Electron. J. Diff. Eqns., Vol. 1997(1997), No. 24, pp. 1-20.

Initial value problems for nonlinear nonresonant delay differential equations with possibly infinite delay

Lance D. Drager & William Layton

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.

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Lance D. Drager
Department of Mathematics and Statistics; Texas Tech University; Lubbock, TX 79409-1042 USA.
e-mail: drager@math.ttu.edu

William Layton
Department of Mathematics; University of Pittsburgh; Pittsburgh, PA 15260 USA.
e-mail: wjl+@pitt.edu

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