Matrix representations of Sturm–Liouville problems with coupled eigenparameter-dependent boundary conditions

We study the matrix representations of Sturm–Liouville problems with coupled eigenparameter-dependent boundary conditions with a finite spectrum. We prove for any positive integer n  , the considered problems have at most n+3n+3 eigenvalues, and show that this kind of Sturm–Liouville problems with coupled eigenparameter-dependent boundary conditions is equivalent to a class of matrix eigenvalue problems in the sense that they have exactly the same eigenvalues.