the upper-lower solutions method, a continuous curve which resolves the distribution of positive solutions is derived.