Asymptotic Integration of a Perturbed Constant Coefficient Differential Equation under Mild Integral Smallness Conditions

Abstract

The problem of asymptotic behavior of solutions of an nth order linear differential equation is reconsidered, and a result obtained by Hartman and Wintner under integral smallness conditions on the perturbing terms which require absolute integrability is shown to hold under weaker integrability conditions requiring only ordinary (perhaps conditional) convergence of some of the improper integrals that occur. The estimates of the asymptotic behavior of solutions of the perturbed equation are also sharper than in the classical result.