Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives

Abstract

In this paper we study the existence of unique positive solutions for the following coupled system:

{Da0 + x(t) + f1(t, x(t), D. 0+ x(t)) + g1(t, y(t)) = 0,

D beta 0+ y(t) + f2(t, y(t), D. 0+ y(t)) + g2(t, x(t)) = 0,

t. (0, 1), n - 1 < a, beta < n;

x(i)(0) = y(i)(0) = 0, i = 0, 1, 2,..., n - 2;

[D. 0+ y(t)] t=1 = k1(y(1)), [D. 0+ x(t)] t=1 = k2(x(1)),

where the integer number n > 3 and 1 =. =. = n - 2, 1 =. =. = n - 2, f1, f2 : [0, 1] xR+ xR+. R+, g1, g2 : [0, 1] xR+. R+ and k1, k2 : R+. R+ are continuous functions, Da0 + and D beta 0+ stand for the Riemann-Liouville derivatives. An illustrative example is given to show the effectiveness of theoretical results.