×

Construction of the general solution of planar linear discrete systems with constant coefficients and weak delay. (English) Zbl 1167.39001

Summary: Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced.

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
34K06 Linear functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baštinec J, Diblík J:Subdominant positive solutions of the discrete equation[InlineEquation not available: see fulltext.].Abstract and Applied Analysis 2004,2004(6):461-470. 10.1155/S1085337504306056 · Zbl 1078.39004 · doi:10.1155/S1085337504306056
[2] Diblík, J.; Khusainov, DYa, Representation of solutions of linear discrete systems with constant coefficients and pure delay, No. 2006, 13 (2006) · Zbl 1139.39027
[3] Elaydi SN: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. 3rd edition. Springer, New York, NY, USA; 2005:xxii+539.
[4] Khusainov DYa, Benditkis DB, Diblík J: Weak delay in systems with an aftereffect.Functional Differential Equations 2002,9(3-4):385-404. · Zbl 1048.34104
[5] Boichuk A, Růžičková M: Solutions of nonlinear difference equations bounded on the whole line.Folia Facultatis Scientiarium Naturalium Universitatis Masarykiana Brunensis. Mathematica 2002, 13: 45-60. · Zbl 1106.39302
[6] Čermák J: On matrix differential equations with several unbounded delays.European Journal of Applied Mathematics 2006,17(4):417-433. 10.1017/S0956792506006590 · Zbl 1128.34050 · doi:10.1017/S0956792506006590
[7] Čermák J, Urbánek M: On the asymptotics of solutions of delay dynamic equations on time scales.Mathematical and Computer Modelling 2007,46(3-4):445-458. 10.1016/j.mcm.2006.11.015 · Zbl 1138.34037 · doi:10.1016/j.mcm.2006.11.015
[8] Diblík J: Anti-Lyapunov method for systems of discrete equations.Nonlinear Analysis: Theory, Methods & Applications 2004,57(7-8):1043-1057. 10.1016/j.na.2004.03.030 · Zbl 1065.39008 · doi:10.1016/j.na.2004.03.030
[9] Kipnis, M.; Komissarova, D., Stability of a delay difference system, No. 2006, 9 (2006) · Zbl 1139.39015
[10] Liz E, Pituk M: Asymptotic estimates and exponential stability for higher-order monotone difference equations.Advances in Difference Equations 2005,2005(1):41-55. 10.1155/ADE.2005.41 · Zbl 1102.39004 · doi:10.1155/ADE.2005.41
[11] Philos ChG, Purnaras IK: An asymptotic result for some delay difference equations with continuous variable.Advances in Difference Equations 2004,2004(1):1-10. 10.1155/S1687183904310058 · Zbl 1079.39011 · doi:10.1155/S1687183904310058
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.