Advertisement

Functional Difference Equations

  • Youssef N. Raffoul
Chapter

Abstract

In this chapter we consider functional difference equations that we apply to all types of Volterra difference equations. Our general theorems will require the construction of suitable Lyapunov functionals, a task that is difficult but possible. As we have seen in Chapter  1, the concept of resolvent can only apply to linear Volterra difference systems. The theorems on functional difference equations will enable us to qualitatively analyze the theory of boundedness, uniform ultimate boundedness, and stability of solutions of vectors and scalar s Volterra difference equations. We extend and prove parallel theorems regarding functional difference equations with finite or infinite delay, and provide many applications. In addition, we will point out the need of more research in delay difference equations. In the second part of the chapter, we state and prove theorems that guide us on how to systematically construct suitable Lyapunov functionals for a specific nonlinear Volterra difference equation. We end the chapter with open problems. Most of the results of this chapter can be found in [37, 38, 128, 133, 135, 141, 147, 181], and [182].

References

  1. 5.
    Agarwal, R., and Pang, P.Y., On a generalized difference system, Nonlinear Anal., TM and Appl., 30(1997), 365–376.zbMATHGoogle Scholar
  2. 36.
    Crisci, M.R., Kolmanovskii, V.B., and Vecchio. A., Boundedness of discrete Volterra equations, J. Math. Analy. Appl. 211(1997), 106–130.Google Scholar
  3. 37.
    Crisci, M.R., Kolmanovskii, V.B., and Vecchio. A., Stability of difference Volterra equations: direct Lyapunov method and numerical procedure, Advances in Difference Equations, II. Comput. Math. Appl. 36 (1998) 10–12, 77–97.Google Scholar
  4. 38.
    Crisci, M.R., Kolmanovskii, V.B., and Vecchio. A., On the exponential stability of discrete Volterra systems, J. Differ. Equations Appl. 6 (2000) 6, 667–680.Google Scholar
  5. 47.
    Diblík, J., and Schmeidel, E., On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence, Appl. Math. Comput. 218(18), 9310–9320 (2012).MathSciNetzbMATHGoogle Scholar
  6. 52.
    Elaydi, S.E., Periodicity and stability of linear Volterra difference systems, J. Math Anal. Appl. 181(1994), 483–492.MathSciNetCrossRefGoogle Scholar
  7. 59.
    Elaydi, S.E., stability and asymptotocity of Volterra difference equations: A progress report, J. Compu. and Appl. Math. 228 (2009) 504–513.CrossRefGoogle Scholar
  8. 61.
    Elaydi, S.E., and Murakami, S., Uniform asymptotic stability in linear Volterra difference equations, Journal of Difference Equations 3(1998), 203–218.MathSciNetCrossRefGoogle Scholar
  9. 65.
    Eloe, P., Islam, M., and Raffoul, Y., Uniform asymptotic stability in nonlinear Volterra discrete systems, Special Issue on Advances in Difference Equations IV, Computers and Mathematics with Applications 45 (2003), pp. 1033–1039.MathSciNetzbMATHGoogle Scholar
  10. 72.
    Gronek, T., and Schmeidel, E., Existence of bounded solution of Volterra difference equations via Darbos fixed-point theorem, J. Differ. Equ. Appl. 19(10), (2013) 1645–1653.MathSciNetCrossRefGoogle Scholar
  11. 83.
    Islam, M., and Raffoul, Y., Uniform asymptotic stability in linear Volterra difference equations, PanAmerican Mathematical Journal, 11 (2001), No. 1, pp. 61–73.MathSciNetzbMATHGoogle Scholar
  12. 85.
    Islam, M., and Raffoul, Y., Exponential stability in nonlinear difference equations, Journal of Difference Equations and Applications, (2003), Vol. 9, No. 9, pp. 819–825.MathSciNetCrossRefGoogle Scholar
  13. 112.
    Matsunaga, H., and Hajiri, C.,Exact stability sets for a linear difference systems with diagonal delay, J. Math. Anal. 369 (2010) 616–622.Google Scholar
  14. 113.
    Medina, R., The asymptotic behavior of the solutions of a Volterra difference equations, Comput. Math. Appl., 181 (1994), no. 1, pp. 19–26.MathSciNetCrossRefGoogle Scholar
  15. 115.
    Medina, R., Stability results for nonlinear difference equations, Nonlinear Studies, Vol. 6, No. 1, 1999.Google Scholar
  16. 116.
    Medina, R., Asymptotic equivalence of Volterra difference systems, Intl. Jou. of Diff. Eqns. and Appl. Vol. 1 No.1(2000), 53–64.Google Scholar
  17. 117.
    Medina, R., Asymptotic behavior of Volterra difference equations, Computers and Mathematics with Applications, 41, (2001) 679–687.MathSciNetCrossRefGoogle Scholar
  18. 128.
    Raffoul, Y., Boundedness and Periodicity of Volterra Systems of Difference Equations, Journal of Difference Equations and Applications, 1998, Vol. 4, pp. 381–393.MathSciNetCrossRefGoogle Scholar
  19. 133.
    Raffoul, Y., General theorems for stability and boundedness for nonlinear functional discrete systems, J. Math. Analy. Appl., 279 (2003), pp. 639–650.MathSciNetCrossRefGoogle Scholar
  20. 135.
    Raffoul, Y., Periodicity in General Delay Nonlinear Difference Equations Using fixed point Theory, Journal of Difference Equations and Applications, (2004). Vol. 10, pp.1229–1242.Google Scholar
  21. 141.
    Raffoul, Y., Necessary and sufficient conditions for uniform boundedness In functional difference equations, EPAM, Volume 2, Issue 2, 2016, Pages 171–180.Google Scholar
  22. 144.
    Raffoul, Y., Lyapunov-Razumikhin conditions that leads to stability and boundedness of functional difference equations of Volterra difference type, preprint.Google Scholar
  23. 145.
    Raffoul, Y., Uniform asymptotic stability and boundedness in functional finite delays difference equations, preprint.Google Scholar
  24. 147.
    Raffoul, Y., Li, W.L., and Liao, X.Y., Boundedness in nonlinear functional difference equations via non-negative Lyapunov functionals with applications to Volterra discrete systems, Nonlinear Studies, 13(2006), No. 1, 1–13.Google Scholar
  25. 181.
    Zhang, S., Stability of infinite delay difference systems, Nonlinear Analysis, Method & Applications, (1994) Vol. 22, No. 9, pp. 1121–1129.Google Scholar
  26. 182.
    Zhang, S., and Chen, M.P., A new Razumikhin theorem for delay difference equations, Computers Math. Applic. (1998) Vol. 36, No. 10–12, pp. 405–412.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Youssef N. Raffoul
    • 1
  1. 1.Department of MathematicsUniversity of DaytonDaytonUSA

Personalised recommendations