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.

本文研究了一类具有非局部初始条件的时空分数式反应扩散方程，并在分数幂空间中构造了一个抽象理论，讨论了与S-渐近\(\omega\)-周期温和解有关的结果。当系数足够小时，在非线性项可以增长任意阶数的条件下，基于算子半群和不动点理论讨​​论了S-渐近\(\omega\) -周期解的存在唯一性定理。此外，我们还使用奇异型 Gronwall 不等式和适当的分数阶微积分考虑了 Mittag-Leffler-Ulam-Hyers 稳定性结果。本文的结果将（Andrade 等人，Proc Edinb Math Soc 59:65–76, 2016）的工作扩展到时空分数非局部反应扩散方程的情况。