Existence and monotonicity of nonlocal boundary value problems: the one-dimensional case

We consider nonlocal equations of the general form \begin{equation} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{equation} By developing a Green's function representation for the solution of the boundary value problem we study existence, uniqueness, and qualitative properties (e.g., positivity or monotonicity) of solutions to these problems. We apply our methods to fractional order differential equations. We also demonstrate an application of our methodology both to convolution equations with nonlocal boundary conditions as well as those with a nonlocal term in the convolution equation itself.