Differential and Integral Equations

The asymptotic behaviour of perturbed evolution families

Valentina Casarino, Lahcen Maniar, and Susanna Piazzera

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Given an evolution family $\mathcal U:={(U(t,s))_{t\geq s}}$ on a Banach space $X$, we present some conditions under which asymptotic properties of $\mathcal U$ are stable under small perturbations by a family $$ \mathcal{B}:=(B(t),D(B(t))_{t\in\mathbb{J}},$$ $\mathbb{J} =\mathbb{R}$ or $\mathbb{R}_+$, of linear closed operators on $X$. Our results concern asymptotic properties like periodicity, (asymptotic) almost periodicity (even in the sense of Eberlein), uniform ergodicity and total uniform ergodicity. We present, moreover, an application of the abstract results to non-autonomous partial differential equations with delay.

Article information

Differential Integral Equations, Volume 15, Number 5 (2002), 567-586.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
Secondary: 34D05: Asymptotic properties 34D10: Perturbations 34G10: Linear equations [See also 47D06, 47D09]


Casarino, Valentina; Maniar, Lahcen; Piazzera, Susanna. The asymptotic behaviour of perturbed evolution families. Differential Integral Equations 15 (2002), no. 5, 567--586. https://projecteuclid.org/euclid.die/1356060830

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