Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends linearly on the spectral parameter. The differential equation is considered on the interval $(0,1)$ with a classical boundary condition at one endpoint and an eigenparameter--dependent boundary condition at the other. Explicit formulas for the inner products and norms of the root functions are obtained. These relations make it possible to analyze the structure of the system of root functions and the corresponding biorthogonal system. Using these results, the minimality of the system of root functions in $L_2(0,1)$ is established. Furthermore, the basis properties of the system of root functions in the spaces $L_p(0,1)$, $1<p<\infty$, are investigated. Necessary and sufficient conditions under which the system forms a basis are derived. Special attention is given to the cases of multiple eigenvalues and the case when the eigenvalue coincides with the critical value $-d/c$. The obtained results reveal a symmetry between different spectral cases and provide a simpler approach that avoids the use of the exit space $L_2(0,1) \oplus \mathbb{C}$. Several examples are presented to illustrate the theoretical results.