## General explicit solution of planar weakly delayed linear discrete systems and pasting its solutions.(English)Zbl 1468.39002

Summary: Planar linear discrete systems with constant coefficients and delays $x(k + 1) = A x(k) + \sum_{l = 1}^n B^l x_l(k - m_l)$ are considered where $$k \in \mathbb Z_0^{\infty} : = \{0,1, \ldots, \infty \}$$, $$m_1, m_2, \ldots, m_n$$ are constant integer delays, $$0 < m_1 < m_2 < \cdots < m_n$$, $$A, B^1, \ldots, B^n$$ are constant $$2 \times 2$$ matrices, and $$x : \mathbb Z_{- m_n}^{\infty} \rightarrow \mathbb{R}^2$$. It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension $$2(m_n + 1)$$ is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

### MSC:

 39A12 Discrete version of topics in analysis

weak delay
Full Text:

### References:

 [1] Diblík, J.; Khusainov, D. Ya.; Šmarda, Z., Construction of the general solution of planar linear discrete systems with constant coefficients and weak delay, Advances in Difference Equations, 2009 (2009) · Zbl 1167.39001 [2] Diblík, J.; Halfarová, H., Explicit general solution of planar linear discrete systems with constant coefficients and weak delays, Advances in Difference Equations, 2013 (2013) · Zbl 1380.39002 [3] Elaydi, S., An Introduction to Difference Equations (2005), New York, NY, USA: Springer, New York, NY, USA · Zbl 1071.39001 [4] Stević, S., More on a rational recurrence relation, Applied Mathematics E-Notes, 4, 80-85 (2004) · Zbl 1069.39024 [5] Stević, S., On a system of difference equations, Applied Mathematics and Computation, 218, 7, 3372-3378 (2011) · Zbl 1242.39017 [6] Stević, S., On some solvable systems of difference equations, Applied Mathematics and Computation, 218, 9, 5010-5018 (2012) · Zbl 1253.39011 [7] Stević, S., On the difference equation $$x_n = x_{n - k} /(b + c x_{n - 1} \cdots x_{n - k})$$, Applied Mathematics and Computation, 218, 11, 6291-6296 (2012) · Zbl 1246.39010 [8] Stević, S.; Diblík, J.; Iričanin, B.; Šmarda, Z., On a third-order system of difference equations with variable coefficients, Abstract and Applied Analysis, 2012 (2012) [9] Khusainov, D. Ya.; Benditkis, D. B.; Diblik, J., Weak delay in systems with an aftereffect, Functional Differential Equations, 9, 3-4, 385-404 (2002) · Zbl 1048.34104 [10] Čermák, J., The stability and asymptotic properties of the $$\Theta$$-methods for the pantograph equation, IMA Journal of Numerical Analysis, 31, 4, 1533-1551 (2011) · Zbl 1230.65074 [11] Čermák, J.; Dvořáková, S., Boundedness and asymptotic properties of solutions of some linear and sublinear delay difference equations, Applied Mathematics Letters, 25, 5, 813-817 (2012) · Zbl 1245.39012 [12] Čermák, J.; Urbánek, M., On the asymptotics of solutions of delay dynamic equations on time scales, Mathematical and Computer Modelling, 46, 3-4, 445-458 (2007) · Zbl 1138.34037 [13] Diblík, J., Anti-Lyapunov method for systems of discrete equations, Nonlinear Analysis A, 57, 7-8, 1043-1057 (2004) · Zbl 1065.39008 [14] Khokhlova, T.; Kipnis, M., Numerical and qualitative stability analysis of ring and linear neural networks with a large number of neurons, International Journal of Pure and Applied Mathematics, 76, 403-420 (2012) · Zbl 1257.34066 [15] Khokhlova, T.; Kipnis, M.; Malygina, V., The stability cone for a delay differential matrix equation, Applied Mathematics Letters, 24, 5, 742-745 (2011) · Zbl 1229.34114 [16] Kipnis, M.; Komissarova, D., Stability of a delay difference system, Advances in Difference Equations, 2006 (2006) · Zbl 1139.39015 [17] Liz, E.; Pituk, M., Asymptotic estimates and exponential stability for higher-order monotone difference equations, Advances in Difference Equations, 1, 41-55 (2005) · Zbl 1102.39004 [18] Philos, Ch. G.; Purnaras, I. K., An asymptotic result for some delay difference equations with continuous variable, Advances in Difference Equations, 1, 1-10 (2004) · Zbl 1079.39011 [19] Faraz, N.; Khan, Y.; Austin, F., An alternative approach to differential-difference equations using the variational iteration method, Journal of Physical Sciences, 65, 12, 1055-1059 (2010) [20] Khan, Y.; Vázquez-Leal, H.; Faraz, N., An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations, Applied Mathematical Modelling, 37, 5, 2702-2708 (2013) · Zbl 1352.65172 [21] Khan, Y.; Wu, Q., Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers & Mathematics with Applications, 61, 8, 1963-1967 (2011) · Zbl 1219.65119
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.