$(\omega ,c)$-periodic solutions for a class of fractional integrodifferential equations

Abstract

In this paper we investigate the following fractional order in time integrodifferential problem $$ \mathbb{D}_{t}^{\alpha}u(t)+Au(t)=f \bigl(t,u(t) \bigr)+ \int _{-\infty}^{t} k(t-s)g \bigl(s,u(s) \bigr)\,ds, \quad t \in \mathbb{R}. $$ D t α u ( t ) + A u ( t ) = f ( t , u ( t ) ) + ∫ − ∞ t k ( t − s ) g ( s , u ( s ) ) d s , t ∈ R . Here, $\mathbb{D}_{t}^{\alpha}$ D t α is the Caputo derivative. We obtain results on the existence and uniqueness of $(\omega ,c)$ ( ω , c ) -periodic mild solutions assuming that −A generates an analytic semigroup on a Banach space X and f, g, and k satisfy suitable conditions. Finally, an interesting example that fits our framework is given.