Asymptotically almost periodic solutions of limit periodic difference systems with coefficients from commutative groups
Keywords
Limit periodicity, almost periodicity, limit periodic sequences, asymptotically almost periodic solutions, difference equationsAbstract
We study the behaviour of solutions of limit periodic difference systems over (infinite) fields with absolute values. The considered systems are described by the coefficient matrices that belong to commutative groups whose boundedness is not required. In particular, we are interested in special systems with solutions which vanish at infinity or which are not asymptotically almost periodic. We obtain a transparent condition on the matrix groups which ensures that the special systems form a dense subset in the space of all considered systems, i.e. that, in any neighbourhood of any considered limit periodic system, there exists asystem which have non-asymptotically almost periodic or vanishing solutions. The presented results improve and extend known ones.References
Z. AlSharawi and J. Angelos, Linear almost periodic difference equations, J. Comput. Math. Optim. 4 (2008), no. 2, 61–91.
J. Andres and D. Pennequin, Semi-periodic solutions of difference and differential equations, Bound. Value Probl. 2012, no. 141, 1–16. DOI: 10.1186/1687-2770-2012-141
L.V. Bel’gart and R.K. Romanovskiı̆, The exponential dichotomy of solutions to systems of linear difference equations with almost periodic coefficients, Russ. Math. 54 (2010), 44–51. DOI: 10.3103/S1066369X10100051
I.D. Berg and A. Wilansky, Periodic, almost-periodic, and semiperiodic sequences, Michigan Math. J. 9 (1962), 363–368.
A.S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.
T. Caraballo and D. Cheban, Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition I, J. Differential Equations 246 (2009), no. 1, 108–128. DOI: 10.1016/j.jde.2008.04.001
T. Caraballo and D. Cheban, Almost periodic and almost automorphic solutions of linear differential equations, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 1857–1882.
M. Chvátal, Non-almost periodic solutions of limit periodic and almost periodic homogeneous linear difference systems, Electron. J. Qual. Theory Differ. Equ. 2014, no. 76, 1–20.
M. Chvátal, Asymptotically almost periodic solutions of limit and almost periodic linear difference systems, Electron. J. Qual. Theory Differ. Equ. 2015, no. 75, 1–17. DOI: 0.14232/ejqtde.2015.1.75
C. Corduneanu, Almost periodic discrete processes, Libertas Math. 2 (1982), 159–169.
C. Corduneanu, Almost Periodic Functions, Chelsea, New York, 1989, 2nd edition (1st edition: John Wiley and Sons, New York, 1968).
C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, 2009. DOI: 10.1007/978-0-387-09819-7
T. Diagana, S. Elaydi and A.-A. Yakubu, Population models in almost periodic environments, J. Difference Equ. Appl. 13 (2007), no. 4, 239–260.
K. Fan, Les fonctions asymptotiquement presque-périodiques d’une variable entière et leur application à l’étude de l’itération des transformations continues, Math. Z. 48 (1942/43), 685–711. DOI: 10.1007/BF01180036
H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601.
K. Gopalsamy, P. Liu and S. Zhang, Almost periodic solutions of nonautonomous linear difference equations, Appl. Anal. 81 (2002), no. 2, 281–301. DOI: 10.1080/0003681021000021961
P. Hasil and M. Veselý, Almost periodic transformable difference systems, Appl. Math. Comput. 218 (2012), no. 9, 5562–5579. DOI: 10.1016/j.amc.2011.11.050
P. Hasil and M. Veselý, Limit periodic linear difference systems with coefficient matrices from commutative groups, Electron. J. Qual. Theory Differ. Equ. 2014, no. 23, 1–25.
P. Hasil and M. Veselý, Limit periodic homogeneous linear difference systems, Appl. Math. Comput. 265 (2015), 958–972. DOI: 10.1016/j.amc.2015.06.008
P. Hasil and M. Veselý, Solution spaces of homogeneous linear difference systems with coefficient matrices from commutative groups, J. Difference Equ. Appl. 23 (2017), no. 8, 1324–1353. DOI: 10.1080/10236198.2017.1326912
O.V. Kirichenova, A.S. Kotyurgina and R.K. Romanovskiı̆, The method of Lyapunov functions for systems of linear difference equations with almost periodic coefficients, Siberian Math. J. 37 (1996), no. 1, 147–150. DOI: 10.1007/BF02104765
J. Kurzweil and A. Vencovská, On a problem in the theory of linear differential equations with quasiperiodic coefficients, Ninth International Conference on Nonlinear Oscillations, Vol. 1 (Kiev, 1981), Naukova Dumka, Kiev, 1984, 214–217.
