Abstract
We prove the existence and the uniqueness of a positive solution to the following combined fractional boundary value problem on the half-line
integral D(alpha)u(t) +a(1)(t)u(sigma 1) + a(2)(t)u(sigma 2) = 0, t is an element of (0, infinity), 1 < alpha < 2,
lim(t -> 0) t(2-alpha)u(t) = 0, lim(t ->infinity)t(1-alpha)u(t) = 0,
where D-alpha is the standard Riemann-Liouville fractional derivative, sigma(1), sigma(2) is an element of (-1, 1), and a(1), a(2) are non negative continuous functions on (0, infinity), which may be singular at t = 0 and satisfying some convenient assumptions related to the Karamata regular variation theory. We also give sharp estimates on such solution. (C) 2016 All rights reserved.