#### 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 *L _{p} *solutions of second order linear differential equations. The question as to when the equation -(q

_{o}(

*x)*y^{}'(x))' +

*q*

_{1}(x)

*y*(x) =

*f*(x), α ≤ x < ∞, admits

*L*solutions

_{p}*y*(x) for arbitrary

*f*(x) in

*L*is investigated. We show the condition Re(

_{p}*q*

_{1}(x)) ≥ 1 or the conditions Re(

*q*

_{}_{1}(x)) ≥ 0 and Im(

*q*

_{1}(x)) ≥ 1 are sufficient for a

*L*solution

_{p}*y*(x) to exist.

Functions that bound a solution of the homogeneous equation -(*q*_{0}(x)*y*' (x))' + *q*_{1}(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 -(

*q*

_{0}

*y*' +

*q*

_{1}

*y*= 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.

#### Recommended Citation

Smith, James C., "On L_{p} Solutions of Second Order Linear Differential Equations. " PhD diss., University of Tennessee, 1995.

https://trace.tennessee.edu/utk_graddiss/4224