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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