Convergence tests for one scalar differential equation with vanishing delay. (English) Zbl 1090.34596

The paper deals with the delay differential equation \[ y'(t)=\alpha (t) [y(t)-y(t-\tau (t))] (1) \] where \(\alpha ,\,\tau \in C^0([t_0,\infty ))\) are positive, \(\tau (t) \leq \tau _0 \in \mathbb R\) and the distance \(t-\tau (t)\) is increasing on \([t_0,\infty )\). The delay \(\tau (t)\) can satisfy \(\lim _{t \to \infty } \tau (t)=0\). A solution \(x(t)\) of (1) defined on \([t_x,\infty ) \subset [t_0,\infty )\) is called to be convergent if it has a finite limit as \(t \to \infty \). The authors present a point test and integral tests for convergence of all solutions of (1). Their results generalized those given by F. V. Atkinson and J. R. Haddock [J. Math. Anal. Appl. 91, 410-423 (1983; Zbl 0529.34065)] and J. Diblík [Ann. Pol. Math. 72, 115–130 (1999; Zbl 0953.34065)].


34K25 Asymptotic theory of functional-differential equations
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