Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation

Abstract

<p style='text-indent:20px;'>This paper deals with the solvability in the semilinear abstract evolution equation with countable time delays,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \dfrac{du}{dt}(t)+Au(t) = F(u(t), (u(t-\tau_n))_{n\in\mathbb{N}}), &amp; t&gt;0, \\ u(t) = u_0(t), &amp; t \in \bigcup\limits_{n \in \mathbb{N}}[-\tau_n,0], \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a Banach space <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ -A $\end{document}</tex-math></inline-formula> is a generator of a <inline-formula><tex-math id="M3">\begin{document}$ C_0 $\end{document}</tex-math></inline-formula>-semigroup with exponential decay and <inline-formula><tex-math id="M4">\begin{document}$ F: X \times X^\mathbb{N} \to X $\end{document}</tex-math></inline-formula> is Lipschitz continuous. Nicaise and Pignotti (J. Evol. Equ.; 2018;18;947–971) established global existence and exponential decay in time for solutions of the above equation with finite time delays in Hilbert spaces under global or local Lipschitz conditions. The purpose of the present paper is to generalize the result to the case of countable time delays in Banach spaces under a global Lipschitz condition.</p>