Monotone iterative method for fractional differential equations

Abstract

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 $0<h<+\infty$, $f\in C([0, h]\times \mathbb{R}, \mathbb{R})$, $D_{0+}^\alpha u (t) $ 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.