J. Nonl. Evol. Equ. Appl. 2024 (5) pp. 67-80, published on July 6, 2024:

(ω,c)-asymptotically periodic mild solutions to semilinear two terms fractional differential equations

¹ Université Joseph KI-ZERBO, Département de Mathématiques, 03 BP 7021 OUA. 03, Burkina Faso.

¹ Ecole Normale Supérieure, Institut des Sciences et de Technologie, 01 BP 1757 OUA. 01, Burkina Faso.

Received on April 27, 2024
Accepted on May 14, 2024

Communicated by Gaston M. N'Guérékata

Abstract. In this article, we first explore new properties of (ω, c)-asymptotically periodic functions. Then using the Banach fixed point principle and the Leray-Schauder alternative theorem, we prove the existence and uniqueness of (ω, c)-asymptotically periodic mild solutions to the abstract semilinear fractional differential equation of the form: D_t^αu(t) = Au(t) + D_t^α−1f(t, u(t)), 1 < α < 2, t ≥ 0, u(0) = u₀ , where A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X, u₀ ∈ X , f : R₊ × X → X is (ω, c)-asymptotically periodic in t ∈ R₊, and D_t^α(·) is the Riemann-Liouville fractional derivative.
Keywords:	(ω, c)-asymptotically periodic, mild solution, fractional differential equation, operator of sectorial type.
2010 Mathematics Subject Classification: 34A08, 47D60, 34C25.