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
Full Text:

References:

 [1] Agarwal, R.P., () [2] Elaydi, S.N., () [3] Kocić, V.L.; Ladas, G., () [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, (), 1-12 [13] Musielak, J., Wstep do analizy funkcjonalnej, (1976), PWN Warszawa, (in Polish) [14] Zeidler, E., Nonlinear functional analysis and its application I, fixed-point theorems, (1986), Springer-Verlag New York, Inc.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.