Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations

In this paper, we study the existence of positive solutions for a class of coupled integral boundary value problems of nonlinear semipositone Hadamard fractional differential equations[D^alpha u(t) + lambda f(t, u(t), v(t)) = 0,quad D^alpha v(t) + lambda g(t, u(t), v(t)) = 0,quad t in (1, e),quad lambda 0] [u^{(j)}(1) = v^{(j)}(1) = 0, 0 leq j leq n - 2; u(e) = mu int^e_1 v(s) frac{ds}{ s} , v(e) = nuint^e_1 u(s) frac{ds}{ s},] where (lambda,mu,nu) are three parameters with 0 mu beta) and (0 nu alpha,quad alpha,betain (n-1, n]) are two real numbers and (n≥3, D^alpha, D^beta) are the Hadamard fractional derivative of fractional order, and f,g are signchanging continuous functions and may be singular at t = 1 or and t = e. First of all, we obtain the corresponding -Green s function for the boundary value problem and some of its properties. Furthermore, by means of the n onlinear alternative of LeraySchauder type and Krasnoselskii s fixed point theorems, we derive an interval of lambda such that the semipositone boundary value problem has one or multiple positive solutions for any lambda lying in this interval. At last, several illustrative examples were given to illustrate the main results.