# Nodal solutions for a second-order *m*-point boundary value problem

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Article

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DOI: 10.1007/s10587-006-0092-7

- Cite this article as:
- Ma, R. Czech Math J (2006) 56: 1243. doi:10.1007/s10587-006-0092-7

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## Abstract

We study the existence of nodal solutions of the where

*m*-point boundary value problem$$\begin{gathered} u'' + f(u) = 0,0 < t < 1, \hfill \\ u'(0) = 0,u(1) = \sum\limits_{i = 1}^{m - 2} {\alpha _i u(\eta i)} \hfill \\ \end{gathered} $$

*η*_{i}∈ ℚ (*i*= 1, 2, ...,*m*− 2) with 0 <*η*_{1}<*η*_{2}< ... <*η*_{m−2}< 1, and*α*_{i}∈ ℝ (*i*= 1, 2, ...,*m*− 2) with*α*_{i}> 0 and \(\sum\nolimits_{i = 1}^{m - 2} {\alpha _i } \) < 1. We give conditions on the ratio*f*(*s*)/*s*at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.### Keywords

multiplicity resultseigenvaluesbifurcation methodsnodal zerosmulti-point boundary value problems## Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2006