Abstract
This research was supported by Fundacao para a Ciencia e a Tecnologia (Portugal), under the Projects UID/MAT/04561/2013 (T. Faria) and UID/MAT/00013/2013 (J. J. Oliveira).
Highlights
We consider a family of scalar non-autonomous delay differential equations (DDEs) with impulses, and establish a criterion for the global asymptotic stability of its trivial solution
In order to establish stability results, the basic approach is to control the growth of the delayed terms by imposing a Yorke-type condition coupled with limitations on the amplitude of the delays
In [10], Yan considered (1.1) with m = 1 and obtained the global attractivity of its zero solution assuming a set of more restrictive hypotheses: again the impulsive functions Ik are required to satisfy (H1) with ak = 1 for all k ∈ N, the Yorke condition (H4) for f = f1 in (2.1) was assumed with a unique function λ1(t) = λ2(t) =: λ(t) providing the left and right growth control of f in
Summary
We consider a family of scalar non-autonomous delay differential equations (DDEs) with impulses, and establish a criterion for the global asymptotic stability of its trivial solution. We shall consider a very general setting for our method, not presenting any theoretical results about existence and global continuation of solutions, since this has been the topic of a variety of papers; see some references given below. It is important to mention that these conditions together with the set of assumptions imposed imply that the initial value problem (1.1)–(1.2) has a unique solution x(t) defined on [t0, ∞), which will be denoted by x(t, t0, φ), see e.g. It is applicable to the study of the global attractivity of other solutions, such as periodic solutions, as illustrated in Section 3 with an example
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