Nodal solutions of multi-point boundary value problems
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.
scientific article; zbMATH DE number 5656718
scientific article