Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 867623, 18 pages
doi:10.1155/2008/867623
Research Article

Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations

1Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy
2Department of Mathematics, Waseda University, 3-4-1 Ohkubo Shinjuku-ku, Tokyo 169-8555, Japan
3Istituto per le Applicazioni del Calcolo “Mauro Picone”, Sede di Napoli, Via P. Castellino 111, 80131 Napoli, Italy

Received 19 June 2007; Revised 14 January 2008; Accepted 5 May 2008

Academic Editor: A. Matsumoto

Copyright © 2008 E. Messina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Y. Song and C. T. H. Baker, “Perturbation theory for discrete Volterra equations,” Journal of Difference Equations and Applications, vol. 9, no. 10, pp. 969–987, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. C. T. H. Baker and Y. Song, “Discrete Volterra operators, fixed point theorems and their application,” Nonlinear Studies, vol. 10, no. 1, pp. 79–101, 2003. View at Zentralblatt MATH · View at MathSciNet
  3. G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, vol. 34 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990. View at Zentralblatt MATH · View at MathSciNet
  4. M. S. Klamki, Ed., Problems in Applied Mathematics: Selections from SIAM Review, M. S. Klamki, Ed., SIAM, Philadelphia, Pa, USA, 1990. View at Zentralblatt MATH · View at MathSciNet
  5. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Stability of difference Volterra equations: direct Liapunov method and numerical procedure,” Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 77–97, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Stability of discrete Volterra equations of Hammerstein type,” Journal of Difference Equations and Applications, vol. 6, no. 2, pp. 127–145, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Asymptotic properties of solutions of Volterra difference equations with finite linear part,” Dynamic Systems and Applications, vol. 8, no. 1, pp. 1–20, 1999. View at Zentralblatt MATH · View at MathSciNet
  8. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “A priori bounds on the solution of a nonlinear Volterra discrete equation,” Stability and Control: Theory and Applications, vol. 3, no. 1, pp. 38–47, 2000. View at Zentralblatt MATH · View at MathSciNet
  9. S. N. Elaydi, V. L. Kocic, and J. Li, “Global stability of nonlinear delay difference equations,” Journal of Difference Equations and Applications, vol. 2, no. 1, pp. 87–96, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005. View at Zentralblatt MATH · View at MathSciNet
  11. Ch. Lubich, “On the numerical solution of Volterra equations with unbounded nonlinearity,” Journal of Integral Equations, vol. 10, no. 1–3, pp. 175–183, 1985. View at Zentralblatt MATH · View at MathSciNet
  12. A. Vecchio, “Boundedness of the global error of some linear and nonlinear methods for Volterra integral equations,” Journal of Integral Equations and Applications, vol. 12, no. 4, pp. 449–465, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “Boundedness of discrete Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 106–130, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Ph. Clément and J. A. Nohel, “Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels,” SIAM Journal on Mathematical Analysis, vol. 12, no. 4, pp. 514–535, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. A. Vecchio, “Volterra discrete equations: summability of the fundamental matrix,” Numerische Mathematik, vol. 89, no. 4, pp. 783–794, 2001. View at Zentralblatt MATH · View at MathSciNet