On an Application of a Banach-Steinhaus Theorem to the Study of the Boundedness of Solutions of Non-Linear Differential and Difference Equations

The purpose of this note is to prove anew and considerably extend an interesting result obtained by Perron, [5]. In the statement and proof we shall use standard vector and matrix notation; the vectors will be n-dimensional column vectors and the matrices n X n square matrices. Norms for vectors and matrices will be written I I I I, and will be, by definition, the sums of the absolute values of the terms. A vector or matrix function will be termed bounded for t > 0 if its norm is bounded for t > 0. Perron's result may now be stated as follows: THEOREM 1: Let the matrix A (t) be continuous and bounded for t ? 0. Then the necessary and suicient condition that the differential equation