On the uniform ergodic for α−times integrated semigroups
DOI:
https://doi.org/10.5269/bspm.41944Abstract
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 $\alpha\geq 0$.
References
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2. N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, (1958).
3. M. Heiber, Laplace transforms and −times integrated semigroups, Forum Math. 3, 595-612, (1991).
4. U. Krengel, Ergodic theorems. de Gruyter Studies in Mathematics, Berlin, New York, (1985).
5. M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc., 43, 337-340, (1974).
6. M. Lin, On the uniform ergodic theorem II, Proc. Amer. Math. Soc., 46, 217-225, (1974).
7. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, (1983).
8. A. Tajmouati, A. El Bakkali and M. A. Ould Mohamed Baba, Spectral inclusions between -times integrated semigroups and their generators, Boletim da Sociedade Paranaense de Matematica, to appear.
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Published
2020-10-10
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