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-vector- functions. 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).