Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions
Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions
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DOI：
10.1017/s0308210506001041
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发表时间：
2008-04
期刊：
影响因子：
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通讯作者：
J. R. L. Webb;G. Infante;D. Franco
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文献类型：
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作者：
J. R. L. Webb;G. Infante;D. Franco
建立了一类四阶非线性方程多个正解的存在性结果。我们考虑了广泛研究的边界条件对应的固支和铰接端和许多非局部边界条件，与一个统一的方法。我们的方法是表明，每个边值问题可以写为相同类型的扰动积分方程，在空间$C[0，1]$，涉及一个线性泛函$\alpha[u]$，但是，虽然我们寻求正解，功能不被假定为所有积极的$u$。结果是新的，即使对于经典的边界条件的夹紧或铰接端时，$\alpha[u]=0$，因为我们得到尖锐的结果存在一个正解;对于多个解决方案，我们寻求最佳值的一些常数发生在理论中，这使我们能够施加较弱的假设上的非线性项比以前的作品。我们的非局部边界条件包含多点问题作为特殊情况下，第一次在四阶问题，我们允许系数的两个符号。
We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.