Positive solutions to differential inclusions with nonlocal, nonlinear boundary conditions

We demonstrate the existence of at least one positive solution to the differential inclusion -u″(t)∈F(t,u(t)),t∈(0,1), equipped with the boundary conditions u(0)=H(φ(u)) and u(1)=0. Here φ is linear functional realized as a Lebesgue-Stieltjes integral and H is a nonlinear function. Consequently, the boundary condition at t=0 can be both nonlocal and nonlinear. By imposing an asymptotic condition on H we provide conditions under which the problem will have at least one positive solution.