Global structure of the positive solution for a class of fourth-order boundary value problems with first derivative. (Chinese. English summary) Zbl 1463.34112
Summary: This paper considers the global structure of positive solutions for a fourth-order boundary value problem with first derivative \[\begin{cases}u^{(4)}(t) = rf(t, u(t), u'(t)),\, t \in (0,1),\\ u(0) = u'(0) = u''(1) = u'''(1) = 0, \end{cases}\] where \(r\) is a positive parameter, \(f:[0,1] \times [0,\infty) \times [0,\infty) \to [0,\infty)\) is continuous, and \(f(t, 0, 0) = 0\). When the parameter \(r\) changes in a certain range, the global structure of positive solutions of the problem is obtained by using the Rabinowitz global bifurcation theorems. The conclusions in this paper generalize and improve the related results.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
