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

where
and
are bounded and
is a finite
Borel measure. Motivated by the nonresonance condition for the
linear case and previous work of the authors, we introduce conditions
on
. 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
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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