On stability for non-instantaneous impulsive delay differential equations (Q6624365)

In this paper, a class of scalar differential equations with delays and non-instantaneous impulses is studied. On the intervals without impulses \([l_i,m_{i+1})\ (l_0=0<m_1,m_i<l_i<m_{i+1},\ i\in\mathbb{N})\), the solution follows a delay differential equation \(x'(t)+a(t)x(t)=f(t,x_t)\), where \(x_t(s)=x(t+s),-\tau(t)\le s\le 0\), \(a(t)\ge 0\) and the nonlinearity \(f(t,\varphi)\) satisfies a Yorke-type condition; the impulses occur at some points \(m_i\), and the solution remains constant for intervals \((m_i,l_i]\ (i\in\mathbb{N})\). By adopting the hypotheses on the impulsive functions and delays and the methodology in [\textit{T. Faria} and \textit{J. J. Oliveira}, Discrete Contin. Dyn. Syst., Ser. B 21, No. 8, 2451--2472 (2016; Zbl 1352.34107)], as well as in [\textit{J. Yan}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 63, No. 1, 66--80 (2005; Zbl 1082.34069)], the authors establish the global asymptotic stability of the zero solution of the system, by treating separately non-oscillatory and oscillatory solutions. Two examples illustrate the results.