Date of Award

5-1995

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Don Hinton

Committee Members

Henry Simpson, R. Childers, N. Alskakos

Abstract

In this dissertation we study the Lp solutions of second order linear differential equations. The question as to when the equation -(qo(x)y'(x))' + q1(x)y(x) = f(x), α ≤ x < ∞, admits Lp solutions y(x) for arbitrary f(x) in Lp is investigated. We show the condition Re(q1(x)) ≥ 1 or the conditions Re(q1(x)) ≥ 0 and Im(q1(x)) ≥ 1 are sufficient for a Lp solution y(x) to exist.

Functions that bound a solution of the homogeneous equation -(q0(x)y' (x))' + q1(x)y(x) = 0, α ≤ x < ∞, either above or below, are given for non-oscillatory equations.

An extensive discussion regarding the linear dimension of the set of Lp solutions of -(q0y' + q1y = 0 is given. The equation -(xβ y'(x))' + (-mxγ)y(x) = 0, 1 ≤ x < ∞, is used as an example to illustrate the results.


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