Nodal solutions of multi-point boundary value problems. (English) Zbl 1195.34028

The authors study a nonlinear multipoint boundary value problem of the form \[ y''+ w(t) f(y)= 0,\qquad t\in (a,b), \]
\[ \cos\alpha\;y(\alpha)- \sin\alpha\;y')a_= 0, \]
\[ y(b)- \sum^m_{i=1} k_i y(\eta_i)= 0. \] Here \(zf(z)> 0\) for all \(z\neq 0\). The coefficient \(w(t)\) is positive. Sufficient conditions are obtained for the existence and nonexistence of nodal solutions, which have specific properties of zeroes of the solution and its first derivative. These sufficient conditions are formulated with the help of eigenvalues of an auxiliary Sturm-Liouville problem.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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