Multiple Positive Solutions to Nonlinear Boundary Value Problems of a System for Fractional Differential Equations

By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by −D (0(+)) (ν(1)) y (1)(t) = λ (1) a (1)(t)f(y (1)(t), y (2)(t)), − D (0(+)) (ν(2)) y (2)(t) = λ (2) a (2)(t)g(y (1)(t), y (2)(t)), where D (0(+)) (ν) is the standard Riemann-Liouville fractional derivative, ν (1), ν (2) ∈ (n − 1, n] for n > 3 and n ∈ N, subject to the boundary conditions y (1) ((i))(0) = 0 = y (2) ((i))(0), for 0 ≤ i ≤ n − 2, and [D (0(+)) (α) y (1)(t)](t=1) = 0 = [D (0(+)) (α) y (2)(t)](t=1), for 1 ≤ α ≤ n − 2, or y (1) ((i))(0) = 0 = y (2) ((i))(0), for 0 ≤ i ≤ n − 2, and [D (0(+)) (α) y (1)(t)](t=1) = ϕ (1)(y (1)), [D (0(+)) (α) y (2)(t)](t=1) = ϕ (2)(y (2)), for 1 ≤ α ≤ n − 2, ϕ (1), ϕ (2) ∈ C([0,1], R). Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.