Continuous Solutions of a Class of Difference-Functional Equations

Authors

  • Тетяна Олександрівна Єрьоміна NTUU KPI, Ukraine

DOI:

https://doi.org/10.20535/1810-0546.2014.4.28227

Keywords:

Difference equations, Functional equations, Difference-functional equations

Abstract

The main object of research in this article is a study of the continuous solutions set structure of difference-functional equations of the form \[x\left ( qt \right )=a\left ( t \right )x\left ( t \right )+b\left ( t \right )x\left ( t+1 \right )+f\left ( t \right ),\] where \[a\left ( t \right ),b\left ( t \right ),f\left ( t \right )\]
– some real functions and q – real constant. Using methods of the theory of differential and difference equations a question of continuous solutions existence of linear difference-functional equations with constant coefficients was investigated and a building method for them was proposed. In particular for homogeneous equations, a continuous bounded solutions family that depends on an arbitrary continuous 1-periodic function with \[0< q< 1,a> 1\] and \[0< a< 1,q> 1,\] – positive was built. Furthermore, if \[0< q< 1,a> 1,\] then continuous solutions existence for heterogeneous equations with an arbitrary real value is proved and continuous bounded solutions family of a heterogeneous equation with an arbitrary nonnegative are built.

Author Biography

Тетяна Олександрівна Єрьоміна, NTUU KPI

Senior lecturer

References

R.P. Agarwal, Difference Equations and Inequalities, Theory, Methods and Applications, 2nd ed. Revised and Expanded, 2000, 972 р.

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”, Ibid, vol. 61, pp. 1–38, 1933.

Мартынюк Д.И. Лекции по качественной теории разностных уравнений. – К.: Наук. думка, 1972. – 248 с.

Миролюбов А.А., Солдатов М.А. Линейные неоднородные разностные уравнения. – М.: Наука, 1986. – 128 с.

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

Пелюх Г.П. К теории систем линейных разностных уравнений с непрерывным аргументом // Докл. АН. – 2006. – 407, № 5 – С. 600–603.

Published

2014-08-19