JOURNAL OF DIFFERENTIAL EQUATIONS 82, 15-27 (1989) On the Monotonicity Property for a Certain Class of Second Order Differential Equations MARIELLA CECCHI Facoltd di Scienze, Vniversitri di Siena, Via del Capitano 15, 53100 Siena, Italy MAURO MARINI Facoltci di Ingegneria, Vniversitti di Firenze, Via di S. Marta 3, 50139 Firenze, Italy AND GABRIELE VILLARI Facolth di Scienze, Vniversith di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy Received October 13. 1988 In the present paper we study the asymptotic behavior of certain solu- tions of the second order differential equations (P(t) x’(t))’ =4(t) X(l) ’ d ( > =z and (P(t) x’(t))’ = 4(t) .Mf)), (N) where p, q: [0, co) + R and S: R + R are continuous, p(t) > 0, q(t) > 0, and z&u) > 0 for 24 # 0. As customary [4, p. 3221 it will be assumed throughout this paper that a solution x=x(t) of (L) C(N)] is a continuously differentiable function such that p(t) x’(t) has a continuous derivative satisfying (L) [(N)]. When the function p is continuous but does not have a continuous derivative, Eqs. (L) and (N) can be interpreted as the first order systems: 15 0022-0396/89 53.00 Copyright 0 1989 by Academic Press, Inc. 505/82/l-2 All rights of reproduction I” any form reserved.