Extremal solutions for some periodic fractional differential equations

Abstract

By using the lower and upper solution method, the existence of an iterative solution for a class of fractional periodic boundary value problems, $$\begin{aligned}& D_{0+}^{\alpha}u(t)=f\bigl(t, u(t)\bigr),\quad t \in(0, h),\\& \lim_{t \to0^{+}}t^{1-\alpha}u(t) = h^{1-\alpha}u(h), \end{aligned}$$ is discussed, where $0< h<+\infty$ , $f\in C([0, h]\times R, R)$ , $D_{0+}^{\alpha}u (t) $ is the Riemann-Liouville fractional derivative, $0<\alpha< 1$ . Different from other well-known results, a new condition on the nonlinear term is given to guarantee the equivalence between the solution of the periodic boundary value problem and the fixed point of the corresponding operator. Moreover, the existence of extremal solutions for the problem is given.