Solutions of Nonlinear Integral Equations and Their Application to Singular Perturbation Problems Thesis By

Sufficient conditions for the existence and uniqueness, and estimates, of a continuous vector solution y = y(t) to the integral equation y(t) = u(t) + EL(t) Ef t M(s) f(s,y(s)) ds + K(t,s) f(s,y(s)) ds, t 0 are derived. A successive approximation technique involving a double sequence is used in the proof . This integral equation result is applied to the second order singular perturbation problem with differential equation Ex"+ p(t,e) x ' + q(t,e) x + r(t,e) + Eh(t,x,x',E) = o, and boundary conditions b1(e) x(o,e) + b2(e) x'(o,e) c 1(e) x(l,E) + c2(e) x'(l,e) = £ (e), 0 O 0+ , "'e obtain an asymptotic expansion with leading term w(t,e) for a solution to this problem. 1. We are concerned with proving the existence and uniqueness of solutions for certain classes of ordinary differential equations which depend in a singular manner on a small positive parameter €. We are further concerned with describing these solutions by obtaining for them asymptotic expansions uniformly valid in the whole interval as € -> 0+ More precisely, let P = P(€) be a positive function of € with the property that P(€) -> 0 as € -> O+, and let t belong to the interval 00 I. The formal sum ~ x. ( t, €) is said to be a uniform asymptotic expansion • 1 ~ ~= with scale P(€) for x(t,€) if there exists a function A= A(t,€) such that for € -> 0+ and m = 1,2, ••• , m x(t,€) ~ i=1 uniformly for t in I. It is the nature of most perturbation problems, where the small ( 1.1) parameter multiplies the highest derivative in the differential equation, to exhibit non-uniform convergence, as € -> O+, in the neighborhood of some point or points in the interval. Such problems are usually called singular perturbation problems . Most of the literature on singular perturbation problems has been concerned with the case when the non-uniformity occurs at one of the end points of the interval. It is the custom in this case to call the region near this end point a boundary layer in analogy with certain hydrodynamic phenomena. Most singular perturbation problems with linear differential equations exhibit boundary layers, and in this case theoretical means exist for determining when a particular end point is part of a boundary layer (see Wasow [1]). Usually, different analytic expressions are developed for the boundary layers and the rest of the interval (see, e.g., Levin and Levinson [2]).