Opuscula Math. 36, no. 2 (2016), 265-278
http://dx.doi.org/10.7494/OpMath.2016.36.2.265

Opuscula Mathematica

# Asymptotic behavior of solutions of discrete Volterra equations

Janusz Migda
Małgorzata Migda

Abstract. We consider the nonlinear discrete Volterra equations of non-convolution type $\Delta^m x_n=b_n+\sum\limits_{i=1}^{n}K(n,i)f\left(i,x_i\right), \quad n\geq 1.$ We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior, especially asymptotically polynomial and asymptotically periodic solutions. We use $$\operatorname{o}(n^s)$$, for a given nonpositive real $$s$$, as a measure of approximation. We also give conditions under which all solutions are asymptotically polynomial.

Keywords: Volterra difference equation, prescribed asymptotic behavior, asymptotically polynomial solution, asymptotically periodic solution, bounded solution.

Mathematics Subject Classification: 39A10, 39A22.

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• Janusz Migda
• Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Umultowska 87, 61-614 Poznań, Poland
• Małgorzata Migda
• Poznan University of Technology, Institute of Mathematics, Piotrowo 3A, 60-965 Poznań, Poland
• Communicated by Josef Diblík.
• Revised: 2015-08-10.
• Accepted: 2015-08-14.
• Published online: 2015-12-18.