Monotone iterative method for fractional differential equations - Zhanbing Bai; Zhang Shuo; Sujing Sun; Chun Yin (Electronic Journal of Differential Equations, 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 $$\displaylines{ D_{0+}^α u(t)=f(t, u(t)), \quad t \in (0, h), \cr t^{1-α}u(t)\big|_{t=0} = u_0 ≠ 0, }$$ where $0<h<+∞$, $f\in C([0, h]× \mathbb{R}, \mathbb{R})$, $D_{0+}^α u (t) $ is the standard Riemann-Liouville fractional derivative, $0<α< 1$. 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.