Gulgowski, Jacek Global bifurcation and multiplicity results for Sturm-Liouville problems. (English) Zbl 1137.34324 NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 5-6, 559-568 (2007). Summary: We prove a multiplicity theorem for the problem \[ u''(t) + \mu u(t) + \varphi (t,u(t),u'(t)) = 0\text{ for a.e. }t \in (a,b), \qquad l(u) = 0, \] with Sturm-Liouville boundary conditions \(l\), and function \(\varphi\) satisfying Carathéodory conditions. We assume that \(\varphi (t, x, y) = Ax + o (|x| + |y|)\) for \(|x| + |y| \rightarrow 0\) and \(\varphi (t, x, y) = Bx + o (|x| + |y|)\) for \(|x| + |y| \rightarrow +\infty\). We prove that the above problem has at least \(2n\) solutions where \(n\) is the number of eigenvalues of the appropriate linear problem, laying between \(\min(A, B)\) and \(\max(A, B)\). Some additional remarks are following. Cited in 3 Documents MSC: 34B24 Sturm-Liouville theory 34C23 Bifurcation theory for ordinary differential equations Keywords:multiple solutions PDFBibTeX XMLCite \textit{J. Gulgowski}, NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 5--6, 559--568 (2007; Zbl 1137.34324) Full Text: DOI