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2018 Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces
Yong-Kui Chang, Rodrigo Ponce
J. Integral Equations Applications 30(3): 347-369 (2018). DOI: 10.1216/JIE-2018-30-3-347

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.

Citation

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Yong-Kui Chang. Rodrigo Ponce. "Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces." J. Integral Equations Applications 30 (3) 347 - 369, 2018. https://doi.org/10.1216/JIE-2018-30-3-347

Information

Published: 2018
First available in Project Euclid: 8 November 2018

zbMATH: 06979944
MathSciNet: MR3874005
Digital Object Identifier: 10.1216/JIE-2018-30-3-347

Subjects:
Primary: 47D06
Secondary: 45D05 , 45M10 , 45N05 , 47J35

Keywords: $C_0$-semigroups , almost periodic , Exponential stability , heat equation with memory , Mild solutions , Volterra equations

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.30 • No. 3 • 2018
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