Construction of Continuous Solutions of Nonlinear Functional-Difference Equations Systems
DOI:
https://doi.org/10.20535/1810-0546.2016.4.72752Keywords:
Functional-difference equations, Continuous limited solutionsAbstract
Background. We study the structure of the set of solutions of functional-difference equations systems
\[
x(t+1)=Ax(t)+F(t,x(qt)),
\]
(1)
under certain assumptions about the matrix A and number q.
Objective. The aim is to build continuous limited solutions for
\[
t \in R^+(R^-)
\]
and study the structure of their set.
Methods. We use the classical methods of the theory of ordinary differential and difference equations.
Results. The existence of the family of continuous limited solutions for
\[
t \ge 0
\]
which depends on arbitrary one-periodic function dimension k is proved. A similar result was obtained for case
\[
t \le 0
\]
(the theorem 2).
Conclusions. New sufficient conditions for the existence of continuous solutions of functional-difference equations systems (1) are established, we developed the method of constructing these solutions and investigated the structure of their set.
References
G.D. Birkhoff, “General theory of linear difference equations”, Trans. Amer. Math. Soc., vol. 12, pp. 243–284, 1911.
W.J. Trjitzinsky, “Analytic theory of linear q-difference equations”, Trans. Amer. Math. Soc., vol. 61, pp. 1–38, 1933.
G.P. Pelyukh and O.A. Sivak, “A study of the structure of the set of continuous solutions to systems of linear functional-difference equations”, Neliniyni Kolyvannya, vol. 12, no. 3, pp. 307–335, 2009 (in Ukrainian).
O.A. Sivak, “The structure of a set of continuous solutions of systems of linear functional difference equations”, Naukovi Visti NTUU KPI, no. 4, pp. 81–87, 2011 (in Ukrainian).
I.V. Betsko, “Investigation of the structure of a set of continuous solutions of difference equations systems”, Naukovi Visti NTUU KPI, no. 4, pp. 7–13, 2015 (in Ukrainian).
I.V. Betsko, “The existence of continuous solutions to systems of difference equations”, Neliniyni Kolyvannya, vol. 19, no. 1, pp. 3–10, 2016 (in Ukrainian).
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