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Asymptotically periodic solutions of Volterra system of difference equations. (English) Zbl 1202.39013

This paper is mainly concerned with the following system of Volterra difference equations:
\[ x_s(n+1)=a_s(n)+b_s(n)x_s(n)+\sum_{p=1}^r\sum_{i=0}^n K_{sp}(n,i)x_p(i),\quad s=1,2,\dots,r, \quad n=0,1,2,\dots. \]
By using Schauder’s fixed point theorem, the authors prove that the above Volterra difference system has at least an asymptotically periodic solution. In addition, the author give several examples to illustrate their main results.

MSC:

39A23 Periodic solutions of difference equations
39A30 Stability theory for difference equations
39A06 Linear difference equations
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References:

[1] Agarwal, R. P., (Difference Equations and Inequalities. Theory, Methods, and Applications. Difference Equations and Inequalities. Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics (2000), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York)
[2] Elaydi, S. N., (An Introduction to Difference Equations. An Introduction to Difference Equations, Undergraduate Texts in Mathematics (2005), Springer: Springer New York) · Zbl 1071.39001
[3] Kocić, V. L.; Ladas, G., (Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications (1993), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht) · Zbl 0787.39001
[4] Elaydi, S. N.; Murakami, S., Uniform asymptotic stability in linear Volterra difference equations, J. Difference Equ. Appl., 3, 203-218 (1998) · Zbl 0891.39013
[5] Morchało, J.; Szmanda, B., Asymptotic properties of solutions of some Volterra difference equations and second-order difference equations, Nonlinear Anal., 63, 801-811 (2005)
[6] Agarwal, R. P.; Popenda, J., Periodic solutions of first order linear difference equations, Math. Comput. Modelling, 22, 11-19 (1995) · Zbl 0871.39002
[7] Popenda, J.; Schmeidel, E., On the asymptotically periodic solution of some linear difference equations, Arch. Math., 35, 1, 13-19 (1999) · Zbl 1051.39010
[8] Popenda, J.; Schmeidel, E., Asymptotically periodic solution of some linear difference equations, Facta Univ. Ser. Math. Inform., 14, 31-40 (1999) · Zbl 1017.39004
[9] Furumochi, T., Periodic solutions of Volterra difference equations and attractivity, Nonlinear Anal., 47, 4013-4024 (2001) · Zbl 1042.39500
[10] Furumochi, T., Asymptotically periodic solutions of Volterra difference equations, Vietnam J. Math., 30, 537-550 (2002) · Zbl 1031.39011
[11] Appleby, J.; Györi, I.; Reynolds, D., On exact convergence rates for solutions of linear systems of Volterra difference equations, J. Difference Equ. Appl., 12, 1257-1275 (2006) · Zbl 1119.39003
[12] Diblík, J.; Růžičková, M.; Schmeidel, E., Asymptotically periodic solutions of Volterra difference equations, (Ruffing, A.; etal., Communications of the Laufen Colloquium on Science. Communications of the Laufen Colloquium on Science, Laufen, Austria, April 1-5, 2007. Communications of the Laufen Colloquium on Science. Communications of the Laufen Colloquium on Science, Laufen, Austria, April 1-5, 2007, Berichte aus der Mathematik, vol. 5 (2007), Shaker: Shaker Aachen), 1-12
[13] Musielak, J., Wstep do analizy funkcjonalnej (1976), PWN: PWN Warszawa, (in Polish)
[14] Zeidler, E., Nonlinear Functional Analysis and its Application I, Fixed-Point Theorems (1986), Springer-Verlag New York, Inc.
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