JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 134, 408425 (1988) On the Stability Region of Scalar Delay- Differential Equations TOSHIAKI YONEYAMA Department of Mathematical Sciences, University of Osaka Prefecture, Sakai 591. Japan AND JITSURO SUGIE* Department of Applied Physics, Osaka University, Suita 565, Japan Submitted by Kenneth L. Cooke Received June 3. 1986 1. INTRODUCTION Consider the scalar delay-differential equation x’(t) = ctx( t) - j?x( t - q), (l-1) where cc, /?E R and q 20. By the theory of zero locations of the charac- teristic equation for (1.1 ), it is known [2, 41 that the zero solution of (1.1) is uniformly stable if and only if (a, /3) is in the region or (See the slightly shaded portion in Fig. 1, which was drawn by a computer and an X- Y plotter.) Consider the equation x’(t)=a(t)x(t)-b(t)x(t-r(r)), (1.3) where a, b : [0, co) + R, r: [O, co) + [0, q] are continuous. In view of the * Current address: Department of Mathematics, Okayama University, Okayama 700, Japan. 408 0022-247X/88 $3.00 Copyright Q 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.