Geodesic
Monotone iterative method for fractional differential equations
Electronic journal of differential equations, Tome 2016 (2016)
In this article, by using the lower and upper solution method, we prove the existence of iterative solutions for a class of fractional initial value problem with non-monotone term

$\displaylines{ D_{0+}^\alpha u(t)=f(t, u(t)), \quad t \in (0, h), \cr t^{1-\alpha}u(t)\big|_{t=0} = u_0 \neq 0, }$

where $0$ is the standard Riemann-Liouville fractional derivative, $0\alpha 1$. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the IVP and the fixed-point of the corresponding operator. Moreover, we show the existence of maximal and minimal solutions.
Classification : 34B15, 34A08
Keywords: fractional initial value problem, lower and upper solution method, existence of solutions 
@article{EJDE_2016__2016__a24,
     author = {Bai,  Zhanbing and Shuo,  Zhang and Sun,  Sujing and Yin,  Chun},
     title = {Monotone iterative method for fractional differential equations},
     journal = {Electronic journal of differential equations},
     year = {2016},
     volume = {2016},
     zbl = {1329.34051},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2016__2016__a24/}
}
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AU  - Yin,  Chun
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%A Shuo,  Zhang
%A Sun,  Sujing
%A Yin,  Chun
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%J Electronic journal of differential equations
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%F EJDE_2016__2016__a24
Bai,  Zhanbing; Shuo,  Zhang; Sun,  Sujing; Yin,  Chun. Monotone iterative method for fractional differential equations. Electronic journal of differential equations, Tome 2016 (2016). http://geodesic.mathdoc.fr/item/EJDE_2016__2016__a24/