Pergamon ivonlme~r .lnoiym, Theory, Merhods & Ap~l~utmns, Vol. 26, No. 4. PP. 761-766, 1996 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/% SlS.OO+ .Otl 0362-546X(94)00315-7 COMPOSITE BOUNDEDNESS AND STABILITY RESULTS BY PERTURBING LYAPUNOV FUNCTIONS DONNA STUTSON and A. S. VATSALA Department of Mathematics, University of Southwestern Louisiana. Lafayette, LA 70504-1010, U.S.A. (Received 14 July 1994; receivedfor publication 27 October 1994) Key words and phrases: Boundedness, Lagrange stability and perturbing Lyapunov functions. 1. INTRODUCTION In [l], nonuniform boundedness properties of a given differential system are discussedunder weaker assumptions using an idea known as perturbing Lyapunov functions. In this approach, when a single Lyapunov function is found that does not satisfy all the desired conditions to obtain a desired property, it is then fruitful to perturb that Lyapunov function rather than discard it. Also, employing this technique, nonuniform boundednessproperties are investigated in [2] by splitting the given differential system into two parts and using two Lyapunov functions in a manner that provided weaker assumptions. It is known that using two different measures rather than the norm, as candidates to obtain boundedness and stability properties, offers a way to unify a variety of results found in the literature [3]. In this paper, we shall discuss boundedness and stability properties in a composite way by combining the ideas involved in the foregoing approach thereby obtaining results in a unified way under much weaker assumptions. 2. PRELlMlNARlES Let us list the following definitions and classes of functions for convenience K = [a E C[(p, oo), R,]: a(u) is strictly increasing and a(u) + 00 as u + 001, CK = [a E C[R+ x [p, CO), R,]: a(t, u) E K for each t E R,], r = [h E C[R+ x R”, I?,]: .‘,“,f. h(t, .u) = 0 for each t E R,]. Consider the differential system x’ = f(t, x) (2.1) x(4,) = 4) 3 I, L 0, where f E C[R+ x IR”, a”]. We shall assume, for convenience, that f is smooth enough to ensure global existence of solutions of (2.1).