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The existence of solutions of differential and integral equation in Banach Spaces is a new branch of maths. It is arising in the physical science, biology and other applied subjects . The theory of function and differential equation is used together to insearch the existence of solutions for nonlinear integral and differential equation in Banach space.The boundary value problems are an important part in the subject of differential equations, which have affluent background information in the classical physics and elec-tricitty. Two-point boundary value problems of differential equations have been studied widely and there are many excellent results. However, there are fewer papers that considered the singular two-point boundary value problems in abstract space and sign-changing solutions problems. We will study these questions in the following papers and obtain some useful results.There are four chapters in the dissertation.In the first chapter, the existence of positive solutions of boundary value problems for nonlinear singular differential equations in Banach spaces is considered. Singular boundary value problems are considered in many problems, but, people can get better results, using theories of function to study these problems. In particularly, Professor Guo have given many excellent results .However, few results have been got in abstract space, Professor Liu have given some better results on the problem in reference[4]. In Chapter 1, the condition of x(0) = x(1) = θ in origine dissertation is replaced by the general condition and a condition of completely continue operator is gave, next, the existence of positive solutions is obtained by using fixed point theorem.Obviously, classical two-point boundary value problems have important application.but, non-local problems of differential equation are studied late because they are more difficul. Kiguradze and Lomtatidze(1984), II’in and Moiseev(1987) studied the existences of solutions of multi-point boundary value problems for second-order linear differential equations boundary value problems .Gupta studied the existences of some three point boundary value problems for second -order non-linear differential equations in 1992. Since then, the existence of positive solutions rathe than sign-changing solutions developed quickly, but the existence of sign-changing solutions have wide application, the following papers of Chapter 2 , 3 ? 4 , we will study the existence of sign-changing solution. In the second chapter, we consider the existence of sign-changing solutions for nonlinear operator equation and its application in abstract space. By defining a special uniformly bounded positive linear operator, that is, let E, X be two Banach spaces, P C E be a normal cone, Q C X be a cone, K : X -> E be linear operator. Assume that 3u* € P° and constant /? > 0, such thatKx>0\\ Kx || u*, Vx e Qthen K is uniformly positive linear operator. Assume that 3ft > 0 such thatKx < ft || x || ?’, Vx 6 Qthen K is uniformly bounded positive linear operator.and it* continue operator, that is, forVe > 0, 35 > 0, let u,v € E and || u - v ||< 5, then, the following conclusion hold:—eu* < Au - Av < eu*using fixed point index method, we get the existence of sign-changing solutions.In the third chapter, we investigate the existences of two sign-changing solutions for some three point boundary value problems for second -order differential operator. This problem which are presented is resulted from ecology problems. By some conditions and using the fixed point theorem, this paper gets the existence of two sign-changing solutions.In the last chapter, two order and three point boundary value problems are dealt with in this paper, we get the existence of two sign-changing solutions by using the fixed point index method .
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