Almost automorphic solutions of difference equations. (English) Zbl 1177.39002

Summary: We study discrete almost automorphic functions (sequences) defined on the set of integers with values in a Banach space \(X\). Given a bounded linear operator \(T\) defined on \(X\) and a discrete almost automorphic function \(f(n)\), we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation \(\Delta u(n)=Tu(n)+f(n)\). We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation \(\Delta u(n)=Tu(n)+g(n,u(n))\) assuming that \(g(n,x)\) is discrete almost automorphic in \(n\) for each \(x\in X\), satisfies a global Lipschitz type condition, and takes values on \(X\).


39A10 Additive difference equations
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[1] Agarwal RP, Popenda J: Periodic solutions of first order linear difference equations.Mathematical and Computer Modelling 1995,22(1):11-19. 10.1016/0895-7177(95)00096-K · Zbl 0871.39002
[2] Corduneanu C: Almost periodic discrete processes.Libertas Mathematica 1982, 2: 159-169. · Zbl 0548.39002
[3] Halanay A: Solutions périodiques et presque-périodiques des systèmes d’équations aux différences finies.Archive for Rational Mechanics and Analysis 1963, 12: 134-149. 10.1007/BF00281222 · Zbl 0112.06106
[4] Pang PYH, Agarwal RP: On periodicity of difference equations of a general type.Journal of Difference Equations and Applications 1996,2(3):271-287. 10.1080/10236199608808062 · Zbl 0884.39005
[5] Sugiyama S: On periodic solutions of difference equations.Bulletin of Science and Engineering Research Laboratory. Waseda University 1971, 52: 89-94.
[6] Elaydi S: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. 3rd edition. Springer, New York, NY, USA; 2005:xxii+539. · Zbl 1071.39001
[7] Agarwal RP: Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics. Volume 155. Marcel Dekker, New York, NY, USA; 1992:xiv+777.
[8] Walther A: Fastperiodische Folgen und Potenzreihen mit fastperiodischen Koeffzienten.Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1928, 6: 217-234. 10.1007/BF02940612 · JFM 54.0370.01
[9] Walther A: Fastperiodische Folgen und ihre Fouriersche Analysis.Atti del Congresso Internazionale dei Matematici 1928, 2: 289-298.
[10] Agarwal RP, O’Regan D, Wong PJY: Constant-sign periodic and almost periodic solutions of a system of difference equations.Computers & Mathematics with Applications 2005,50(10-12):1725-1754. · Zbl 1087.39004
[11] Hamaya Y: Existence of an almost periodic solution in a difference equation with infinite delay.Journal of Difference Equations and Applications 2003,9(2):227-237. 10.1080/1023619021000035836 · Zbl 1033.39009
[12] Ignatyev AO, Ignatyev OA: On the stability in periodic and almost periodic difference systems.Journal of Mathematical Analysis and Applications 2006,313(2):678-688. 10.1016/j.jmaa.2005.04.001 · Zbl 1096.39005
[13] Song Y: Almost periodic solutions of discrete Volterra equations.Journal of Mathematical Analysis and Applications 2006,314(1):174-194. · Zbl 1088.39007
[14] Song, Y., Periodic and almost periodic solutions of functional difference equations with finite delay, No. 2007, 15 (2007) · Zbl 1148.39013
[15] Song Y, Tian H: Periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay.Journal of Computational and Applied Mathematics 2007,205(2):859-870. 10.1016/j.cam.2005.12.042 · Zbl 1122.39007
[16] Zhang S, Liu P, Gopalsamy K: Almost periodic solutions of nonautonomous linear difference equations.Applicable Analysis 2002,81(2):281-301. 10.1080/0003681021000021961 · Zbl 1022.39014
[17] Minh NV, Naito T, Nguerekata G: A spectral countability condition for almost automorphy of solutions of differential equations.Proceedings of the American Mathematical Society 2006,134(11):3257-3266. 10.1090/S0002-9939-06-08528-5 · Zbl 1120.34044
[18] Fatajou S, Minh NV, N’Guérékata GM, Pankov A: Stepanov-like almost automorphic solutions for nonautonomous evolution equations.Electronic Journal of Differential Equations 2007,2007(121):1-11. · Zbl 1146.47029
[19] Minh NV, Dat TT: On the almost automorphy of bounded solutions of differential equations with piecewise constant argument.Journal of Mathematical Analysis and Applications 2007,326(1):165-178. 10.1016/j.jmaa.2006.02.079 · Zbl 1115.34068
