Initial value problems for Caputo fractional equations with singular nonlinearities




Webb, Jeffrey

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Texas State University, Department of Mathematics


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.



Fractional derivatives, Volterra integral equation, Weakly singular kernel, Gronwall inequality


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.


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