Asymptotic Behavior of Polynomially Bounded Solutions of Linear Fractional Differential Equations

TL;DR

This paper investigates the long-term behavior of solutions to fractional differential equations with polynomial growth, establishing conditions under which solutions decay to zero at infinity based on spectral properties.

Contribution

It develops a spectral theory for polynomially bounded functions and characterizes the asymptotic decay of solutions to fractional differential equations with unbounded operators.

Findings

Solutions decay to zero if spectral set intersects imaginary axis countably.

Spectral conditions determine asymptotic behavior of solutions.

New spectral framework for polynomial growth functions in fractional calculus.

Abstract

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^{\alpha}_Cu(t)=Au(t)+f(t)$ on the half line, where $D^{\alpha}_Cu(t)$ 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 $\lim_{h\downarrow 0} \sup_{t\ge 0} \| u(t+h)-u(t)\|/(1+t)^n=0$, and $\sup_{t\ge 0} \| u(t)\| /(1+t)^n <\infty$, then, $\lim_{t\to\infty} u(t)/(1+t)^n =0$ provided that the spectral set $\Sigma (A,\alpha )\cap i\R$ is countable, where $\Sigma (A,\alpha )$ is defined to be the set of complex numbers $\xi$ such that $\lambda^{\alpha -1} (\lambda^\alpha -A)^{-1}$ is analytic in a neighborhood of $\xi$, and $u$ satisfies some ergodic