Initial value problems for Caputo fractional equations with singular nonlinearities

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...

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Published inElectronic journal of differential equations Vol. 2019; no. 117; pp. 1 - 32
Main Author Jeffrey R. L. Webb
Format Journal Article
LanguageEnglish
Published Texas State University 30.10.2019
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ISSN1072-6691

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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.
AbstractList 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.
Author Jeffrey R. L. Webb
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Snippet 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...
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SubjectTerms fractional derivatives
gronwall inequality
volterra integral equation
weakly singular kernel
Title Initial value problems for Caputo fractional equations with singular nonlinearities
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