In this paper we study the existence of unique positive solutions for the following
coupled system:
⎧
⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
Dα
0+ x(τ ) + f1(τ , x(τ ),Dη
0+ x(τ )) + g1(τ , y(τ )) = 0,
Dβ
0+ y(τ ) + f2(τ , y(τ ),Dγ
0+ y(τ )) + g2(τ , x(τ )) = 0,
τ ∈ (0, 1), n –1< α,β < n;
x(i)
(0) = y(i)
(0) = 0, i = 0, 1, 2, ... , n – 2;
[Dξ
0+ y(τ )]τ=1 = k1(y(1)), [Dζ
0+ x(τ )]τ=1 = k2(x(1)),
where the integer number n > 3 and 1 ≤ γ ≤ ξ ≤ n – 2, 1 ≤ η ≤ ζ ≤ n – 2,
f1, f2 : [0, 1] × R+ × R+ → R+, g1, g2 : [0, 1] × R+ → R+ and k1, k2 : R+ → R+ are
continuous functions, Dα
0+ and Dβ
0+ stand for the Riemann–Liouville derivatives. An
illustrative example is given to show the effectiveness of theoretical results.