[20] N’Guérékata GM: Topics in Almost Automorphy. Springer, New York, NY, USA; 2005:xii+168. · Zbl 1073.43004
[21] N’Guérékata GM: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic/Plenum Publishers, New York, NY, USA; 2001:x+138. · Zbl 1001.43001
[22] Bochner S, von Neumann J: On compact solutions of operational-differential equations. I.Annals of Mathematics 1935,36(1):255-291. 10.2307/1968678 · JFM 61.0442.01
[23] Bochner S: Uniform convergence of monotone sequences of functions.Proceedings of the National Academy of Sciences of the United States of America 1961, 47: 582-585. 10.1073/pnas.47.4.582 · Zbl 0103.05304
[24] Bochner S: A new approach to almost periodicity.Proceedings of the National Academy of Sciences of the United States of America 1962, 48: 2039-2043. 10.1073/pnas.48.12.2039 · Zbl 0112.31401
[25] Bochner S: Continuous mappings of almost automorphic and almost periodic functions.Proceedings of the National Academy of Sciences of the United States of America 1964, 52: 907-910. 10.1073/pnas.52.4.907 · Zbl 0134.30102
[26] Araya D, Lizama C: Almost automorphic mild solutions to fractional differential equations.Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3692-3705. 10.1016/j.na.2007.10.004 · Zbl 1166.34033
[27] Goldstein JA, N’Guérékata GM: Almost automorphic solutions of semilinear evolution equations.Proceedings of the American Mathematical Society 2005,133(8):2401-2408. 10.1090/S0002-9939-05-07790-7 · Zbl 1073.34073
[28] Zaidman S: Almost automorphic solutions of some abstract evolution equations.Istituto Lombardo, Accademia di Scienze e Lettere 1976,110(2):578-588. · Zbl 0374.34042
[29] Zaidman S:Behavior of trajectories of[InlineEquation not available: see fulltext.]-semigroups.Istituto Lombardo, Accademia di Scienze e Lettere 1980, 114: 205-208. · Zbl 0513.47030
[30] Zaidman S: Existence of asymptotically almost-periodic and of almost-automorphic solutions for some classes of abstract differential equations.Annales des Sciences Mathématiques du Québec 1989,13(1):79-88. · Zbl 0711.34055
[31] Zaidman S: Topics in abstract differential equations.Nonlinear Analysis: Theory, Methods & Applications 1994,23(7):849-870. 10.1016/0362-546X(94)90124-4 · Zbl 0818.34035
[32] Zaidman S: Topics in Abstract Differential Equations, Pitman Research Notes in Mathematics Series. Volume 304. Longman Scientific & Technical, Harlow, UK; 1994:xii+188. · Zbl 0806.34001
[33] Zaki M: Almost automorphic solutions of certain abstract differential equations.Annali di Matematica Pura ed Applicata 1974,101(1):91-114. 10.1007/BF02417100 · Zbl 0304.42028
[34] Veech WA: Almost automorphic functions on groups.American Journal of Mathematics 1965, 87: 719-751. 10.2307/2373071 · Zbl 0137.05803
[35] Gal SG, N’Guérékata GM: Almost automorphic fuzzy-number-valued functions.Journal of Fuzzy Mathematics 2005,13(1):185-208. · Zbl 1093.42009
[36] Gal CS, Gal SG, N’Guérékata GM: Almost automorphic functions in Fréchet spaces and applications to differential equations.Semigroup Forum 2005,71(2):201-230. 10.1007/s00233-005-0508-y · Zbl 1085.43004
[37] Gal CG, Gal SG, N’Guérékata GM:Almost automorphic functions with values in a[InlineEquation not available: see fulltext.]-Fréchet space.Electronic Journal of Differential Equations 2008, 21: 1-18. · Zbl 1178.45014
[38] Corduneanu C: Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, no. 2. John Wiley & Sons, New York, NY, USA; 1968:x+237.
[39] Veech WA: Almost automorphic functions.Proceedings of the National Academy of Sciences of the United States of America 1963, 49: 462-464. 10.1073/pnas.49.4.462 · Zbl 0173.33402
[40] Basit BR: A generalization of two theorems of M. I. Kadec on the indefinite integral of abstract almost periodic functions.Matematicheskie Zametki 1971, 9: 311-321. · Zbl 0208.37904
[41] Liu J, N’Guérékata GM, Minh NV: A Massera type theorem for almost automorphic solutions of differential equations.Journal of Mathematical Analysis and Applications 2004,299(2):587-599. 10.1016/j.jmaa.2004.05.046 · Zbl 1081.34054
[42] Ezzinbi K, Nelson V, N’Guérékata GM:[InlineEquation not available: see fulltext.]-almost automorphic solutions of some nonautonomous differential equations.Cubo 2008,10(2):61-74. · Zbl 1168.47033
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