On a delay integro-differential equation in Banach space

Abstract

<p style='text-indent:20px;'>Consider in a Banach space <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula> the following delay integro-differential equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u^{\prime}(t) = Au(t)+\int_{-r}^0g(s, u_t(s))\, ds +f(t), \ \ 0\le t\le T, \nonumber \ $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;r \le \infty $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ A:D(A)\subset X \to X $\end{document}</tex-math></inline-formula> is a linear operator generating a strongly continuous semigroup, while <inline-formula><tex-math id="M4">\begin{document}$ g $\end{document}</tex-math></inline-formula> is an unbounded nonlinear function which is dominated in some sense by <inline-formula><tex-math id="M5">\begin{document}$ A $\end{document}</tex-math></inline-formula>. Global existence and uniqueness are derived for the initial value problem associated with the above equation by using a fixed point approach combined with properties of linear operator semigroups. Continuity of solutions with respect to initial data as well as successive approximations of solutions will be also addressed. We illustrate our results with an application to the heat equation with nonlinear memory.</p>