Electron. J. Differential Equations, Vol. 2021 (2021), No. 80, pp. 1-22.

A fractional Gronwall inequality and the asymptotic behaviour of global solutions of Caputo fractional problems

Jeffrey R. L. Webb

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.
DOI: https://doi.org/10.58997/ejde.2021.80

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Jeffrey R. L. Webb
School of Mathematics and Statistics
University of Glasgow
Glasgow G12 8SQ, UK
email: jeffrey.webb@glasgow.ac.uk

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