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 <tex-math id="M1">{document}σ,\; \; 0<σ<1{document}</tex-math> and the space fractional derivative is of order <tex-math id="M2">{document}α,β >0{document}</tex-math>. In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen <tex-math id="M3">{document}α, β{document}</tex-math>. 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 <tex-math id="M4">{document}Lᵖ{document}</tex-math> between the regularized solution and the sought solution is obtained.