Abstract
The present paper deals with the dependence of eigenvalues of 2nth order boundary value transmission problems on the problem. The eigenvalues depend not only continuously but also smoothly on the problem. Some new differential expressions of eigenvalues with respect to an endpoint, a coefficient, the weight function, boundary conditions, and transmission conditions, are given.
Highlights
It is well known that boundary value transmission problems are of great importance for their wide applications in physics and engineering
To deal with interior discontinuities, some conditions are imposed on the discontinuous points, which are often called transmission conditions, interface conditions, or point interactions
We study the dependence of eigenvalues of nth order boundary value transmission problems on the problem
Summary
It is well known that boundary value transmission problems are of great importance for their wide applications in physics and engineering. Proof Let λ be an eigenvalue of L and u (x) be the corresponding eigenfunction. To study the continuity of eigenvalues and eigenfunctions on the problem, is assumed to be a subset of X and inherits its norm from X on which the convergence in depends. Based on the space X, the set and Lemma , we obtain that the eigenvalues of nth order boundary value transmission problems depend continuously on the problem. The above discussion illustrates that for every self-adjoint boundary value transmission problem and every eigenvalue λ(ω), the eigenfunction u(·, ω) and its quasi-derivatives u[ ](·, ω), . Pn) and λ eigenvalue of operator L connected with ω, and let u = u(·, ω) be the corresponding eigenfunction.
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