Maximal regularity for fractional integro-differential equations in Besov spaces

The theory of operator-valued Fourier multipliers is used to obtain well-posedness results for some degenerate multi-term fractional differential equations with infinite delay in general Banach spaces. We use the right-sided Liouville fractional derivative on the real axis and the Besov spaces $ B^s_{p, q}(\mathbb{R}, X) $, $ 1\le p, q\le \infty $, $ s>0 $, in particular the spaces of Hölder continuous functions $ C^s(\mathbb{R}, X), \, 0<s<1 $, with $ X $ a given Banach space. The results are used for the treatment of some concrete integro-differential equations which appear in several models in the applied sciences, notably physics, biological sciences, rheology, and material science. In many more areas, fractional models are increasingly used to refine traditional models based on partial differential equations of evolution.