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

In this paper, the approximate controllability for Sobolev type \begin{document}$ \psi - $\end{document} Hilfer fractional backward perturbed integro-differential equations with \begin{document}$ \psi - $\end{document} fractional non local conditions in a Hilbert space are studied. A new set of sufficient conditions are established by using semigroup theory, \begin{document}$ \psi - $\end{document}Hilfer fractional calculus and the Schauder's fixed point theorem. The results are obtained under the assumption that the associate backward \begin{document}$ \psi - $\end{document} fractional linear system is approximately controllable. Finally, an example is given to illustrate the obtained results.