Existence of positive solutions for a coupled system of nonlinear fractional differential equations

Abstract

We study the following nonlinear boundary-value problems for fractional differential equations $$D^{\alpha} u(t) = f(t, v(t),D^{\beta - 1}v(t)), t > 0,\\ D^{\beta} v(t) = g(t, u(t),D^{\alpha - 1}u(t)), t > 0,\\ u > 0,\; v > 0 \in (0,\infty), \lim_{t\rightarrow 0+} u(t) = \lim_{t\rightarrow 0+} v(t) = 0,$$ where $1 < \alpha \leq 2$ and $1 < \beta \leq 2$. Under certain conditions on $f$ and $g$, the existence of positive solutions is obtained by applying the Schauder fixed-point theorem.