On the uniform ergodic for α−times integrated semigroups

Abdelaziz Tajmouati; Abdeslam El Bakkali; Fatih Barki; Mohamed Ahmed Ould Mohamed Baba

doi:10.5269/bspm.41944

Abdelaziz Tajmouati Sidi Mohamed Ben Abdellah University https://orcid.org/0000-0003-1572-1241
Abdeslam El Bakkali Chouaib Doukkali University
Fatih Barki Sidi Mohamed Ben Abdellah University
Mohamed Ahmed Ould Mohamed Baba Sidi Mohamed Ben Abdellah University

DOI: https://doi.org/10.5269/bspm.41944

Abstract

Let $A$ be a generator of an $\alpha-$times integrated semigroup
$(S(t))_{t\geq 0}$. We study the uniform ergodicity of $(S(t))_{t\geq 0}$ and we show that the range of $A$ is closed if and only if $\lambda R(\lambda,A)$ is uniformly ergodic.
Moreover, we obtain that $(S(t))_{t\geq 0}$ is uniformly ergodic if and only if $\alpha=0$. Finally, we get that $\frac{1}{t^{\alpha+1}}\int_{0}^{t}S(s)ds$ converge uniformly for all $\alpha\geq 0$.

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