On L\u3csub\u3ep\u3c/sub\u3e Solutions of Second Order Linear Differential Equations

In this dissertation we study the Lp solutions of second order linear differential equations. The question as to when the equation -(qo(x)y\u27(x))\u27 + q1(x)y(x) = f(x), α ≤ x \u3c ∞, 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\u27 (x))\u27 + q1(x)y(x) = 0, α ≤ x \u3c ∞, 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\u27 + q1y = 0 is given. The equation -(xβ y\u27(x))\u27 + (-mxγ)y(x) = 0, 1 ≤ x \u3c ∞, is used as an example to illustrate the results