Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space
Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats
Abstract
<p style='text-indent:20px;'>This paper aims to establish sufficient conditions for the exact controllability of the nonlocal Hilfer fractional integro-differential system of Sobolev-type using the theory of propagation family <inline-formula><tex-math id="M1">\begin{document}$ \{P(t), \; t\geq0\} $\end{document}</tex-math></inline-formula> generated by the operators <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ R $\end{document}</tex-math></inline-formula>. For proving the main result we do not impose any condition on the relation between the domain of the operators <inline-formula><tex-math id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ R $\end{document}</tex-math></inline-formula>. We also do not assume that the operator <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> has necessarily a bounded inverse. The main tools applied in our analysis are the theory of measure of noncompactness, fractional calculus, and Sadovskii's fixed point theorem. Finally, we provide an example to show the application of our main result.</p>