J. Kurzweil and A. Vencovská, Linear differential equations with quasiperiodic coefficients, Czechoslovak Math. J. 37 (112) (1987), no. 3, 424–470.
B.M. Levitan and V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982.
A.V. Lipnitskiı̆, On the singular and higher characteristic exponents of an almost periodic linear differential system that depends affinely on a parameter, Differ. Equ. 42 (2006), no. 3, 369–379.
A.V. Lipnitskiı̆, On the discontinuity of Lyapunov exponents of an almost periodic linear differential system affinely dependent on a parameter, Differ. Equ. 44 (2008), no. 8, 1072–1081.
W. Marzantowicz and J. Signerska, Firing map of an almost periodic input function, Discrete Contin. Dyn. Syst. 2011 (2011); Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference. Suppl. Vol. II, 1032–1041.
Y. Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, Afr. Diaspora J. Math. 13 (2012), no. 1, 1–45.
M.G. Nerurkar and H.J. Sussmann, Construction of minimal cocycles arising from specific differential equations, Israel J. Math. 100 (1997), 309–326. DOI: 10.1007/BF02773645
M.G. Nerurkar and H.J. Sussmann, Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form, J. Mod. Dyn. 1 (2007), no. 2, 205–253. DOI: 10.3934/jmd.2007.1.205
W. Schwarz and J. Spilker, Arithmetical Functions. An Introduction to Elementary and Analytic Properties of Arithmetic Functions and to Some of Their Almost-Periodic Properties, Cambridge University Press, Cambridge, 1994.
V.I. Tkachenko, Linear almost periodic difference equations with bounded solutions, Asymptotic Solutions of Nonlinear Equations with a Small Parameter, Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1991, 121–124.
V.I. Tkachenko, On linear systems with quasiperiodic coefficients and bounded solutions, Ukrainian Math. J. 48 (1996), no. 1, 122–129. DOI: 10.1007/BF02390989
V.I. Tkachenko, On linear almost periodic systems with bounded solutions, Bull. Austral. Math. Soc. 55 (1997), no. 2, 177–184. DOI: 10.1017/S0004972700033852
V.I. Tkachenko, On unitary almost periodic difference systems, Advances in Difference Equations (Veszprém, 1995), Gordon and Breach, Amsterdam, 1997, 589–596.
M. Veselý, On orthogonal and unitary almost periodic homogeneous linear difference systems, Proceedings of Colloquium on Differential and Difference Equations (Brno, 2006), Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math., vol. 16, Masaryk Univ., Brno, 2007, 179–184.
M. Veselý, Construction of almost periodic sequences with given properties, Electron. J. Differential Equations 2008, no. 126, 1–22.
M. Veselý, Construction of almost periodic functions with given properties, Electron. J. Differential Equations 2011, no. 29, 1–25.
M. Veselý, Almost periodic homogeneous linear difference systems without almost periodic solutions, J. Difference Equ. Appl. 18 (2012), no. 10, 1623–1647. DOI: 10.1080/10236198.2011.585984
M. Veselý, Almost periodic skew-symmetric differential systems, Electron. J. Qual. Theory Differ. Equ. 2012, no. 72, 1–16.
M. Veselý and P. Hasil, Values of limit periodic sequences and functions, Math. Slovaca 66 (2016), no. 1, 43–62. DOI: 10.1515/ms-2015-0114
R. Yuan and J. Hong, The existence of almost periodic solutions for a class of differential equations with piecewise constant argument, Nonlinear Anal. 28 (1997), no. 8, 1439–1450. DOI: 10.1016/0362-546X(95)00225-K
S. Zhang, Almost periodic solutions of difference systems, Chinese Sci. Bull. 43 (1998), no. 24, 2041–2046. DOI: 10.1007/BF03183502
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0