Positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders

In this article, we study the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders {-D(0+)(alpha)u(t) = f(t,v(t)), 0 < t < 1, D(0+)(beta)v(t) = g(t,u(t)), 0 < t < 1, u(0) = u(1) = u'(0) = v(0) = v(1) = v'(0) = v'(1) = u(1) = 0, where 2 < alpha <= 3, 3 <beta = 4, D-0+(alpha), D-0+(beta) are the standard Riemann-Liouville fractional derivative, and f, g : [0, 1] x [0, +infinity) -> [0, +infinity) are given continuous functions, f(t, 0) = 0, g(t, 0) = 0. Our analysis relies on fixed point theorems on cones. Some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established. As an application, examples are presented to illustrate the main results.