In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in $C^{2}[0,1]$, and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients.

中文翻译：

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Green函数的不等式和非局部边值问题的正解

在本文中，我们讨论了在包含Stieltjes积分的非局部边界条件下具有非线性（包括斜率和弯矩）的非线性梁方程的正解。我们在非线性上提出一些不等式条件，并在相关的线性算子上提出谱半径条件。这些条件意味着非线性具有超线性或亚线性增长。正定解的存在是通过圆锥上的不动点索引在$ C ^ {2} [0,1] $中获得的，并给出了在混合积分和多点边界条件下且符号改变的系数的梁方程的一些示例。 。