Equivalence of the minimality conditions for the root functions of Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

We study the minimality of the system of root functions associated with a Sturm--Liouville problem whose boundary condition depends linearly on the eigenparameter.

Two different criteria for minimality were previously obtained using independent approaches.

In this paper, we establish the equivalence of these criteria and provide a unified characterization of the exceptional cases in which the removal of certain associated functions fails to preserve minimality.

The theoretical results are illustrated by several examples involving multiple eigenvalues, demonstrating the consistency of the two approaches and clarifying the structure of the corresponding root function systems.