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A Nagumo-like uniqueness theorem for fractional differential equations. (English) Zbl 1229.26013

The Nagumo uniqueness theorem for first-order ordinary differential equations

${x}^{\text{'}}+f\left(t,x\right)=0,\phantom{\rule{1.em}{0ex}}t>0,$

established by M. Nagumo [“Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung”, Japanese Journ. of Math. 3, 107–112 (1926; JFM 52.0438.01)] has recently been generalized by A. Constantin [Proc. Japan Acad., Ser. A 86, No. 2, 41–44 (2010; Zbl 1192.34014)], where the conditions imposed on $f$ have been replaced by

$|f\left(t,x\right)|\le \frac{w\left(|x|\right)}{t},\phantom{\rule{1.em}{0ex}}\text{where}\phantom{\rule{1.em}{0ex}}{\int }_{0}^{r}\frac{w\left(s\right)}{s}\phantom{\rule{0.166667em}{0ex}}ds\le r$

and $w:\left[0,+\infty \right)\to \left[0,+\infty \right)$ is a continuous, increasing function, with $w\left(0\right)=0$.

In the paper under review, after a brief introduction to the subject related with Nagumo’s uniqueness result and various extensions of it, the authors give the gist of the proof of Nagumo’s result in the classical case. In the following, they obtain a variant of the classical and generalized (in the sense of Constantin) uniqueness theorem for fractional differential equations, i.e., they show uniqueness results for

$\left\{\begin{array}{cc}& {}_{0}{D}_{t}^{\alpha }x+f\left(t,x\right)=0,\phantom{\rule{1.em}{0ex}}t>0\hfill \\ & \underset{t↘0}{lim}\left[{t}^{1-\alpha }x\left(t\right)\right]={x}_{0}\in ℝ\hfill \end{array}\right\$

with $\alpha \in \left(0,1\right)$, where ${}_{0}{D}_{t}^{\alpha }$ is the Riemann-Liouville fractional derivative given by

$\left({}_{0}{D}_{t}^{\alpha }x\right)\left(t\right)=\frac{1}{{\Gamma }\left(1-\alpha \right)}·\frac{d}{dt}\left[{\int }_{0}^{t}\frac{x\left(s\right)}{{\left(t-s\right)}^{\alpha }}\phantom{\rule{0.166667em}{0ex}}ds\right],\phantom{\rule{1.em}{0ex}}t>0·$

MSC:
 26A33 Fractional derivatives and integrals (real functions) 34K37 Functional-differential equations with fractional derivatives