Xuping Zhang, Yongwei Jia, and Jiahui An. Monotone Iterative Method for Initial Value Problem to Fractional Differential Equations of Variable-Order of Variable-Order. Dynamic Systems and Applications 30 (2021) No.2, 263-275

https://doi.org/10.46719/dsa20213027

ABSTRACT.
In this paper, by using monotone iterative method based on lower and upper solutions, we study the existence and uniqueness of solutions to initial value problem of fractional differential equations of variable-order

$$
\left\{\begin{array}{ll}
D^{q(t)}_{0+}x(t)=f(t,x(t)),\quad t\in(0,T],\\[8pt]
t^{2-q(t)}x(t)|_{t=0}=t^{2-q(t)}x^{‘}(t)|_{t=0}=0,
\end{array}
\right.
$$
where~$1<q(t)<2$~, $0<T<+\infty$, $D^{q(t)}_{0+}$ is the Riemann-Liouvile fractional derivative of variable-order q(t).

 AMS (MOS) Subject Classification. 26A33; 34A08.