Investigation of the Structure of a Set of Continuous Solutions of Difference Equations Systems

Іванна Володимирівна Бецко

Abstract


Background. We consider the structure of a set of continuous solutions of equations systems

[x(t+1)=A(t)x(t)+B(t)x(qt)+F(t)\] (1)

in a number of cases depending on the hypotheses for the matrices AB, number q and their properties.

Objective. To study existence of continuous limited solutions for \[t\in \mathbb{R}\], study the structure of their set and also developing the method of their construction.

Methods. We use methods of the theory of differential and difference equations.

Results. The existence of the family of continuous limited solutions for [t\geqslant 0\] which depends on \[\bar{n}=\sum_{i=1}^{k}n_{i}\] arbitrary continuous one-periodic functions at some conditions is proved in theorem 1. Similar theorem is proved for case \[t\leq 0\] (the theorem 2), and is proved the theorem 3 about the existence of the continuous limited solution of homogeneous system of the equation (1) is also proved.

Conclusions. New conditions for the existence of continuous solutions of difference equations systems (1) are established, we proposed the method of constructing these solutions and investigated the structure of their set.


Keywords


Difference equations; Continuous limited solutions

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).

G.P. Pelyukh and O.A. Sivak, “On the structure of the set of continuous solutions of functional-difference equations with linearly transformed argument”, Neliniyni Kolyvannya, vol. 13, no. 1, pp. 75–95, 2010 (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).


GOST Style Citations


  1. Birkhoff G.D. General theory of linear difference equations // Trans. Amer. Math. Soc. – 1911. – 12. – P. 243–284.

  2. Trjitzinsky W.J. Analytic theory of linear q-difference equations // Trans. Amer. Math. Soc. – 1933. – 61. – P. 1–38.

  3. Пелюх Г.П., Сівак О.А. Дослідження структури множини неперервних розв’язків систем лінійних функціонально-різницевих рівнянь // Нелінійні коливання. – 2009. – 12, № 3. – С. 307–335.

  4. Пелюх Г.П., Сівак О.А. Про структуру множини неперервних розв’язків функціонально-різницевих рівнянь з лінійно перетвореним аргументом // Нелінійні коливання. – 2010. – 13, № 1. – С. 75–95.

  5. Сівак О.А. Структура множини неперервних розв’язків систем лінійних функціонально-різницевих рівнянь // Наукові вісті НТУУ “КПІ”. – 2011. – № 4. – С. 81–87.




DOI: http://dx.doi.org/10.20535/1810-0546.2015.4.50445

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