Bounds on the norm of a solution of a general differential equation.

Abstract

Upper bounds obtained by application of Bellman's lemma [1, p. 35] and its generalization by Bihari [2] have been much used in the study of solutions of equations such as (1.1). Similar methods permit the determination of analogous lower bounds which seem to be unknown until now. In ?2 we present a result which might indicate why this is the case. Concerning (1.1) we make the following assumptions: (1) x is a real variable, z and F are finite dimensional complex vectors with n components zi and Fi respectively, (2) F is continuous in (x, z) for all z and all xE [a, b], i.e. a _x < b with a<b, (3) for some norm, say I z zi I, F satisfies (1.2) IF(x, z) | v(x)g( I )