Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces

Abstract

Let $\mathbb {X}$ be a Banach space. Let $A$ be the generator of an immediately norm continuous $C_0$-semigroup defined on $\mathbb {X}$. We study the uniform exponential stability of solutions of the Volterra equation

$u'(t) = Au(t)+\displaystyle \int _0^t a(t-s)Au(s)\,ds,\quad t\geq 0,\ u(0)=x$,

\noindent where $a$ is a suitable kernel and $x\in \mathbb {X}$. Using a matrix operator, we obtain some spectral conditions on $A$ that ensure the existence of constants $C,\omega >0$ such that $\|u(t)\|\leq Ce^{-\omega t}\|x\|$, for each $x\in D(A)$ and all $t\geq 0$. With these results, we prove the existence of a uniformly exponential stable resolvent family to an integro-differential equation with infinite delay. Finally, sufficient conditions are established for the existence and uniqueness of bounded mild solutions to this equation.