Initial value problems for Caputo fractional equations with singular nonlinearities
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Date
2019-10-30
Authors
Webb, Jeffrey
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We consider initial value problems for Caputo fractional equations of the form DαCu = ƒ where ƒ can have a singularity. We consider all orders and prove equivalences with Volterra integral equations in classical spaces such as Cm [0, T]. In particular for the case 1 < α < 2 we consider nonlinearities of the form t-γ ƒ(t, u, DβCu) where 0 < β ≤ 1 and 0 ≤ γ < 1 with ƒ continuous, and we prove results on existence of global C1 solutions under linear growth assumptions on ƒ(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.
Description
Keywords
Fractional derivatives, Volterra integral equation, Weakly singular kernel, Gronwall inequality
Citation
Webb, J. R. L. (2019). Initial value problems for Caputo fractional equations with singular nonlinearities. <i>Electronic Journal of Differential Equations, 2019</i>(117), pp. 1-32.
Rights
Attribution 4.0 International