Coupled systems of boundary value problems with nonlocal boundary conditions

We consider the coupled system −x″=λ1f(t,y(t))−x″=λ1f(t,y(t)), −y″=λ2g(t,x(t))−y″=λ2g(t,x(t)), t∈(0,1)t∈(0,1), subject to the coupled boundary conditions x(0)=H1(φ1(y))x(0)=H1(φ1(y)), x(1)=0x(1)=0 and y(0)=H2(φ2(x))y(0)=H2(φ2(x)), y(1)=0y(1)=0. Since H1H1 and H2H2 are nonlinear functions and φ1φ1 and φ2φ2 are linear functionals realized as Stieltjes integrals, the boundary conditions may be nonlocal and nonlinear in character. By assuming that φ1φ1 and φ2φ2 satisfy a particular decomposition hypothesis together with some growth assumptions on H1H1 and H2H2 at 00 and +∞+∞, we show that this system can possess at least one positive solution even if no growth conditions are imposed on ff and gg.