An existence result of ($\omega$,c)-periodic mild solutions to some fractional differential equation

Abstract

We first investigate in this paper further properties of the new concept of $(\omega,c)$-periodic functions; then we apply the results to study the existence of $(\omega,c)$-periodic mild solutions of the fractional differential equations $D_t^{\alpha}(u(t)-F_1(t,u(t)))=A(u(t)-F_1(t,u(t)))+D_t^{\alpha-1}F_2(t,u(t)),\;t\in\mathbb{R},$ where $1<\alpha<2,\;A:D(A)\subseteq X\to X$ is a linear densely defined operator of sectorial type on a complex Banach space $X$, $F_1, F_2:\mathbb{R}\times X\to X$ are two $(\omega, c)$-periodic functions satisfying suitable conditions in the second variable. The fractional derivative is understood in the sense of Riemann-Liouville.