On Dirac systems with multi-eigenparameter-dependent transmission conditions

Abstract

In this work, we investigate a Dirac system which has discontinuities at finite interior points and contains eigenparameter in both boundary and transmission conditions. By defining a suitable Hilbert space H associated with the problem, we generate a self-adjoint operator T such that the eigenvalues of the considered problem coincide with those of T. We construct the fundamental system of solutions of the problem and get the asymptotic formulas for the fundamental solutions, eigenvalues and eigen-vectorfunctions. Also, we examine the asymptotic behaviour for the norm of eigenvectors corresponding to the operator T. We construct Green?s matrix, and derive the resolvent of the operator T in terms of Green?s matrix. Finally, we estimate the norm of resolvent of the operator T. In the special case, when our problem has no eigenparameter in both boundary and transmission conditions, the obtained results coincide with the corresponding results in Tharwat (Boundary Value Problems, DOI:10.1186/s13661-015-0515-1, 2016).