Singular Boundary Value Problems

In this dissertation, we are concerned with singular focal boundary value problems. We begin by studying the n^{th} order ordinary differential equationsy^{(n) } + f(x,y,y^',...,y^ {(n-2)}) = 0, 0 < x < 1,or y^{(n)} + h(x,y,y^ ',...,y^{(n-1)}) = 0, 0 < x < 1,where f(x,y_1,y _2 ,... ,y_{n-1}) has singularities at y_{i} = 0, 1 <= i<= n - 1, and h(x,y _1{,}y_2,... ,y_{n }) has singularities at y_{i } = 0, 1 <= i<= n. We seek solutions of these differential equations on (0,1), which satisfy homogeneous two-point focal boundary conditions at x = 0 and x = 1. A fixed point theorem in a cone is applied to appropriate integral operators in order to find the solutions of the boundary value problems. Next, we are concerned with solutions of the n^{th} order differential equationy^{(n)} + psi(x)f(x,y,y ^',...,y^{(n-1)}) = 0on the interval (0, 1) satisfying homogeneous two-point focal boundary value conditions, as well as non -homogeneous two-point focal boundary conditions at x = 0 and x = 1. Different constraints on the function f are assumed depending on whether homogeneous or non-homogeneous boundary conditions are specified. In both cases, psi (x) has singularities at x = 0, x = 1, and f(x,y _1,y_2 ,... ,y_{n}) has singularities at y_{j} = 0, 1 <= j <= n. Homotopy methods are used to obtain the solutions of these singular boundary value problems.