Initial Value Problems for Nonlinear Nonresonant Delay Differential Equations with Possibly Infinite Delay
Date
1997-12-19
Authors
Drager, Lance D.
Layton, William
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We study initial value problems for scalar, nonlinear, delay differential equations with distributed, possibly infinite, delays. We consider the initial value problem
{ x(t) = φ(t), t ≤ 0
x'(t) + ∫∞0 g(t, s, x(t), x(t − s)) dµ(s) = ƒ(t), t ≥ 0,
where φ and ƒ are bounded and µ 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 µ satisfies an exponential decay condition, globally exponentially asymptotically stable.
Description
Keywords
Delay differential equation, Infinite delay, Initial value problem, Nonresonance, Asymptotic stability, Exponential asymptotic stability
Citation
Drager, L. D., & Layton, W. (1997). Initial value problems for nonlinear nonresonant delay differential equations with possibly infinite delay. <i>Electronic Journal of Differential Equations, 1997</i>(24), pp. 1-20.
Rights
Attribution 4.0 International