S-Asymptotically Periodic Solutions for Time-Space Fractional Evolution Equation

Abstract

This paper discusses the abstract time-space fractional evolution equation with the Caputo derivative of order $$\alpha \in (0,1)$$ and fractional power operator $$-A^{\beta }$$ , $$\beta \in (0,1)$$ , where $$-A$$ generates a $$C_{0}$$ -semigroup on a Banach space. The compactness and exponential stability of the semigroup which is generated by fractional power operator $$-A^{\beta }$$ are investigated. With the aid of the properties of the semigroup, the existence and global asymptotic behavior of S-asymptotically periodic solutions are obtained by some fixed point theorems and related inequalities. An example to the time-space fractional diffusion equation with fractional Laplacian will be shown.