Existence of positive solutions to a higher order singular boundary value problem with fractional q-derivatives

The authors study the singular boundary value problem with fractional q-derivatives -(D(q)(nu)u)(t) = f(t, u), t is an element of (0,1), (D(q)(i)u)(0) = 0, i = 0, ... , n - 2, (D(q)u)(1) = Sigma(m)a(j)(D(q)u)(t(j)) + lambda, where q is an element of (0, 1), m >= 1 and n >= 2 are integers, n - 1 < nu <= n, lambda >= 0 is a parameter, f: (0, 1] x (0, infinity) -> [0, infinity) is continuous, a(i) >= 0 and t(i) is an element of (0, 1) for i = 1, ... , m, and D-q(nu) is the q-derivative of Riemann-Liouville type of order nu. Sufficient conditions are obtained for the existence of positive solutions. Their analysis is mainly based on a nonlinear alternative of Leray-Schauder.