On asymptotic periodic solutions of fractional differential equations and applications

2023/04/10 by Vu Trong Luong, Nguyễn Đức Huy, Luong, Vu Trong +5 · 1 citation
Mathematics · Computer Science · #Nonlinear Differential Equations Analysis #Differential Equations and Numerical Methods #Advanced Mathematical Modeling in Engineering

paper · doi:10.48550/arxiv.2304.04850

Abstract

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form D^α_Cu(t)=Au(t)+f(t), u(0)=x, 0<α≤1, ( *) where D^α_Cu(t) is the derivative of the function u in the Caputo's sense, A is a linear operator in a Banach space \X that may be unbounded and f satisfies the property that lim_{t→ ∞} (f(t+1)-f(t))=0 which we will call asymptotic 1-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1-periodic, or there exists an asymptotic mild solution that is asymptotic 1-periodic.

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• Variational Formulations for Fractional Integral and Differential Equations

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