A note on the convergence of the solutions of a linear functional differential equation. (English) Zbl 0693.45012

Krisztin, T.

J. Math. Anal. Appl. 145, No. 1, 17-25 (1990).

The paper deals with the equation \((1)\quad x'(t)=-\mu_ 0x(t)+\int^{0}_{-\infty}x(t+s)d\mu (s),\) where \(\mu_ 0\in R\) and \(\mu\) : (-\(\infty,\infty)\to R\) is a nondecreasing function and \(\mu_ 0=\int^{0}_{-\infty}d\mu\). The author proves that any solution to (1) belonging to a bounded continuous initial function converges to a finite limit as \(t\to \infty\) if and only if \(\int^{\infty}_{-\infty}| s| d\mu (s)<\infty\) holds.

Reviewer: M.Tvrdý

MSC:

45J05	Integro-ordinary differential equations
45M05	Asymptotics of solutions to integral equations

Keywords:

convergence; linear functional differential equation; functional- differential equation; critical case; asymptotic behaviour

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