Global solutions to a time-fractional Cauchy problem

In the current work, we study a Cauchy problem for time-fractional pseudo-parabolic equation with globally Lipschitz source term. We prove unique existence of mild solution to problem, by common Banach fixed point theorem. This is then verified that exists in time Gr¨onwall’s inequality. Compare previous works about similar issuse, approach way does not require using weighted spaces. Although our share spirit studies, method seems be more precise and natural.