Volume 70, № 1, 2018
Antoniouk A. Vict., Berezansky Yu. M., Boichuk О. A., Gutlyanskii V. Ya., Khruslov E. Ya., Kochubei A. N., Korolyuk V. S., Kushnir R. M., Lukovsky I. O., Makarov V. L., Marchenko V. O., Nikitin A. G., Parasyuk I. O., Pastur L. A., Perestyuk N. A., Portenko N. I., Ronto M. I., Sharkovsky O. M., Tkachenko V. I., Trofimchuk S. I.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 3-6
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 7-28
We study the problems of existence and representations of the solutions bounded on the entire axis for both linear and nonlinear differential equations with unbounded operator coefficients in the Fr´echet and Banach spaces under the condition of exponential dichotomy on the semiaxes of the corresponding homogeneous equation.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 29-39
The stability of global attractor is proved for an impulsive infinite-dimensional dynamical system. The obtained abstract results are applied to a weakly nonlinear parabolic equation whose solutions are subjected to impulsive perturbations at the times of intersection with a certain surface of the phase space.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 40-62
We present an overview of development of the direct Lyapunov method in the global analysis of chaotic systems and describe three directions in which the Lyapunov functions are applied: in the methods of localization of global attractors, where the estimates of dissipativity in a sense of Levinson are obtained, in the problems of existence of homoclinic trajectories, and in the estimation of the dimension of attractors. The efficiency of construction of Lyapunov-type functions is demonstrated. In particular, the Lyapunov dimension formula is proved for the attractors of the Lorentz system.
Variational method for the solution of nonlinear boundary-value problems of the dynamics of bounded volumes of liquid with variable boundaries
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 63-78
We study nonlinear boundary-value problems of the hydrodynamic type formulated for the domains whose boundaries are unknown in advance. The investigations are based on the variational formulations of these problems with the help of specially introduced integral functionals with variable domains of integration. It is shown that the required solutions of the boundary-value problems are, in a certain sense, equivalent to finding stationary points of the analyzed functionals. These are pairs formed by the families of domains and the functions defined in these domains. On an example of the problem of space motion of a vessel with elastic walls partially filled with an ideal incompressible liquid, we propose a method for the construction in the analytic form both of the required solutions and of the boundaries of the domains (deformed walls and the shape of the perturbed free surface of the liquid).
Exact and approximate solutions of spectral problems for the Schrödinger operator on (−∞,∞) with polynomial potential
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 79-93
New exact representations for the solutions of numerous one-dimensional spectral problems for the Schr¨odinger operator with polynomial potential are obtained by using a technique based on the functional-discrete (FD) method. In cases where the ordinary FD-method is divergent, we propose to use its modification, which proved to be quite efficient. The obtained theoretical results are illustrated by numerical examples.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 94-114
We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear boundary-value problems for systems of differential equations under the condition that the components of solutions vanish at some unknown points. The technique can be applied to nonlinearities involving the signs of absolute value and positive or negative parts of functions under various types of boundary conditions.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 115-129
This paper deals with forced frequency locking, i.e., the behavior of periodic solutions to autonomous differential equations under the influence of small periodic forcings. We show that, although the forcings are allowed to be discontinuous (e.g., step-function-like) or even distributional (e.g., Dirac-function-like), the forced frequency locking happens as in the case of smooth forcings, and we derive formulas for the locking cones and for the asymptotic phases as in the case of smooth forcings.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 130-148
We study the existence of $L^p$ -solutions for a class of Hammerstein integral equations and neutral functional differential equations involving abstract Volterra operators. Using compactness-type conditions, we establish the global existence of solutions. In addition, a global existence result for a class of nonlinear Fredholm functional integral equations involving abstract Volterra equations is given.