S-asymptotically $$\omega $$ -periodic solutions for time-space fractional nonlocal reaction-diffusion equation with superlinear growth nonlinear terms

This paper study a class of time-space fractional reaction-diffusion equations with nonlocal initial conditions and construct an abstract theory in fractional power spaces to discuss the results related S-asymptotically \(\omega \)-periodic mild solutions. When the coefficients are sufficiently small, under the condition that the nonlinear term can grow any number of orders, we discuss the existence and uniqueness of S-asymptotically \(\omega \)-periodic solutions based on the theory of operator semigroups and fixed point theorem. In addition, we considered the Mittag-Leffler-Ulam-Hyers stability results using the singular type Gronwall inequality and appropriate fractional calculus. The results in the present paper extended the work of (Andrade et al. in Proc Edinb Math Soc 59:65–76, 2016) to the case of time-space fractional nonlocal reaction-diffusion equations.