Dependence of eigenvalues for higher odd-order differential operator with transmission conditions

Abstract

In this paper, we consider the dependence of eigenvalues on the parameter for higher odd-order differential operator with three types of boundary conditions and corresponding transmission conditions. By defining an appropriate Hilbert space and its inner product, we prove that the eigenvalues of the operators associated with the problem are real and discrete, and give the resolvent operators related to the differential operators. In addition, we show that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, and give the corresponding differential expressions. In particular, giving the Frechet derivatives of the eigenvalues concerning the coefficient functions q0, q1, · · · , qn and the derivatives of the eigenvalues with respect to the left and right sides of the inner discontinuity point c.