Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space

Abstract

<p style='text-indent:20px;'>In this paper, the approximate controllability for Sobolev type <inline-formula><tex-math id="M2">\begin{document}$ \psi - $\end{document}</tex-math></inline-formula> Hilfer fractional backward perturbed integro-differential equations with <inline-formula><tex-math id="M3">\begin{document}$ \psi - $\end{document}</tex-math></inline-formula> fractional non local conditions in a Hilbert space are studied. A new set of sufficient conditions are established by using semigroup theory, <inline-formula><tex-math id="M4">\begin{document}$ \psi - $\end{document}</tex-math></inline-formula>Hilfer fractional calculus and the Schauder's fixed point theorem. The results are obtained under the assumption that the associate backward <inline-formula><tex-math id="M5">\begin{document}$ \psi - $\end{document}</tex-math></inline-formula> fractional linear system is approximately controllable. Finally, an example is given to illustrate the obtained results.</p>