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Asymptotic behaviour of non-autonomous Caputo fractional differential equations with a one-sided dissipative vector field

Authors: T. S. Doan, P. E. Kloeden
Keywords: Non-autonomous Caputo fractional differential equations, skew-product flows, attractor, entire solution,Volterra integral equations

Abstract

A non-autonomous Caputo fractional differential equation of order $\alpha\in(0,1)$ in $\mathbb{R}^d$ with a driving system $\{\vartheta_t\}_{t\in \mathbb{R}}$ on a compact base space $P$ generates a skew-product flow on $\mathfrak{C}_{\alpha}\times P$, where $\mathfrak{C}_{\alpha}$ is the space of continuous functions $f$ $:$ $\mathbb{R}^+$ $\to$ $\mathbb{R}^d$ with a weighted norm giving uniform convergence on compact time subsets. It was shown by Cui \& Kloeden \cite{CK} to have an attractor when the vector field of the Caputo FDE satisfies a uniform dissipative vector field. {{This attractor is closed, bounded and invariant in $\mathfrak{C}_{\alpha}\times P$ and attracts bounded subsets of $\mathfrak{C}_{\alpha}$ consisting of constant initial functions. }} The structure of this attractor is investigated here in detail for an example with a vector field satisfying a stronger one-sided dissipative Lipschitz condition. In particular, the component sets of the attractor are shown to be singleton sets corresponding to a unique entire solution of the skew-product flow. Its evaluation on $\mathbb{R}^d$ is a unique entire solution of the Caputo FDE, which is both pullback and forward attracting.

T.S. Doan
Institute of Mathematics, Vietnam Academy of Science
and Technology, 18 Hoang Quoc Viet, Hanoi, Viet Nam
E-mail:

P.E. Kloeden
Mathematisches Institut, Universitat Tubingen, D-72076
T¨ubingen, Germany
E-mail:

DOI

https://doi.org/10.56415/basm.y2024.i1-2.p44

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