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 in | Electronic journal of differential equations Vol. 2019; no. 117; pp. 1 - 32 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
Texas State University
30.10.2019
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Subjects | |
Online Access | Get full text |
ISSN | 1072-6691 |
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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. |
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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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