Affiliation:
1. Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
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).
Publisher
National Library of Serbia