S-asymptotically omega-periodic mild solutions to fractional differential equations

Abstract

<p>This article concerns the existence of mild solutions to the semi-linear fractional differential equation</p> <pre>D^α_tu(t) = Au(t) + D^α-1_t ƒ(t, u(t)), t ≥ 0</pre> <p>with nonlocal conditions u(0) = u₀ + g(u) where D^α_t(‧) (1 < α < 2) is the Riemann-Liouville derivative, A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X, ƒ : ℝ⁺ x X → X is S-asymptotically ω-periodic with respect to the first variable. We use the Krsnoselskii's theorem to prove our main theorem. The results obtained are new even in the context of asymptotically ω-periodic functions. An application to fractional relaxation-oscillation equations is given.</p>