arXiv subjects

Paulo Mendes Carvalho-Neto

Publications and source records attributed to Paulo Mendes Carvalho-Neto.

2 recordsLinked to original sources

arXiv2022

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

In this work we study the Riemann-Liouville fractional integral of order $α\in(0,1/p)$ as an operator from $L^p(I;X)$ into $L^{q}(I;X)$, with $1\leq q\leq p/(1-pα)$, whether $I=[t_0,t_1]$ or $I=[t_0,\infty)$ and $X$ is a Banach space. Our main result give necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral from $L^p(t_0,t_1;X)$ into $L^{q}(t_0,t_1;X)$, when $1\leq q< p/(1-pα)$.

math.FA↗

arXiv2021

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

In this paper we study the Riemann-Liouville fractional integral of order $α>0$ as a linear operator from $L^p(I,X)$ into itself, when $1\leq p\leq \infty$, $I=[t_0,t_1]$ (or $I=[t_0,\infty)$) and $X$ is a Banach space. In particular, when $I=[t_0,t_1]$, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a $C_0-$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.

math.FA↗