Existence solutions for nonlocal fractional differential equation with nonlinear boundary conditions

In this paper, by employing the Guo-Krasnoselskii fixed point theorem in a cone, we study the existence of positive solutions to the following nonlocal fractional boundary value problems { (c)D(0)(+)(alpha)u(t) = f(t, u(t)), t is an element of (0,1), u(t) + u'(0) = 1/2[H-1 (phi(u)) + integral(E) H-2 (s, u(s))ds], u (1) + u'(1) = 0, where D-c(0)+(alpha) is the standard Caputo derivative of order alpha,1 < alpha < 2, E subset of (0,1) is some measurable set. We provide conditions on f, H-1, H-2 and phi such that the problem exhibits at least one positive solution.