Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions

In this paper, we consider four-point coupled boundary value problem for systems of the nonlinear semipositone fractional differential equation {D-0+(alpha)+u + lambda f(t, u, v) = 0, 0 < t < 1, lambda > 0, D-0+(alpha)+v + lambda g(t, u, v) = 0, u((i))(0) = v((i))(0) = 0, 0 <= i <= n -2, u(1) = av(xi), v(1) = bu(eta), xi, eta is an element of (0,1) where lambda is a parameter, a, b, xi, eta satisfy xi, eta is an element of (0, 1), 0 < ab xi eta < 1, alpha is an element of (n - 1, n] is a real number and n >= 3, and D-0+(alpha) is the Riemann-Liouville's fractional derivative, and f, g are continuous and semipositone. We derive an interval on lambda such that for any lambda lying in this interval, the semipositone boundary value problem has multiple positive solutions.