On a Nonlocal BVP with Nonlinear Boundary Conditions

We consider the existence of at least one positive solution of the problem -y '' (t) = f(t,y(t)), y(0) = H-1 (phi(y)) + integral(E) H-2 (s, y(s)) ds, y(1) = 0, where y(0) = H-1 (phi(y)) + integral(E) H-2 (s, y(s)) represents a nonlinear, nonlocal boundary condition. We show by imposing some relatively mild structural conditions on f, H-1, H-2, and phi that this problem admits at least one positive solution. Finally, our results generalize and improve existing results, and we give a specific example illustrating these generalizations and improvements.