We are concerned with the existence of nodal solutions for the n-th order boundary value problem \( {\left\{ \begin{array}{ll} u^{(n)}(t)+a(t) f(u(t))=0, & t\in (0,\pi ),\\ u^{(i)}(0)=0,\ & i\in \{i_1,...,i_{n-m}\},\\ u^{(j)}(\pi )=0,\ & j\in \{j_1,...,j_{m}\}, \end{array}\right. } \) where \(n\ge 2\) , \(1\le m\le n-1,\) \(\{i_1,...,i_{n-m}\},\) \(\{j_1,...,j_{m}\}\) are subsets of the set of integers in \(\{0,..., n-1\}\) ; \(a:[0,\pi ]\rightarrow (-1)^m\mathbb {R}^{-}\) is continuous and does not vanish identically on any subinterval of \([0, \pi ]\) ; \(f: \mathbb {R}\rightarrow \mathbb {R}\) is a continuous function. The proof of our main result is based upon bifurcation techniques.
