Opuscula Math. 36, no. 2 (2016), 265-278

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.

Full text (pdf)

  1. C.T.H. Baker, Y. Song, Periodic solutions of non-linear discrete Volterra equations with finite memory, J. Comput. Appl. Math. 234 (2010) 9, 2683-2698.
  2. M.R. Crisci, V.B. Kolmanovskii, E. Russo, A. Vecchio, Boundedness of discrete Volterra equations, J. Math. Anal. Appl. 211 (1997), 106-130.
  3. V.B. Demidovič, A certain criterion for the stability of difference equations, Diff. Urav. 5 (1969), 1247-1255 [in Russian].
  4. J. Diblík, M. Růžičková, E. Schmeidel, Asymptotically periodic solutions of Volterra difference equations, Tatra Mt. Math. Publ. 43 (2009), 43-61.
  5. J. Diblík, M. Růžičková, L.E. Schmeidel, M. Zbaszyniak, Weighted asymptotically periodic solutions of linear Volterra difference equations, Abstr. Appl. Anal. (2011), Art. ID 370982, 14 pp.
  6. J. Diblík, E. Schmeidel, On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence, Appl. Math. Comput. 218 (2012) 18, 9310-9320.
  7. T. Gronek, E. Schmeidel, Existence of bounded solution of Volterra difference equations via Darbo's fixed-point theorem, J. Difference Equ. Appl. 19 (2013) 10, 1645-1653.
  8. I. Győri, E. Awwad, On the boundedness of the solutions in nonlinear discrete Volterra difference equations, Adv. Difference Equ. 2 (2012), 1-20.
  9. I. Győri, F. Hartung, Asymptotic behavior of nonlinear difference equations, J. Difference Equ. Appl. 18 (2012) 9, 1485-1509.
  10. I. Győri, L. Horvath, Asymptotic representation of the solutions of linear Volterra difference equations, Adv. Difference Equ. (2008), ID 932831, 22 pp.
  11. I. Győri, D.W. Reynolds, On asymptotically periodic solutions of linear discrete Volterra equations, Fasc. Math. 44 (2010), 53-67.
  12. V. Kolmanovskii, L. Shaikhet, Some conditions for boundedness of solutions of difference Volterra equations, Appl. Math. Lett. 16 (2003), 857-862.
  13. R. Medina, Asymptotic behavior of Volterra difference equations, Comput. Math. Appl. 41 (2001) 5-6, 679-687.
  14. J. Migda, Asymptotic properties of solutions of nonautonomous difference equations, Arch. Math. (Brno) 46 (2010), 1-11.
  15. J. Migda, Asymptotically polynomial solutions of difference equations, Adv. Difference Equ. 92 (2013), 16 pp.
  16. J. Migda, Approximative solutions of difference equations, Electron. J. Qual. Theory Differ. Equ. 13 (2014), 1-26.
  17. J. Migda, Approximative full solutions of difference equations, Int. J. Difference Equ. 9 (2014), 111-121.
  18. M. Migda, J. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations, Nonlinear Anal. 47 (2001) 7, 4687-4695.
  19. M. Migda, J. Migda, Bounded solutions of nonlinear discrete Volterra equations, accepted for publication in Math. Slovaca.
  20. M. Migda, J. Morchało, Asymptotic properties of solutions of difference equations with several delays and Volterra summation equations, Appl. Math. Comput. 220 (2013), 365-373.
  21. J. Morchało, Volterra summation equations and second order difference equations, Math. Bohem. 135 (2010) 1, 41-56.
  22. J. Popenda, Asymptotic properties of solutions of difference equations, Proc. Indian Acad. Sci. Math. Sci. 95 (1986) 2, 141-153.
  23. A. Zafer, Oscillatory and asymptotic behavior of higher order difference equations, Math. Comput. Modelling 21 (1995) 4, 43-50.
  • 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.
Opuscula Mathematica - cover

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

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.