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October  2014, 19(8): 2681-2690. doi: 10.3934/dcdsb.2014.19.2681

## On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type

 1 University of Bialystok, ul. Akademicka 2, 15-267 Białystok 2 Poznan University of Technology, ul. Piotrowo 3A, 60-965 Poznań, Poland, Poland

Received  November 2013 Revised  May 2014 Published  August 2014

A Volterra difference equation of the form $$x(n+2)=a(n)+b(n)x(n+1)+c(n)x(n)+\sum\limits^{n+1}_{i=1}K(n,i)x(i)$$ where $a, b, c, x \colon\mathbb{N} \to\mathbb{R}$ and $K \colon \mathbb{N}\times\mathbb{N}\to \mathbb{R}$ is studied. For every admissible constant $C \in \mathbb{R}$, sufficient conditions for the existence of a solution $x \colon\mathbb{N} \to\mathbb{R}$ of the above equation such that $x(n)\sim C \, n \, \beta(n),$ where $\beta(n)= \frac{1}{2^n}\prod\limits_{j=1}^{n-1}b(j)$, are presented. As a corollary of the main result, sufficient conditions for the existence of an eventually positive, oscillatory, and quickly oscillatory solution $x$ of this equation are obtained. Finally, a conditions under which considered equation possesses an asymptotically periodic solution are given.
Citation: Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681
##### References:
  R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications,, Second edition, (2000). Google Scholar  J. Appleby, I. Györi and D. Reynolds, On exact convergence rates for solutions of linear systems of Volterra difference equations,, J. Difference Equ. Appl., 12 (2006), 1257. doi: 10.1080/10236190600986594.  Google Scholar  J. Diblík, M. Růžičková and E. Schmeidel, Existence of asymptotically periodic solutions of scalar Volterra difference equations,, Tatra Mt. Math. Publ., 43 (2009), 51. doi: 10.2478/v10127-009-0024-7.  Google Scholar  J. Diblík, M. Růžičková, E. Schmeidel and M. Zbąszyniak, Weighted asymptotically periodic solutions of linear Volterra difference equations,, Abstr. Appl. Anal., (2011). doi: 10.1155/2011/370982.  Google Scholar  J. Diblík and E. Schmeidel, On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence,, Appl. Math. Comput., 218 (2012), 9310. doi: 10.1016/j.amc.2012.03.010.  Google Scholar  S. N. Elaydi, An Introduction to Difference Equations,, Third edition, (2005). Google Scholar  T. Gronek and E. Schmeidel, Existence of a bounded solution of Volterra difference equations via Darbo's fixed point theorem,, J. Differ. Equations Appl., 19 (2013), 1645. doi: 10.1080/10236198.2013.769974.  Google Scholar  I. Györi and L. Horváth, Asymptotic representation of the solutions of linear Volterra difference equations,, Adv. Difference Equ., (2008). Google Scholar  I. Györi and D. Reynolds, On asymptotically periodic solutions of linear discrete Volterra equations,, Fasc. Math., 44 (2010), 53. Google Scholar  W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications,, Academic Press, (2001). Google Scholar  V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, Mathematics and its Applications, (1993). doi: 10.1007/978-94-017-1703-8.  Google Scholar  M. Migda and J. Morchało, Asymptotic properties of solutions of difference equations with several delays and Volterra summation equations,, Appl. Math. Comput., 220 (2013), 365. doi: 10.1016/j.amc.2013.06.032.  Google Scholar  J. Morchało, Perturbation theory for discrete Volterra equation,, Int. J. Pure Appl. Math., 68 (2011), 371. Google Scholar  J. Morchało, Volterra summation equations and second order difference equations,, Math. Bohem., 135 (2010), 41. Google Scholar  J. Morchało and M. Migda, Boundedness of solutions of difference systems with delays,, Comput. Math. Appl., 64 (2012), 2233. doi: 10.1016/j.camwa.2012.01.075.  Google Scholar  J. Musielak, Wstęp do Analizy Funkcjonalnej,, (in Polish) PWN, (1976). Google Scholar  E. Schmeidel, Properties of Solutions of Higher Order Difference Equations,, Publishing House of Poznan University of Technology, (2010). Google Scholar

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##### References:
  R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications,, Second edition, (2000). Google Scholar  J. Appleby, I. Györi and D. Reynolds, On exact convergence rates for solutions of linear systems of Volterra difference equations,, J. Difference Equ. Appl., 12 (2006), 1257. doi: 10.1080/10236190600986594.  Google Scholar  J. Diblík, M. Růžičková and E. Schmeidel, Existence of asymptotically periodic solutions of scalar Volterra difference equations,, Tatra Mt. Math. Publ., 43 (2009), 51. doi: 10.2478/v10127-009-0024-7.  Google Scholar  J. Diblík, M. Růžičková, E. Schmeidel and M. Zbąszyniak, Weighted asymptotically periodic solutions of linear Volterra difference equations,, Abstr. Appl. Anal., (2011). doi: 10.1155/2011/370982.  Google Scholar  J. Diblík and E. Schmeidel, On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence,, Appl. Math. Comput., 218 (2012), 9310. doi: 10.1016/j.amc.2012.03.010.  Google Scholar  S. N. Elaydi, An Introduction to Difference Equations,, Third edition, (2005). Google Scholar  T. Gronek and E. Schmeidel, Existence of a bounded solution of Volterra difference equations via Darbo's fixed point theorem,, J. Differ. Equations Appl., 19 (2013), 1645. doi: 10.1080/10236198.2013.769974.  Google Scholar  I. Györi and L. Horváth, Asymptotic representation of the solutions of linear Volterra difference equations,, Adv. Difference Equ., (2008). Google Scholar  I. Györi and D. Reynolds, On asymptotically periodic solutions of linear discrete Volterra equations,, Fasc. Math., 44 (2010), 53. Google Scholar  W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications,, Academic Press, (2001). Google Scholar  V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,, Mathematics and its Applications, (1993). doi: 10.1007/978-94-017-1703-8.  Google Scholar  M. Migda and J. Morchało, Asymptotic properties of solutions of difference equations with several delays and Volterra summation equations,, Appl. Math. Comput., 220 (2013), 365. doi: 10.1016/j.amc.2013.06.032.  Google Scholar  J. Morchało, Perturbation theory for discrete Volterra equation,, Int. J. Pure Appl. Math., 68 (2011), 371. Google Scholar  J. Morchało, Volterra summation equations and second order difference equations,, Math. Bohem., 135 (2010), 41. Google Scholar  J. Morchało and M. Migda, Boundedness of solutions of difference systems with delays,, Comput. Math. Appl., 64 (2012), 2233. doi: 10.1016/j.camwa.2012.01.075.  Google Scholar  J. Musielak, Wstęp do Analizy Funkcjonalnej,, (in Polish) PWN, (1976). Google Scholar  E. Schmeidel, Properties of Solutions of Higher Order Difference Equations,, Publishing House of Poznan University of Technology, (2010). Google Scholar
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