Electron. J. Differential Equations, Vol. 2019 (2019), No. 117, pp. 1-32.

Initial value problems for Caputo fractional equations with singular nonlinearities

Jeffrey R. L. Webb

Abstract:
We consider initial value problems for Caputo fractional equations of the form $D_{C}^{\alpha}u=f$ where f can have a singularity. We consider all orders and prove equivalences with Volterra integral equations in classical spaces such as $C^{m}[0,T]$. In particular for the case $1<\alpha<2$ we consider nonlinearities of the form $t^{-\gamma}f(t,u,D^{\beta}_{C}u)$ where $0<\beta \leq 1$ and $0\leq \gamma<1$ with f continuous, and we prove results on existence of global $C^1$ solutions under linear growth assumptions on f(t,u,p) in the u,p variables. With a Lipschitz condition we prove continuous dependence on the initial data and uniqueness. One tool we use is a Gronwall inequality for weakly singular problems with double singularities. We also prove some regularity results and discuss monotonicity and concavity properties.

Submitted November 20, 2018. Published October 30, 2019.
Math Subject Classifications: 34A08, 34A12, 26A33, 26D10.
Key Words: Fractional derivatives; Volterra integral equation; weakly singular kernel; Gronwall inequality.

An addendum was posted on August 5, 2020. It provides additional information and corrects some typos. See the last page of this article.

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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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