On time fractional pseudo-parabolic equations with nonlocal integral conditions

In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order \begin{document}$ \sigma,\; \; 0<\sigma<1 $\end{document} and the space fractional derivative is of order \begin{document}$ \alpha,\beta >0 $\end{document} . In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen \begin{document}$ \alpha, \beta $\end{document} . The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in \begin{document}$ L^p $\end{document} between the regularized solution and the sought solution is obtained.