Left-definite system of first-order equations together with eigenparameter-dependent boundary conditions

Abstract

This paper provides some information on the eigenvalues and eigenfunctions of some left-definite system of first-order differential equations subject to eigenparameter-dependent boundary conditions. Namely, we show that the pair of solutions of the system of equations satisfying some initial conditions exists and is unique, and this pair is analytic in the spectral parameter of order 1/2. We also introduce Lagrange's formula for the left-definite equation. Using some Prüfer angels, we investigate oscillation of zeros of eigenfunctions and asymptotics equations for the eigenvalues of the problem. Moreover, we share some ordinary and Fréchet derivatives of eigenvalues and eigenfunctions with respect to some elements of data.