Asymptotic Behavior of Polynomially Bounded Solutions of Linear Fractional Differential Equations

2019/10/18 by Nguyễn Văn Minh, Van Minh, Nguyen, Vu Trong Luong +1
Engineering · Mathematics · #34G10 #34K30 #34K37 #45J05 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1910.08609

openalex publication_date 2019/10/18 · openalex created_date 2019/10/25 · openalex updated_date 2026/07/28

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) on the half line, where D^α_Cu(t) is the derivative of the function u in Caputo's sense, A is generally an unbounded closed operator, f is polynomially bounded. To this end we develop a spectral theory for functions of polynomial growth on the half line. Our main result claims that if u is mild solution of the Cauchy problem such that lim_{h\downarrow 0} sup_{t≥ 0} ‖ u(t+h)-u(t)‖/(1+t)ⁿ=0, and sup_{t≥ 0} ‖ u(t)‖ /(1+t)ⁿ

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