Abstract:
We study the conditions on
right-hand side of a system that guarantee the convergence of
Euler's broken lines to the funnel of solutions of the system for
sufficiently small diameter of partition; in particular, the
condition that lets us select a subsequence from any sequence of
Euler's broken lines that would converge to the solution on a
given time interval. We obtain the condition that guarantees the
convergence of Euler's broken lines to the funnel of solutions of
the system as the diameter of partitions corresponding to the
broken lines tends to zero. The condition is specified for a given
explicit constant such that for any mapping that is Liepshitz
continuous with this constant and maps onto the phase plane, the
set of points of discontinuity has the zero Lebesgue measure (on
the graph of this mapping). Several examples are given to
demonstrate that this condition cannot be relaxed; specifically,
there may be no convergence even if, for each trajectory generated
by the system, the restriction of the dynamics function to that
graph is Riemann integrable; the constant from the condition above
can never be decreased either.
\qquad In the paper, Euler's broken lines are embedded into the
family of solutions of delay integral equations of the special
form, for which, in its own turn, the main result of the paper is
proved. It is due to this fact that the results of the paper hold
for a broader class of numerical methods, for example, for broken
lines with countable number of segments.