On the Rate of Convergence of Solutions of Functional Differential

Funkcialaj Ekvacioj, 29 (1986) 1-10

On the Rate of Convergence of Solutions of Functional Differential Equations By

T. KRISZTIN (University of Szeged, Hungary)

Introduction

§1.

In case of many models described by differential equations the most characteristic propety of certain phase coordinates is the convergence to a constant as the time , [ , [9] the question of tends to infinity (see e.g. [2], [3], [11]). In papers [5], convergence of solutions is examined for delay differential equations, where the righthand side of the equation is the sum of an ordinary and a functional part of the same order. While asymptotic stability may be impossible (as in the case where any constant is a solution of the equation), the problem of convergence of solutions is of interest. The rate of convergence of solutions is also a natural question and it is very important for the applications. Since asymptotic stability results cannot be applied for the above-mentioned type of equations, the literature on the question of rapidity of convergence is sparse. The goal of the present paper is to estimate the rate of convergence of solutions of these delay differential equations. We use Liapunov-Razumikhin type technics. Section 2 gives the preliminary results. Section 3 contains the main result. Applications are given in Section 4. Among others we . prove the exponential rapidity of convergence of solutions of an equation (see (28) which arose in the study of the motion of a classical radiating electron ([6], [11]). $ 8¥rfloor$

$¥mathrm{f}¥mathrm{f}¥mathrm{l}$

$¥mathrm{e}¥mathrm{q}¥mathrm{u}$

$)$

§2.

Preliminaries

the real, the non-negative real numbers, respectively, and by Denote by $R$ , a norm in . the set of all ordered -tuples. Denote be a continuous, non-decreasing function such that the funcLet denote by . For given tion t?l(t) is non-decreasing and $[t¥ lambda(t), t]$ into . If the space of continuous functions which map the interval is defined by $x_{t}(s)=x(s)$ , is a continuous function, then , :[ $F$ ¥ ¥ ¥ as follows: for function . Define the function , function , and for any continuous function : [ maps from into $R^{+}$

$R^{n}$

$R^{n}$

$|¥cdot|$

$¥mathrm{n}$

$¥lambda:R^{+}¥rightarrow(0, ¥infty)$

$t¥geqq 0$

$¥lim_{t¥rightarrow¥infty}[t-¥lambda(t)]=¥infty$

$R^{n}$

$S_{t}$

$x$

$-¥lambda(0)$

$x_{t}¥in S_{t}$

$¥infty)¥rightarrow R^{n}$

$t- lambda(t) leqq s leqq t$

$S_{t}$

$t¥geqq 0$

$t¥geqq 0$

$x$

$R^{n}$

is continuous in on a prototype of $F$.

$F(t, x_{t})$

$t$

$R^{+}$

.

$-¥lambda(0)$

For example, the function

$F(f_{ },¥cdot)$

$¥infty)¥rightarrow R^{n}$

$-x(t)+¥int_{t-¥lambda(t)}^{t}x(s)ds$

is

T. KRISZTIN

Consider the functional differential equation

(1)

$¥chi^{¥prime}(i)=F(i, x_{t})$

.

Let , be given. A function is said to be a solu$ 0¥overline{¥mathrm{V}}(t, x_{t})$

imp fies

(3)

,

$D_{(1)}^{+}¥mathrm{V}(t, x_{t})¥leqq f_{1}(t)g_{1}(¥mathrm{V}(t, x(t)),¥overline{¥mathrm{V}}(t, x_{t}))$

$¥mathrm{V}$

be a Liapunov

Convergence

where

$h_{1}$

:

$¥int_{t-¥lambda(t)}^{t}f_{1}(s)ds¥leqq K_{1}$

$R¥times R¥rightarrow R$

$>0$

$¥underline{7}$

is continuous, monotone increasing and $h_{1}(s)>s$ for continuous and there is a $K_{l}¥in R_{+}such$ that

(4) :

FDE

$R¥rightarrow R$

$¥infty)¥rightarrow R^{+}is$

$g_{1}$

of solutions of

is continuous,

$g_{1}(u, ¥cdot)$

$(t ¥in R^{+})$

$s¥in R$

,

$f_{1}$

:

$[-¥lambda(0)$

,

is monotone non-decreasing,

$u