On L_p Solutions of Second Order Linear Differential Equations

In this dissertation we study the L_p solutions of second order linear differential equations. The question as to when the equation -(q_o(x)y'(x))' + q₁(x)y(x) = f(x), α ≤ x < ∞, admits L_p solutions y(x) for arbitrary f(x) in L_p 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 L_p 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 L_p solutions of -(q₀y' + q₁y = 0 is given. The equation -(x^β y'(x))' + (-mx^γ)y(x) = 0, 1 ≤ x < ∞, is used as an example to illustrate the results.