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On asymptotically periodic solutions of linear discrete Volterra equations. (English) Zbl 1210.39009

Let $K$ denote either of the fields $ℝ$ or $ℂ$. The authors study the asymptotic behavior of solutions of the Volterra equations

$x\left(n+1,\xi \right)=A\left(n\right)x\left(n,\xi \right)+\sum _{j=0}^{n}K\left(n,j\right)x\left(j,\xi \right)+f\left(n\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}n\ge 0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

which satisfy the initial condition $x\left(0,\xi \right)=\xi \in {K}^{d}$. Under suitable conditions the authors establish the existence of an asymptotically periodic solution of equation (1).

From an applied perspective asymptotically periodic systems describe our world more realistically and more accurately than periodic ones. There is much interest in developing the qualitative theory and numerical methods of such systems.

F. Wei and K. Wang [Appl. Math. Comput. 182, No. 1, 161–165 (2006; Zbl 1113.92062)] investigated the asymptotically periodic Lotka-Volterra cooperative systems. Forced asymptotically periodic solutions of predator-prey systems with or without hereditary effects have been examined by J. M. Cushing [SIAM J. Appl. Math. 30, 665–674 (1976; Zbl 0331.93078)]. F. Wei and K. Wang [Nonlinear Anal., Real World Appl. 7, No. 4, 591–596 (2006; Zbl 1114.34340)] studied some ecosystem with asymptotically periodic coefficients. They obtained the existence and uniqueness of asymptotically periodic solutions of asymptotically periodic ecosystems. B. de Andrade and C. Cuevas [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 6, 3190–3208 (2010; Zbl 1205.34074)] have studied the existence and uniqueness of asymptotically periodic solutions for control systems and partial differential equations.

##### MSC:
 39A23 Periodic solutions (difference equations) 39A06 Linear equations (difference equations) 39A22 Growth, boundedness, comparison of solutions (difference equations) 39A12 Discrete version of topics in analysis