(ω, c)-periodic and asymptotically (ω, c)-periodic mild solutions to fractional Cauchy problems

Abstract

In this paper, we establish some new properties of $(\omega ,c)$-periodic and asymptotically $(\omega ,c)$-periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) $\{\begin{array}{rl}{}^c{D}_t^{\alpha }u(t)& =Au(t){+}^c{D}_t^{\alpha -1}f(t,u(t)),1<\alpha <2,t\in \mathbb{R},\\ u(0)& =0\end{array}$(1) and (2) $\{\begin{array}{rl}{}^c{D}_t^{\alpha }u(t)& =Au(t){+}^c{D}_t^{\alpha -1}f(t,u(t-h)),1<\alpha <2,t,h\in {\mathbb{R}}_{+},\\ u(0)& =0\end{array}$(2) where ${}^c{D}_t^{\alpha }(\cdot )(1<\alpha <2)$ stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space $\mathbb{X}$ and the function $f(t,x)$ is $(\omega ,c)$-periodic or asymptotically $(\omega ,c)$-periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations.

Keywords:

• Leray–Schauder alternative theorem
• Arzela–Ascoli theorem
• Schauder theorem
• fractional differential equation
• $(\omega c)$-periodic
• asymptotically $(\omega c)$-periodic
• mild solutions

AMS Subject Classifications:

• 26A33
• 34C25
• 34C27
• 34K14
• 35B15
• 47D06

Acknowledgments

The authors would like to express their sincere gratitude to the referees for their careful reading and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).