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.
  • Received: 2015-02-22.
  • Revised: 2015-08-10.
  • Accepted: 2015-08-14.
  • Published online: 2015-12-18.
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Cite this article as:
Janusz Migda, Małgorzata Migda, Asymptotic behavior of solutions of discrete Volterra equations, Opuscula Math. 36, no. 2 (2016), 265-278, http://dx.doi.org/10.7494/OpMath.2016.36.2.265

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