Electronic Journal of Differential Equations,
Vol. 1997(1997), No. 24, pp. 1-20.
Title: Initial value problems for nonlinear nonresonant delay
differential equations with possibly infinite delay
Authors: Lance D. Drager (Texas Tech Univ., TeXas, USA)
William Layton (Univ. of Pittsburgh, Pennsylvania, USA)
Abstract:
We study initial value problems for scalar, nonlinear, delay
differential equations with distributed, possibly infinite, delays.
We consider the
initial value problem
$$\cases{
x(t) = \varphi(t), & $t \leq 0$\cr
x'(t)+\int_0^\infty g(t, s, x(t), x(t-s))\, d \mu(s) = f(t),
& $t\geq 0$,\cr}
$$
where $\varphi$ and $f$ are bounded and $\mu$ is a finite
Borel measure. Motivated by the nonresonance condition for the
linear case and previous work of the authors, we introduce conditions
on $g$. Under these conditions, we prove an existence and uniqueness
theorem. We show that under the same
conditions, the solutions are globally
asymptotically stable and, if $\mu$ satisfies an exponential decay
condition, globally exponentially asymptotically stable.
Submitted August 14, 1997. Published December 19, 1997.
Math Subject Classification: 34K05, 34K20, 34K25
Key Words: Delay differential equation; infinite delay; initial value problem;
nonresonance; asymptotic stability; exponential asymptotic stability.