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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 and Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681
##### References:
 [1] R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, Second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker, Inc., New York, 2000. [2] 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), 1257-1275. doi: 10.1080/10236190600986594. [3] J. Diblík, M. Růžičková and E. Schmeidel, Existence of asymptotically periodic solutions of scalar Volterra difference equations, Tatra Mt. Math. Publ., 43 (2009), 51-61. doi: 10.2478/v10127-009-0024-7. [4] J. Diblík, M. Růžičková, E. Schmeidel and M. Zbąszyniak, Weighted asymptotically periodic solutions of linear Volterra difference equations, Abstr. Appl. Anal., (2011), Art. ID 370982, 14 pp. doi: 10.1155/2011/370982. [5] 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), 9310-9320. doi: 10.1016/j.amc.2012.03.010. [6] S. N. Elaydi, An Introduction to Difference Equations, Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005. [7] 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), 1645-1653. doi: 10.1080/10236198.2013.769974. [8] I. Györi and L. Horváth, Asymptotic representation of the solutions of linear Volterra difference equations, Adv. Difference Equ., (2008), Art. ID 932831, 22 pp. [9] I. Györi and D. Reynolds, On asymptotically periodic solutions of linear discrete Volterra equations, Fasc. Math., 44 (2010), 53-67. [10] W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, 2001. [11] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, 256, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8. [12] M. Migda and J. Morchało, Asymptotic properties of solutions of difference equations with several delays and Volterra summation equations, Appl. Math. Comput., 220 (2013), 365-373. doi: 10.1016/j.amc.2013.06.032. [13] J. Morchało, Perturbation theory for discrete Volterra equation, Int. J. Pure Appl. Math., 68 (2011), 371-385. [14] J. Morchało, Volterra summation equations and second order difference equations, Math. Bohem., 135 (2010), 41-56. [15] J. Morchało and M. Migda, Boundedness of solutions of difference systems with delays, Comput. Math. Appl., 64 (2012), 2233-2240. doi: 10.1016/j.camwa.2012.01.075. [16] J. Musielak, Wstęp do Analizy Funkcjonalnej, (in Polish) PWN, Warszawa 1976. [17] E. Schmeidel, Properties of Solutions of Higher Order Difference Equations, Publishing House of Poznan University of Technology, 2010.

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##### References:
 [1] R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, Second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker, Inc., New York, 2000. [2] 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), 1257-1275. doi: 10.1080/10236190600986594. [3] J. Diblík, M. Růžičková and E. Schmeidel, Existence of asymptotically periodic solutions of scalar Volterra difference equations, Tatra Mt. Math. Publ., 43 (2009), 51-61. doi: 10.2478/v10127-009-0024-7. [4] J. Diblík, M. Růžičková, E. Schmeidel and M. Zbąszyniak, Weighted asymptotically periodic solutions of linear Volterra difference equations, Abstr. Appl. Anal., (2011), Art. ID 370982, 14 pp. doi: 10.1155/2011/370982. [5] 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), 9310-9320. doi: 10.1016/j.amc.2012.03.010. [6] S. N. Elaydi, An Introduction to Difference Equations, Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005. [7] 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), 1645-1653. doi: 10.1080/10236198.2013.769974. [8] I. Györi and L. Horváth, Asymptotic representation of the solutions of linear Volterra difference equations, Adv. Difference Equ., (2008), Art. ID 932831, 22 pp. [9] I. Györi and D. Reynolds, On asymptotically periodic solutions of linear discrete Volterra equations, Fasc. Math., 44 (2010), 53-67. [10] W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, 2001. [11] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, 256, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8. [12] M. Migda and J. Morchało, Asymptotic properties of solutions of difference equations with several delays and Volterra summation equations, Appl. Math. Comput., 220 (2013), 365-373. doi: 10.1016/j.amc.2013.06.032. [13] J. Morchało, Perturbation theory for discrete Volterra equation, Int. J. Pure Appl. Math., 68 (2011), 371-385. [14] J. Morchało, Volterra summation equations and second order difference equations, Math. Bohem., 135 (2010), 41-56. [15] J. Morchało and M. Migda, Boundedness of solutions of difference systems with delays, Comput. Math. Appl., 64 (2012), 2233-2240. doi: 10.1016/j.camwa.2012.01.075. [16] J. Musielak, Wstęp do Analizy Funkcjonalnej, (in Polish) PWN, Warszawa 1976. [17] E. Schmeidel, Properties of Solutions of Higher Order Difference Equations, Publishing House of Poznan University of Technology, 2010.
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