The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I

Abstract

<p style='text-indent:20px;'>In this paper we study the Riemann-Liouville fractional integral of order <inline-formula><tex-math id="M1">\begin{document}$ \alpha&gt;0 $\end{document}</tex-math></inline-formula> as a linear operator from <inline-formula><tex-math id="M2">\begin{document}$ L^p(I,X) $\end{document}</tex-math></inline-formula> into itself, when <inline-formula><tex-math id="M3">\begin{document}$ 1\leq p\leq \infty $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ I=[t_0,t_1] $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M5">\begin{document}$ I=[t_0,\infty) $\end{document}</tex-math></inline-formula>) and <inline-formula><tex-math id="M6">\begin{document}$ X $\end{document}</tex-math></inline-formula> is a Banach space. In particular, when <inline-formula><tex-math id="M7">\begin{document}$ I=[t_0,t_1] $\end{document}</tex-math></inline-formula>, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a <inline-formula><tex-math id="M8">\begin{document}$ C_0- $\end{document}</tex-math></inline-formula>semigroup but does not defines a uniformly continuous semigroup. We close this study by presenting lower and higher bounds to the norm of this operator.</p>