Electron. J. Differential Equations,
Vol. 2019 (2019), No. 117, pp. 132.
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
where f can have a singularity.
We consider all orders and prove equivalences with Volterra integral equations in
classical spaces such as
.
In particular for the case
we consider nonlinearities of the form
where
and
with f continuous, and we prove results on
existence of global
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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