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Asymptotic behavior of delay differential equations with instantaneously terms. (English) Zbl 1068.34076

This is an interesting paper in which the author extends the famous $3/2$ results by Yorke and Yoneyama to the functional-differential equation with instantaneous linear term

${x}^{\text{'}}\left(t\right)=-cr\left(t\right)x\left(t\right)+F\left(t,{x}_{t}\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $c>-1$ is a constant, and $r\left(t\right)$ is a piecewise continuous function.

However, many related references are missed, perhaps due to the large delay between submitting and publishing. Indeed, at least for constant $r\left(t\right)$ and constant delay $\tau$, the main results for $c>0$ are not new. For example, in this case, condition (2.6) in Theorem 2.3 reads

$r\tau <-\frac{1}{c}ln\left[\frac{1}{c}ln\left(\frac{1+c}{1+{c}^{2}}\right)\right]·$

As it was noticed in [A. Ivanov, E. Liz and S. Trofimchuk, Tohoku Math. J., II. Ser. 54, No. 2, 277–295 (2002; Zbl 1025.34078)], under such a condition, the convergence of all solutions of (1) to zero was already proved by S. E. Grossman in a report from 1969. The sharp nature of the condition and its relations with the $3/2$ theorems were shown in the mentioned paper by Ivanov et al. Moreover, such a condition implies the global attractivity of the equilibrium under a condition much more general than the usual Yorke condition, as it was proved in [E. Liz, V. Tkachenko and S. Trofimchuk, SIAM J. Math. Anal. 35, No. 3, 596–622 (2003; Zbl 1069.34109)].

In any case, the results of the paper by Tang are a good contribution to the subject, since he considers the case of nonautonomous instanteneous term, and some results for $c\in \left[-1,0\right)$. When $c=0$, a more general result was recently proved in [T. Faria, E. Liz, J. J. Oliveira and S. Trofimchuk, Discrete Contin. Dyn. Syst. 12, 481–500 (2005; Zbl 1074.34069)].

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34K20 Stability theory of functional-differential equations