JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 135, 309-325 (1988) On a Family of Nonoscillatory Equations y” = G(x) y ’ H. GINGOLD Department of Mathematics, West Virginia University, 218 Eiesland Hall, Morgantown, West Virginia 26506 Submitted by V. Lakshmikantham Received August 22, 1986 1. INTRODUCTION Consider a second-order linear differential equation Y” =4(x) Y (1.1) on an interval [a, b], where b is allowed to be unbounded. There are interesting and important questions regarding the oscillation and non- oscillation of the equation above which attracted a considerable amount of attention. A record of the voluminous literature on this subject area can be found in Hille [lo, Chaps. 8, 91, Kreith [lS], and Swanson [24]. While the literature on oscilation or nonoscillation of solutions of (1.1) is very large for b(x) and y real-valued functions, the same is not true for problems where 4(x) and y are complex-valued. This article is concerned with both for a limited class of differential equations. Before we explain what the nature of our results are we will avoid ambiguity by adopting the following CONVENTION 1.1. We say that the equation (1.1) is nonoscillatory at b - if no nontrivial solution of (1.1) on [a, b) possesses an infinite number of zeros on [a, b) which have an accumulation point at b. A nontrivial solution of (1.1) is said to be oscillatory at b- if it possesses an infinite number of zeros on [a, b) with accumulation point at b-. (In an analogous manner one could define oscillation and nonoscillation of (1.1) and its solutions at a+.) The fact that a differential equation (1.1) is nonocillatory at b ~ means that there is a “small” left-hand side neighbourhood of b on which Eq. ( 1.1) is disconjugate. Namley, there is a small left-hand side neighbourhood of b on which no nontrivial solution of (1.1) possesses more than one zero. ’ This research was supported in part by a grant from the National Aeronautics and Space Administration NAG-1-741. 309 0022-247X/88 $3.00 Copyright 0 1988 by Academic Press, Inc All rights of rcproductmn in any form reserved.