,Von!incnr Analysu. Theor). Methods & Applic~~n~. Vol. 8. NO. 12. pp. 1395-1108. 1984. Oj62-546X8-1 S 00 - .OO Printed m Great Bntain. @ 19s Pcrgamon Press Ltd. ESTIMATES REGARDING THE DECAY OF SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS J. R. HADDOCK* Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152, U.S.A and T. KRISZTIN~ University of Szeged, Bolyai Institute, 6720 Szeged, Hungary zyxwvutsrqponmlkjihgfedcba (Received.12 August 1983; received for publication 23 January 1984) Key words and phrases: Asymptotic behavior, exponential and nonexponential decay, comparison equation, Razumikhin methods. 1. INTRODUCTION THE PURPOSE of this paper is to study the rapidity of convergence (to zero) of solutions of functional differential equations (FDE) with finite delay. Our main techniques involve dif- ferential inequalities and comparison theorems, and the principal intent is to provide results that can be applied in a direct manner. Investigations that deal with convergence properties of solutions of differential equations (ordinary, partial and functional) are quite often concerned with exponential decay. Likewise, the results presented here frequently yield exponential convergence. On the other hand, we are sometimes able to estimate asymptotic behavior that is not exponential (example 4.2), while in other instances convergence that is faster than exponential can be deduced (example 4.5). In Section 2, we provide the general setting and underlying assumptions. Section 3 contains a statement of the main result, with applications and comments given in Section 4. In order to facilitate the exposition of the paper, the proofs have been delayed until Section 5. 2. PRELIMINARIES Let r 3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0 be given and let C = C([-r, 01, R” ) denote the space of continuous functions that map the interval [--I, 0] into R” . For $ E C, define the norm of # by 1) @/I = _z;+, 1 q(s) 1, . . where ( . 1 denotes any convenient norm in R” : If x: [to - r, to + A) + R” is continuous on [to - r, to+A), O<A=S~, then for each t E [to, to + A),q E C is defined by x&s) = x(t + s), --r =Z s g 0. Consider the (nonlinear, nonautonomous) functional differential equation x’(t) = F(t, x,)7 (2.1) where F:Rf X CF+ R”. (R+ and R- denote the set of nonnegative and nonpositive real numbers, respectively, CF C C and x’(t) denotes the right derivative of x with respect to t.) l This paper was supported in part by the National Science Foundation under grant 8301304. t This paper was written while the second author was a visiting faculty member at Memphis State University. 1395