Existence theorems for a class of nonlocal differential equations

A class of nonlocal second-order ordinary differential equations of the formy″(x)=f(x,y(x),(y○λ)(x),y′(x)) for continuous f and λ is studied. The equation is supplemented with none, one, or two Robin boundary conditions depending on whether the interval of interest I is finite, semi-infinite or infinite. The only other restriction on the function λ is that it maps I into itself. Sufficient conditions for the existence of a solution are found, which include the assumption of the existence of 'upper' and 'lower' solutions. The upper and lower solutions provide bounds for the solution on I.