k-component disconjugacy for systems of ordinary differential equations

Abstract

Disconjugacy of the kth component of the mth order system of nth order differenttal equations Y(n)=f(x,Y,Y′,…,Y(n−1)), (1.1), is defined, where f(x,Y1,…,Yn), ∂f∂yij(x,Y1,…,Yn):(a,b)×Rmn→Rm are continuous. Given a solution Y0(x) of (1.1), k-component disconjugacy of the variational equation Z(n)=∑i=1nfYi(x,Y0(x),…,Y0(n−1)(x))Z(i−1), (1.2), is also studied. Conditions are given for continuous dependence and differentiability of solutions of (1.1) with respect to boundary conditions, and then intervals on which (1.1) is k-component disconjugate are characterized in terms of intervals on which (1.2) is k-component disconjugate.