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We study a Sturm–Liouville problem in the cross-section of the ocean waveguide covered by pack ice. We prove the basis properties of the eigenfunctions, the convergence of the corresponding Fourier type series, orthogonality relations for the eigenfunctions, and study the dispersion relations of the leading modes for two maximal eigenvalues analytically, numerically, and asymptotically. We prove the continuity and monotonicity of the eigenvalues with respect to the frequency and the speed of propagation, their differentiability with respect to the frequency, and the existence of the cut-off frequency. We prove analytically that these eigenvalues are strictly greater than the eigenvalues for the case of a waveguide with a free surface. Assuming that the speed of propagation varies within the given limits, we find the minimum and maximum of the wavenumbers of these leading modes. We develop a numerical algorithm based on the formalism for the layered media that allows, for a given continuous profile of the speed, to study dispersion relations for the leading eigenvalues. The results of the numerical experiments are in complete agreement with the analytical results. Finally, we outline our results that would be of interest in connection with the models of ice other than pack ice. [ABSTRACT FROM AUTHOR]

We derive Opial-type inequalities for a class of real functions satisfying nonhomogenous boundary conditions and determine the best constant and extremals. The results are then used to obtain generalized Lyapunov and De la Valée Poussin inequalities.

The problem considered is to give necessary and sufficient conditions for perturbations of a second-order ordinary differential operator to be either relatively bounded or relatively compact. Such conditions are found for three classes of operators. The conditions are expressed in terms of integral averages of the coefficients of the perturbing operator.