Geodesic
Nodal solutions for a second-order $m$-point boundary value problem
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1243-1263.

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We study the existence of nodal solutions of the $m$-point boundary value problem \[ u^{\prime \prime }+ f(u)=0, \quad 01, u^{\prime }(0)=0, \quad u(1)=\sum ^{m-2}_{i=1} \alpha _i u(\eta _i) \] where $\eta _i\in \mathbb{Q}$ $(i=1, 2, \cdots , m-2)$ with $0\eta _1\eta _2\cdots \eta _{m-2}1$, and $\alpha _i\in \mathbb{R}$ $(i=1, 2, \cdots , m-2)$ with $\alpha _i>0$ and $0\sum \nolimits ^{m-2}_{i=1} \alpha _i 1$. 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.
Classification : 34B10, 34C23, 34G20, 34L20, 47J15, 47N20
Keywords: multiplicity results; eigenvalues; bifurcation methods; nodal zeros; multi-point boundary value problems 
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     author = {Ma, Ruyun},
     title = {Nodal solutions for a second-order $m$-point boundary value problem},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1243--1263},
     publisher = {mathdoc},
     volume = {56},
     number = {4},
     year = {2006},
     mrnumber = {2280807},
     zbl = {1164.34329},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006__56_4_a12/}
}
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Ma, Ruyun. Nodal solutions for a second-order $m$-point boundary value problem. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1243-1263. http://geodesic.mathdoc.fr/item/CMJ_2006__56_4_a12/