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

Abstract

We study initial value problems for scalar, nonlinear, delay differential equations with distributed, possibly infinite, delays. We consider the initial value problem $$cases{ x(t) = varphi(t), & $t leq 0$cr x'(t)+int_0^infty g(t, s, x(t), x(t-s)), d mu(s) = f(t), & $tgeq 0$,cr} $$ 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.

Keywords

• Delay differential equation
• infinite delay
• initial value problem
• nonresonance
• asymptotic stability
• exponential asymptotic stability.