Zhang, Yali Global structure of the positive solution for a class of fourth-order boundary value problems with first derivative. (Chinese. English summary) Zbl 1463.34112 J. Shandong Univ., Nat. Sci. 55, No. 8, 102-110 (2020). 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 Keywords:fourth-order boundary value problems; positive solution; principal eigenvalue; bifurcation theorem PDFBibTeX XMLCite \textit{Y. Zhang}, J. Shandong Univ., Nat. Sci. 55, No. 8, 102--110 (2020; Zbl 1463.34112) Full Text: DOI