Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 80, pp. 1-22.

Title: A fractional Gronwall inequality and the asymptotic behaviour of global
      solutions of Caputo fractional problems
       
Author: Jeffrey R. L. Webb (Univ. of Glasgow, Glasgow, UK)

Abstract:
We study the asymptotic behaviour of global solutions of some nonlinear integral equations
related to some Caputo fractional initial value problems.
We consider problems of fractional order between 0 and 1 and of order between 1 and 2,
each in two cases: when the nonlinearity depends only on the function, and when the
nonlinearity also depends on fractional derivatives of lower order.
Our main tool is a new Gronwall inequality for integrals with singular kernels,
which we prove here, and a related boundedness property of a fractional integral of
an $L^1[0,\infty)$ function.

Submitted July 23, 2021. Published September 20, 2021.
Math Subject Classifications: 34A08, 34A12, 26A33, 26D10.
Key Words: Fractional derivatives; asymptotic behaviour; Gronwall inequality;
           weakly singular kernel.