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Zbl 1068.34076
Tang, X.H.
Asymptotic behavior of delay differential equations with instantaneously terms.
(English)
[J] J. Math. Anal. Appl. 302, No. 2, 342-359 (2005). ISSN 0022-247X

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 $$x'(t)=-c r(t) x(t)+F(t,x_t), \tag1$$ where $c>-1$ is a constant, and $r(t)$ 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 $r(t)$ 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 [{\it A. Ivanov, E. Liz} and {\it 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 {\it 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 [{\it E. Liz, V. Tkachenko} and {\it S. Trofimchuk}, SIAM J. Math. Anal. 35, No. 3, 596--622 (2003; Zbl 1069.34109)]. \par 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 $c\in [-1,0)$. When $c=0$, a more general result was recently proved in [{\it T. Faria, E. Liz, J. J. Oliveira} and {\it S. Trofimchuk}, Discrete Contin. Dyn. Syst. 12, 481--500 (2005; Zbl 1074.34069)].
[Eduardo Liz (Vigo)]
MSC 2000:
*34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations

Keywords: delay differential equations; asymptotic behavior; $3/2$-stability conditions

Citations: Zbl 1025.34078; Zbl 1069.34109; Zbl 1074.34069

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