Nonlinear Boundary Value Problems Existence Of Multiple Positive Solutions,

Nonlinear functional analysis is an important branch of mathmatics, and itcan explain several kinds of natural phenomena. The boundary value problems(B VPs) for nonlinear differential equations arise in a variety of areas of appliedmathematics, physics and variational problems of control theory, it is at presentone of the most active fields in analyse mathematics. Among them, muliti-point BVPs come from a lot of branches of applied mathematics and physics,and it is very meaningful in both practical and theoretical aspects. The presentpaper employs the cone theory, fixed point theory, topological degree theoryand upper-lower solutions method and so on, to investigate the existence ofpositove solutions to multi-point BVPs of some kinds of nonlinear 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, by using of the Guo-Krasnoselskii fixed point theorem, theauthor considers the existence of positive solutions of nonlinear multi-pointboundary value problem for systems of the following second-order delay differ-ential equationsThe main results in this paper improves and generalizes the results of [13,15].Three examples are then presented to demonstrate the application of our mainresults.In Chapter 2, by using of the Leggett-William fixed point theorem, theauthor establishes sufficient conditions for the existence of triple positive so- lutions of the following nonlinear two point boundary value problem with ap-Laplacian operator:Where f contains operators Tu and Su, a(t) may be singular at 0 and 1. Wealso show that the results can be applied to study certain higher order mixed;boundary value problems. An example is also given to study the solutionofαp-Laplacian boundary value problem using the theorem developed. Themain results in this paper improves and generalizes the results of [20-23]. Anexample is then presented to demonstrate the application of our main results.In Chapter 3, by using of the Leggett-Williams fixed point theorem, theauthor studies the existence of positive solutions for a class of fourth-ordersingular boundary value problemWhere a≥0, b≥0, c≥0, d≥0, ac + bc + ad > 0. The main results in thispaper improves and generalizes the results of [34-37,39-41].