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.

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非局部边值问题的存在性和单调性：一维情况

我们考虑一般形式的非局部方程\begin{方程} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{方程}通过开发用于解决边值问题的格林函数表示，我们研究了这些问题的解决方案的存在性、唯一性和定性属性（例如，正性或单调性）。我们将我们的方法应用于分数阶微分方程。我们还展示了我们的方法在具有非局部边界条件的卷积方程以及在卷积方程本身中具有非局部项的卷积方程的应用。