Asymptotic stability in differential equations with unbounded delay

T. A. Burton, Northwest Research Institute, Port Angeles, WA, U.S.A.
A. Somolinos, Mercy College, Dobbs Ferry, NY, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 13. (1999), pp. 1-19.

Abstract: In this paper we consider a functional differential equation of the form
$$x'=F(t,x,\int_0^t C(at-s) x(s)\,ds)$$
where $a$ is a constant satisfying $0<a<\infty$. Thus, the integral represents the memory of past positions of the solution $x$. We make the assumption that $\int_0^\infty |C(t)|\, dt<\infty$ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of $a$. Very different properties of solutions emerge as we vary $a$ and we are interested in developing an approach which handles them in a unified way.

See also: EJQTDE, No. 3. (2001)

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