Krein S. G.
Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 323–325
Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 424–432
The Volterra integral equation of the second order with a regular singularity is considered. Under the conditions that a kernel K(x,t) is a real matrix function of order n×n with continuous partial derivatives up to order N+1 inclusively and K(0,0) has complex eigenvalues ν±i μ (ν>0), it is shown that if ν>2|‖K|‖ C -N-1, then a given equation has two linearly independent solutions.
Ukr. Mat. Zh. - 1990. - 42, № 3. - pp. 388-390
Ukr. Mat. Zh. - 1989. - 41, № 9. - pp. 1222–1225
Ukr. Mat. Zh. - 1982. - 34, № 1. - pp. 31-42
Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 78-82
On some transformations of integral equation kernels and their influence on the spectra of these equations
Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 12-38
In recent publications by the authors (1, 2) and V. I. Matsayev (3,4) it was shown that the study of the abstract triangular representation of Volterra operators by the Brodsky integral naturally leads to a series of relations between the eigenvalues of Hermitian components of Volterra operators.
Though some specific properties of the triangular truncation transformation while deducing were used, these results admit in the main a generalization for any transformations (linear continuous operators acting in the Hilbert space of the Hilbert-Schmidt operators). By this generalization both the nature of relations under consideration and their proofs are simplified.
This generalization (§§ 2, 3, 4) is perhaps of interest as it leads to some new applications; it permits us, in particular, to obtain a number of precise estimations for the central stability zone for various Hamiltonian systems of linear differential equations with periodic coefficients (§ §5, 6, 7).