Electron. J. Differential Equations, Vol. 2020 (2020), No. 30, pp. 1-12.

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

Darin Brindle, Gaston M. N'Guerekata

Abstract:
This article concerns the existence of mild solutions to the semilinear fractional differential equation

with nonlocal conditions $u(0)=u_0 + g(u)$ where $D_t^\alpha(\cdot)$ ($1< \alpha < 2$) is the Riemann-Liouville derivative, $A: D(A) \subset X \to X$ is a linear densely defined operator of sectorial type on a complex Banach space $X$, $f:\mathbb{R}^+\times X\to X$ is S-asymptotically $\omega$-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 $\omega$-periodic functions. An application to fractional relaxation-oscillation equations is given.

Submitted August 11, 2019. Published April 7, 2020.
Math Subject Classifications: 34G20, 34G10.
Key Words: S-asymptotically omega-periodic sequence; fractional semilinear differential equation.

An addendum was posted on April 18, 2020. It corrects Theorem 2.9 and its proof. See the last page of this article.
DOI: 10.58997/ejde.2020.30

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Darin Brindle
Department of Mathematics
Morgan State University
Baltimore, MD 21251, USA
email: Darin.Brindle@morgan.edu
Gaston M. N'Guérékata
Department of Mathematics
Morgan State University
Baltimore, MD 21251, USA
email: Gaston.N'Guerekata@morgan.edu

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