Positive solutions to fractional boundary value problems with nonlinear boundary conditions

We consider the existence of at least one positive solution of the problem -D(0+)(alpha)u(t) = f (t, u(t)), 0 < t < 1, under the circumstances that u(0) = 0, u(1) = H-1(phi(u)) + integral(E) H-2(s, u(s)) ds, where 1 < alpha < 2, D-0+(alpha) is the Riemann-Liouville fractional derivative, and u(1) = H-1(phi(u)) + integral H-E(2)(s, u(s)) ds represents a nonlinear nonlocal boundary condition. By imposing some relatively mild structural conditions on f, H-1, H-2, and phi, one positive solution to the problem is ensured. Our results generalize the existing results and an example is given as well.