Boundedness properties for linear ordinary differential equations

Abstract

wheref is always at least locally integrable. T. Yoshizawa [7], S. R. Bernfeld [1], and others have studied the properties of uniform boundedness and uniform ultimate boundedness for (1), primarily in connection with trying to preserve these properties under perturbations. On the other hand, W. A. Coppel [3], [4], R. Conti [2], V. A. Staikos [5], and P. Talpalaru [6] have studied the stability and boundedness properties inherited by (1) when one requires that each memberf of various function spaces yield a bounded solution u of (2). Little is known about the connections between these two schools of thought. An exception: Coppel [4, Theorem 1, p. 131] has shown that uniform boundedness for (1) (see [1, Definition 2.1]) is equivalent to the requirement that iff is in Y' [R+, Y] then every solution u of (2) is bounded. We shall obtain further connections. We shall say that (1) is quasi-uniformly-bounded (QUB) if and only if whenever B is a bounded subset of Y there is a number ,C such that if c is