A linear Volterra difference equation of the form $x(n+1)=a\left(n\right)+b\left(n\right)x\left(n\right)+{{\displaystyle \sum}}_{i=0}^{n}K(n,i)x\left(i\right),$ where $x:{\mathbb{N}}_{0}\to \mathbb{R}$, $a:{\mathbb{N}}_{0}\to \mathbb{R}$, $K:{\mathbb{N}}_{0}\times {\mathbb{N}}_{0}\to \mathbb{R}$ and $b:{\mathbb{N}}_{0}\to \mathbb{R}\setminus \left\{0\right\}$ is $\omega $-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on ${{\displaystyle \prod}}_{j=0}^{\omega -1}b\left(j\right)$ is assumed. The results generalize some of the recent results.

## References

J. Diblík, M. R\accent23užičková, and E. Schmeidel, “Asymptotically periodic solutions of Volterra difference equations,”

*Tatra Mountains Mathematical Publications*, vol. 43, pp. 43–61, 2009. 1180.39022 J. Diblík, M. R\accent23užičková, and E. Schmeidel, “Asymptotically periodic solutions of Volterra difference equations,”*Tatra Mountains Mathematical Publications*, vol. 43, pp. 43–61, 2009. 1180.39022 R. P. Agarwal,

*Difference Equations and Inequalities*, vol. 228 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. 1160.34313 MR1740241 R. P. Agarwal,*Difference Equations and Inequalities*, vol. 228 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. 1160.34313 MR1740241 V. L. Kocić and G. Ladas,

*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256 of*Mathematics and Its Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1993. MR1247956 V. L. Kocić and G. Ladas,*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256 of*Mathematics and Its Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1993. MR1247956 J. A. D. Appleby, I. Győri, and D. W. Reynolds, “On exact convergence rates for solutions of linear systems of Volterra difference equations,”

*Journal of Difference Equations and Applications*, vol. 12, no. 12, pp. 1257–1275, 2006. 1119.39003 MR2277654 10.1080/10236190600986594 J. A. D. Appleby, I. Győri, and D. W. Reynolds, “On exact convergence rates for solutions of linear systems of Volterra difference equations,”*Journal of Difference Equations and Applications*, vol. 12, no. 12, pp. 1257–1275, 2006. 1119.39003 MR2277654 10.1080/10236190600986594 S. Elaydi and S. Murakami, “Uniform asymptotic stability in linear Volterra difference equations,”

*Journal of Difference Equations and Applications*, vol. 3, no. 3-4, pp. 203–218, 1998. 0891.39013 MR1616085 10.1080/10236199808808097 S. Elaydi and S. Murakami, “Uniform asymptotic stability in linear Volterra difference equations,”*Journal of Difference Equations and Applications*, vol. 3, no. 3-4, pp. 203–218, 1998. 0891.39013 MR1616085 10.1080/10236199808808097 I. Győri and L. Horváth, “Asymptotic representation of the solutions of linear Volterra difference equations,”

*Advances in Difference Equations*, vol. 2008, Article ID 932831, 22 pages, 2008. MR2407437 1146.39011 I. Győri and L. Horváth, “Asymptotic representation of the solutions of linear Volterra difference equations,”*Advances in Difference Equations*, vol. 2008, Article ID 932831, 22 pages, 2008. MR2407437 1146.39011 I. Győri and D. W. Reynolds, “On asymptotically periodic solutions of linear discrete Volterra equations,”

*Fasciculi Mathematici*, no. 44, pp. 53–67, 2010. MR2722631 I. Győri and D. W. Reynolds, “On asymptotically periodic solutions of linear discrete Volterra equations,”*Fasciculi Mathematici*, no. 44, pp. 53–67, 2010. MR2722631 Y. Song and C. T. H. Baker, “Admissibility for discrete Volterra equations,”

*Journal of Difference Equations and Applications*, vol. 12, no. 5, pp. 433–457, 2006. 1107.39012 MR2241386 10.1080/10236190600563260 Y. Song and C. T. H. Baker, “Admissibility for discrete Volterra equations,”*Journal of Difference Equations and Applications*, vol. 12, no. 5, pp. 433–457, 2006. 1107.39012 MR2241386 10.1080/10236190600563260 L. Berg and S. Stević, “Periodicity of some classes of holomorphic difference equations,”

*Journal of Difference Equations and Applications*, vol. 12, no. 8, pp. 827–835, 2006. 1103.39004 MR2248789 10.1080/10236190600761575 L. Berg and S. Stević, “Periodicity of some classes of holomorphic difference equations,”*Journal of Difference Equations and Applications*, vol. 12, no. 8, pp. 827–835, 2006. 1103.39004 MR2248789 10.1080/10236190600761575 S. Stević, “On global periodicity of a class of difference equations,”

*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 23503, 10 pages, 2007. 1180.39005 MR2272419 S. Stević, “On global periodicity of a class of difference equations,”*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 23503, 10 pages, 2007. 1180.39005 MR2272419 S. Stević and K. S. Berenhaut, “The behavior of positive solutions of a nonlinear second-order difference equation ${x}_{n}=f({x}_{n-2})/g({x}_{n-1})$,”

*Abstract and Applied Analysis*, vol. 2008, Article ID 653243, 8 pages, 2008. 1146.39018 MR2393108 S. Stević and K. S. Berenhaut, “The behavior of positive solutions of a nonlinear second-order difference equation ${x}_{n}=f({x}_{n-2})/g({x}_{n-1})$,”*Abstract and Applied Analysis*, vol. 2008, Article ID 653243, 8 pages, 2008. 1146.39018 MR2393108 E. Zeidler,

*Nonlinear Functional Analysis and Its Applications. I*, Springer, New York, NY, USA, 1986. MR816732 0583.47050 E. Zeidler,*Nonlinear Functional Analysis and Its Applications. I*, Springer, New York, NY, USA, 1986. MR816732 0583.47050 J. Diblík, E. Schmeidel, and M. R\accent23užičková, “Asymptotically periodic solutions of Volterra system of difference equations,”

*Computers & Mathematics with Applications*, vol. 59, no. 8, pp. 2854–2867, 2010. 1202.39013 MR2607992 J. Diblík, E. Schmeidel, and M. R\accent23užičková, “Asymptotically periodic solutions of Volterra system of difference equations,”*Computers & Mathematics with Applications*, vol. 59, no. 8, pp. 2854–2867, 2010. 1202.39013 MR2607992 J. Diblík, E. Schmeidel, and M. R\accent23užičková, “Existence of asymptotically periodic solutions of system of Volterra difference equations,”

*Journal of Difference Equations and Applications*, vol. 15, no. 11-12, pp. 1165–1177, 2009. 1180.39022 MR2569139 10.1080/10236190802653653 J. Diblík, E. Schmeidel, and M. R\accent23užičková, “Existence of asymptotically periodic solutions of system of Volterra difference equations,”*Journal of Difference Equations and Applications*, vol. 15, no. 11-12, pp. 1165–1177, 2009. 1180.39022 MR2569139 10.1080/10236190802653653