| Of concern is the linear or semilinear differential equation{dollar}{dollar} usp{lcub}primeprime{rcub}(t) = F(t,u(t),uspprime(t))eqno(1){dollar}{dollar}for u mapping {dollar}lbrack 0,infty){dollar} to a Banach space {dollar}chi.{dollar} F can be linear or not. Suppose the initial value problem (1) and{dollar}{dollar}u(0) = fsb{lcub}0{rcub}, uspprime(0) = gsb0eqno(2){dollar}{dollar}is well posed and let {dollar}usb0{dollar} be its solution. Consider the associated family of two-point boundary value problems (1) and{dollar}{dollar}u(0)= fsb0, u(varepsilon) = fsb{lcub}varepsilon{rcub}.eqno(3varepsilon){dollar}{dollar}Let {dollar}usb{lcub}varepsilon{rcub}{dollar} be its solution (which we assume exists and is unique). We take the same {dollar}fsb0{dollar} in both (2) and (3{dollar}varepsilon{dollar}) and show that if{dollar}{dollar}{lcub}fsb{lcub}varepsilon{rcub} - fsb0over varepsilon{rcub} {lcub}buildrel{lcub}varepsilonto 0{rcub}over{lcub}longrightarrow{rcub}{rcub} gsb0,{dollar}{dollar}then{dollar}{dollar}usb{lcub}varepsilon{rcub}(t) {lcub}buildrel{lcub}varepsilon to 0{rcub}over{lcub}longrightarrow{rcub}{rcub} usb0 (t){dollar}{dollar}uniformly for t in bounded arbitrary intervals, under certain conditions. The study is first performed on a Hilbert space, using a version of the Spectral Theorem. Working on a Banach space, we obtain an extension of the Hilbert space results. |
