Solutions On A Class Of Singular Lidstone Boundary Value Problem In Banach Spaces

In the last thirty years, singular boundary value problem in Banach space has become one of the most important problems that attract the attention of more and more mathematicians and other technicians (see. for instance, [9]-[11],[15] and their references ) since it has much practical value in the fields of gas dynamics fluid mechanics, the theory of boundary layer, nonlinear optics and so on. The approaches used to study such problems are the method of upper and lower solutions, topological degree, cone theory, and the method of approximation, etc. What they obtained is the existence of solution. On the basis of the above references, this paper attempts to make use of cone compression and expansion fixed point theorem, M(o|¨)nch fixed point theorem and fixed point index theorem to discuss a special kind of singular boundary value problem-Lidstone boundary value problem.Chapter 1 investigates the existence of positive solutions of the following 2n-order singular Lidstone value problem in Bananch spacewhere f(t,x) is singular at t = 0,1.θis the zero element of Banach space E. By constructing a special cone and using cone compression and expansion fixed point theorem, we get the existence of positive solutions.In chapter 2, we consider the same singular Lidstone value problem as in chapter 1. By using M(o|¨)nch fixed point theorem, we get the existence of positive solutions. Comparing with chapter 1, the uniform continuity of f is deleted.In chapter 3, we study the 2n-order singular Lidstone boundary value problemwhere f(t,x) is singular at t = 0,1, and x =θ. By constructing a special cone and using the fixed point index theorem, we presents the existence of multiple positive solutions for such boundary value problem.