Rynne, Bryan P. Global bifurcation for 2\(m\)th-order boundary value problems and infinitely many solutions of superlinear problems. (English) Zbl 1029.34015 J. Differ. Equations 188, No. 2, 461-472 (2003). The author studies a boundary value problem associated to a \(2m\)th-order ordinary differential equation of the form \[ Lu(x)=p(x)u(x) + g(x,u^{(0)}(x), \dots, u^{(2m-1)}(x))u(x), \] where \(L\) is a selfadjoint, disconjugate operator on \([0,\pi]\) and the boundary conditions are separated. It is assumed that \(g\) is “superlinear at infinity” and that \(\lim_{|\xi|\to 0} g(x,\xi)=0\). It is proved the existence of infinitely many solutions having specified nodal properties.The main result represents a generalization to higher-order problems of a result by P. Hartman [J. Differ. Equations 26, 37-53(1997; Zbl 0365.34032)]. In the proof, it is used a generalization of the Rabinowitz global bifurcation theorem together with general results on the nodal properties of the solutions to the linear eigenvalue problem \(Lu=\mu p u\).Related results for fourth-order problems have been given, among others, by M. Conti, S. Terracini and G. Verzini [Infinitely many solutions to fourth order superlinear periodic problems (preprint)] and by M. Henrard and F. Sadyrbaev [Nonlinear Anal., Theory Methods Appl. 33, 281-302 (1998; Zbl 0937.34020)]. Reviewer: Anna Capietto (Torino) Cited in 1 ReviewCited in 38 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 47J15 Abstract bifurcation theory involving nonlinear operators 34C23 Bifurcation theory for ordinary differential equations Keywords:higher-order boundary value problem; nodal properties; bifurcation Citations:Zbl 0365.34032; Zbl 0937.34020 PDFBibTeX XMLCite \textit{B. P. Rynne}, J. Differ. Equations 188, No. 2, 461--472 (2003; Zbl 1029.34015) Full Text: DOI References: [1] M. Conti, S. Terracini, G. Verzini, Infinitely many solutions to fourth order superlinear periodic problems, preprint.; M. Conti, S. Terracini, G. Verzini, Infinitely many solutions to fourth order superlinear periodic problems, preprint. · Zbl 1074.34047 [2] Capietto, A.; Mawhin, J.; Zanolin, F., A continuation approach to superlinear periodic boundary value problems, J. Differential Equations, 88, 347-395 (1990) · Zbl 0718.34053 [3] Capietto, A.; Henrard, M.; Mawhin, J.; Zanolin, F., A continuation approach to some forced superlinear Sturm-Liouville boundary value problems, Topol. Methods Nonlinear Anal., 3, 81-100 (1994) · Zbl 0808.34028 [4] Coppel, W. A., Disconjugacy, Lecture Notes in Mathematics, Vol. 220 (1971), Springer: Springer Berlin · Zbl 0224.34003 [5] Dancer, E. N., On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23, 1069-1076 (1974) · Zbl 0276.47051 [6] Elias, U., Eigenvalue problems for the equation \(Ly + λp(x) y = 0\), J. Differential Equations, 29, 28-57 (1978) · Zbl 0351.34014 [7] Hartman, P., On boundary value problems for superlinear second order differential equations, J. Differential Equations, 26, 37-53 (1977) · Zbl 0365.34032 [8] Mawhin, J.; Zanolin, F., A continuation approach to fourth order superlinear periodic boundary value problems, Topol. Methods Nonlinear Anal., 2, 55-74 (1993) · Zbl 0799.34024 [9] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504 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.