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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 ' (t)=-cr(t)x(t)+F(t,x t ),(1)

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 τ, the main results for c>0 are not new. For example, in this case, condition (2.6) in Theorem 2.3 reads

rτ<-1 cln1 cln1+c 1+c 2 ·

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[-1,0). 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:
34K25Asymptotic theory of functional-differential equations
34K20Stability theory of functional-differential equations