JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 97, 202-213 (1983) Nonoscillation Criteria for Forced Second Order Nonlinear Equations* LYNN H. ERBE AND V. SREE HARI RAO+ Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada Submitted by Alex McNabb This paper is concerned with nonoscillation criteria and asymptotic behaviour for forced second order nonlinear differential equations. These results improve and include several well-known results. 1. INTRODUCTION Consider the second order nonlinear equation (p(t) Y’)’ + 4w f(Y) = r(t), (1.1) where p,q,rE C([t,, co).R), p> 0, fE C(R,R), R =(-co, a). Our primary concern here is to obtain criteria under which all continuable solutions of (1.1) are nonoscillatory. and also to study the asymptotic behaviour of solutions. We recall that a solution y(t) of (1.1) is said to be nonoscillatory on [to, co) in case there exists t, > t, such that y(t) # 0 on [tr, co). The solution y(t) is said to be oscillatory if for each t, > t, there exist t,, 1, with t, < t, < t, and y(tJ < 0, y(t3) > 0. Finally, a solution y(t) is said to be a Z-type solution if it has arbitrarily large zeros but is ultimately nonnegative or nonpositive. Equation (1.1) (with r(t) = 0) has been studied extensively, in particular for the Casey(y) = yy, y > 0 is the quotient of odd positive integers, with much of the impetus stemming from the well-known paper of Atkinson [ 11. We refer to the survey papers of Wong ] 13, 14 ] and Kartsatos [9] for detailed discussion of (1.1) and its generalizations. The nonoscillation problem for (1.1) has received much less attention, in particular for the case r(t) f 0. That is, criteria which guarantee that all solutions of (1.1) are nonoscillatory, are much more scarce in the literature than those which guarantee that all solutions are oscillatory. For the former, we refer to the recent papers of Graef and Spikes [4-6 and the references * Research supported by grants from NSERC, Canada. ’ On leave from Osmania University, Hyderabad 500007, India. 202 0022-247X183 $3.00 Copyright 8 1983 by Academic Press, Inc. All rights of reproductmn in any form reserved.