Ma, Ruyun Nodal solutions of boundary value problems of fourth-order ordinary differential equations. (English) Zbl 1098.34012 J. Math. Anal. Appl. 319, No. 2, 424-434 (2006). Summary: We study the existence of nodal solutions of the fourth-order two-point boundary value problem \[ y''''+\beta(t)y''=a(t)f(y),\;0<t<1,\qquad y(0)=y(1)=y''(1)=0, \] where \(\beta\in C[0,1]\) with \(\beta(t)<\pi^2\) on \([0,1]\), \(a\in C[0,1]\) with \(a\geq 0\) on \([0,1]\) and \(a(t)\equiv 0\) on any subinterval of \([0,1]\), and \(f\in C(\mathbb{R})\) satisfies \(f(u)u>0\) for all \(u\neq 0\). We give conditions on the ratio \(f(s)/s\) at infinity and zero that guarantee the existence of nodal solutions. The proof of our main results is based upon bifurcation techniques. Cited in 23 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:multiplicity results; eigenvalues; disconjugate; bifurcation methods; nodal solutions PDFBibTeX XMLCite \textit{R. Ma}, J. Math. Anal. Appl. 319, No. 2, 424--434 (2006; Zbl 1098.34012) Full Text: DOI References: [1] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368 (2002) · Zbl 1006.34023 [2] Elias, U., Eigenvalue problems for the equation \(L y + \lambda p(x) y = 0\), J. Differential Equations, 29, 28-57 (1978) · Zbl 0351.34014 [3] Eloe, P. W.; Henderson, J., Singular boundary value problems for quasi-differential equations, Int. J. Math. Math. Sci., 18, 571-578 (1995) · Zbl 0886.34016 [4] Ma, R.; Wang, H., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019 [5] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504 [6] Rynne, B. P., Infinitely many solutions of superlinear fourth order boundary value problems, Topol. Methods Nonlinear Anal., 19, 303-312 (2002) · Zbl 1017.34015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.