On the Solvability of a Periodic Problem for a Two-dimensional System of Ordinary Differential Equations of the Second Order
Author(s):
Ergashboy Mukhamadiev
Doctor of Physical and Mathematical Sciences,Professor,
Professor of the Department of Mathematics,
Vologda State University
emuhamadiev@rambler.ru
Alizhon Nabidjanovich Naimov
Doctor of Physical and Mathematical Sciences,Professor,
Professor of the Department of Mathematics,
Vologda State University
naimovan@vogu35.ru
Abstract:
In this paper we study the periodic problem with a period equal to 1
for a two-dimensional system of second-order ordinary differential equations,
in which the main nonlinear part is generated by a polynomial in one
complex variable. It is proven that if the convex hull of the roots
of the generating polynomial does not contain numbers that are multiples
of 2 i*pi, then there is an a priori estimate for solutions to the
periodic problem. Under the conditions of an a priori estimate,
using methods for calculating the mapping degree of vector fields,
the solvability of the periodic problem for any perturbation from a given class
is proven. The system of equations under consideration does not reduce to
a similar system of first-order equations with the main positive
homogeneous nonlinear part. For systems of first-order equations,
the periodic problem was studied in the works of V.A. Pliss, M.A. Krasnoselskii
and their followers using methods of a priori estimation and calculation of
the mapping degree of vector fields. It is known that an a priori estimate of solutions to boundary value problems
for systems of nonlinear ordinary second order differential equations is fraught with difficulties associated
with an estimate of the first-order derivative of the solution when the solution itself is bounded. In this paper, using the example of
a periodic problem for the considered system of second-order equations,
it is established that the a priori estimate is deducible if
we combine methods for studying similar systems of first-order equations
and methods for qualitative research of singularly perturbed systems of equations.
The results obtained can be further generalized for
multidimensional systems of second-order equations,
applying the idea of the directing function method.
Keywords
- a priori estimate
- periodic problem
- the mapping degree of vector field
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