Abstract
In the paper, we consider the problem on eigenvalues of a loaded differential operator of the first order with a periodic boundary condition on the interval [–1; 1], that is, equation contains a load at the point (–1) and the function of bounded variation (t), with the condition Φ(−1) = Φ(1) = 1 . A characteristic determinant of spectral problem is constructed for the considered loaded differentiation operator, which is an entire analytical function on the spectral parameter. On the basis of the characteristic determinant formula, conclusions are proved about the asymptotic behavior of the spectrum and eigenfunctions of the loaded spectral problem for the differentiation operator, the characteristic determinant of which is an entire analytic function of the spectral parameter l. A theorem on the location of eigenvalues on the complex plane l is formulated, where the regular growth of an entire analytic function is indicated. A theorem is proved on the asymptotics of the zeros of an entire function, that is, the eigenvalues of the original considered spectral problem for a loaded differential operator of differentiation, and the asymptotic properties of an entire function with distribution of roots are studied.