Relative boundedness and second order differential equations

Abstract

Given two positive continuous functions a and β, necessary and sufficient conditions are given for the system u"(t) = f(t) + A(t)u(t) to have an α-bounded solution u for each –β-bounded forcing function f. Applications are given to a nonlinear perturbation problem: u"(t) = A(t)u(t) + F(t, u(t)). Indications are given on how to extend these ideas to nth order equations. © 1973 All rights reserved under the universal copyright convention.