Solutions Of Singular Boundary Value Problems In Banach Spaces

In recent years, because of the high practical value in the fields of gas dynamics fluid mechanics, the theory of boundary layer, nonlinear optics and so on, singular boundary value problem in Banach space becomes one of the important problems that attract the attention of mathematicians and other'technicians gradually(such as [11][13],[17]). Along with the problem study thoroughly, the method of upper and lower solutions, topological degree and cone theory or method of approximation were gradually used to demonstrate the existcnce results of positive solution of singular boundary value promblcm. This paper attcmps to make use of Sadovskii fixed point theorem, fixed point index, cone compression and expansion fixed point theorcm to discuss such problems more generally on the basis of abovc references.Chaptcr 1 considers the existence of positive solutions of second order singular initial value problem of impulsive differential equations in Bananch space wheref(t, x(t), x′(t), (Tx)(t), (Sx)(t))is singularity at x =θ, x′=θ,θis the zero element of Banach space E. But as far as we know, only paper [6] investigated the existcncc of solutions for the problem that f(t, x(t), x′(t), (Tx)(t), (Sx)(t))is singularity at x =θ, x′=θ. By constructing a special cone and using the Sadovskii fixed point theorem, we get the existcncc of positive solutions. In chapter 2, we consider the existence of positive solutions of singular boundary value problem in Banach space where In paper [8], Professor Yansheng Liu has discussed this problem in scalar space, where the author get the existence of positive solutions in sublinear case. We mainly use the fixed point index, discuss the existence of positive solutions for singular boundary value problems on the half-line in Banach space.In chapter 3, we study the 2n-order boundary value problems with the nonlinerities depending on higher-order derivatives By using dominate function, cone compression and expansion fixed point theorem, this chapter presents the existence of infinitely many positive solutions of boundary value problems, which cxpandcs the result of [8].In chapter 4, wc considcr the equation of chapter 3 under the boundary condition u²ⁱ(0) = u²ⁱ(1) = 0, 0≤i≤n—1, and h(t)is singular at t = 0, t = 1. Mainly through definition (A_ju)(t) = ds, the solution of 2n-order equation is equivalent to the solution of 2-order boundary value problems By using dominate function, cone compression and expansion fixed point theorem, we presents the existence of multiple, even infinitely many positive solutions of boundary value problems.