On the asymptotic integration of a class of sublinear fractional differential equations

TL;DR

This paper investigates the long-term growth behavior of solutions to a class of nonlinear fractional differential equations, extending classical results to fractional orders and providing optimal growth estimates.

Contribution

It introduces asymptotic growth estimates for solutions of nonlinear fractional differential equations, generalizing classical Bihari results to fractional orders with optimality demonstrations.

Findings

Solutions grow slower than any polynomial of degree $a\,\alpha$ as $t\to\infty$

The growth estimate $x(t)=o(t^{a\alpha})$ is optimal in some cases

The method uses a triple interpolation inequality on the real line

Abstract

We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations $D_{0+}^{\alpha}(x-x_0) =f(t,x)$ which includes $D_{0+}^{\alpha}(x-x_0) =H(t)x^{\lambda}$ with $\lambda\in(0,1)$ for the case of slowly-decaying coefficients $H$. The proof is based on the triple interpolation inequality on the real line and the growth estimate reads as $x(t)=o(t^{a\alpha})$ when $t\to+\infty$ for $1>\alpha>1-a>\lambda>0$. Our result can be thought of as a non--integer counterpart of the classical Bihari asymptotic integration result for nonlinear ordinary differential equations. By a carefully designed example we show that in some circumstances such an estimate is optimal.