Stability and Boundedness

  • Youssef N. Raffoul


In this chapter we provide a brief introduction to difference calculus including basic material on Volterra difference equations. Using the z-transform we state some known theorems regarding stability of the zero solution of Volterra difference equations of convolution types. We move on to introducing Lyapunov functions for autonomous difference equations and state some known results concerning stability and boundedness. In Section 1.3 we introduce the concept of total stability and its correlation with uniform asymptotic stability for perturbed Volterra difference equations.


  1. 3.
    Adivar, M., and Raffoul, Y., Qualitative analysis of nonlinear Volterra integral equations on time scales using and Lyapunov functionals, Applied Mathematics and Computation 273 (2016) 258–266.MathSciNetCrossRefGoogle Scholar
  2. 5.
    Agarwal, R., and Pang, P.Y., On a generalized difference system, Nonlinear Anal., TM and Appl., 30(1997), 365–376.zbMATHGoogle Scholar
  3. 12.
    Bakke, V.L., and Jackiewicz, Z., Boundedness of solutions of difference equations and applications to numerical solutions of Volterra difference equations of the second kind, J. Math. Anal. Appl. 115 (1986), pp. 592–605.MathSciNetCrossRefGoogle Scholar
  4. 19.
    Brunner, H., The numerical analysis of functional integral and integro-differential equations of Volterra type, Acta Numerica, 13 (2004), pp.55–145.Google Scholar
  5. 21.
    Burton, T. A., Uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc. 68(1978), 195–199.MathSciNetCrossRefGoogle Scholar
  6. 27.
    Burton, T.A. fixed point s, Volterra equations, and Becker’s resolvent, Acta. Math. Hungar. 108(2005), 261–281.MathSciNetCrossRefGoogle Scholar
  7. 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
  8. 43.
    Cushing, J.M., An operator equation and bounded solutions of integro-differential systems, SIAM J. Math. Anal. 6 (1975) 433–445.Google Scholar
  9. 44.
    Cushing, J.M., Integro-differential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Vol. 20, Springer, Berlin, New York, 1977.Google Scholar
  10. 52.
    Elaydi, S.E., Periodicity and stability of linear Volterra difference systems, J. Math Anal. Appl. 181(1994), 483–492.MathSciNetCrossRefGoogle Scholar
  11. 56.
    Elaydi, S.E., Stability of Volterra difference equations of convolution type, Dynamical Systems. Nankai Ser. Pure Appl. Math. Theoret. Phys., 4 (1993), 66–72.MathSciNetzbMATHGoogle Scholar
  12. 57.
    Elaydi, S.E., An Introduction to Difference Equations, Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1999.Google Scholar
  13. 58.
    Elaydi, S.E., Stability and asymptoticity of Volterra difference equations, A progress report, J. Comput. Appl. Math. 228 (2009) 2, 504?513.Google Scholar
  14. 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
  15. 60.
    Elaydi, S.E., and Murakami, S., Asymptotic stability versus exponential stability in linear Volterra difference equations of convolution type, Journal of Difference Equations 2(1996), 401–410.MathSciNetCrossRefGoogle Scholar
  16. 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
  17. 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
  18. 73.
    Hakim, N., Kaufmann, J., Cerf, G., and Meadows, H., Nonlinear time series prediction with a discrete-time recurrent neural network model, Neural Networks, 1991., IJCNN-91-Seattle International Joint Conference on Neural Networks.Google Scholar
  19. 82.
    Hino, Y., and Murakami, S., Stabilities in linear integro-differential equations, Lecture Notes in Numerical and Applied Analysis, Kinokuniya, Tokyo, 15(1996), 31–46.zbMATHGoogle Scholar
  20. 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
  21. 84.
    Islam, M., and Raffoul, Y., Stability properties of linear Volterra integrodifferential equations with nonlinear perturbation, Communication of Applied Analysis, Vol. 7, No. 3 (2003) 406–416.MathSciNetzbMATHGoogle Scholar
  22. 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
  23. 92.
    Kelley, W., and Peterson, A., Difference equations an introduction with applications, Academic Press, 2000.zbMATHGoogle Scholar
  24. 100.
    Lauwerier, H., Mathematical models of epidemics, Math. Centrum, Amsterdam, 1981.Google Scholar
  25. 106.
    Linh, N., and Phat, V., Exponential stability of nonlinear time-varying differential equations and applications, Electronic Journal of Differential Equations, 34 (2001), 1–13.MathSciNetzbMATHGoogle Scholar
  26. 109.
    Lubich, C., On the stability of linear multistep methods for Volterra integro-differential equations, IMA J. Numer. Anal., 10 (1983), pp. 439–465.Google Scholar
  27. 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
  28. 114.
    Medina, R., Solvability of discrete Volterra equations in weighted spaces, Dynamic Systems and Appl. 5(1996), 407–422.MathSciNetzbMATHGoogle Scholar
  29. 115.
    Medina, R., Stability results for nonlinear difference equations, Nonlinear Studies, Vol. 6, No. 1, 1999.Google Scholar
  30. 117.
    Medina, R., Asymptotic behavior of Volterra difference equations, Computers and Mathematics with Applications, 41, (2001) 679–687.MathSciNetCrossRefGoogle Scholar
  31. 124.
    Mohler, R., Rajkumar, V., and Zakrzewski, R.,Nonlinear time-series-based adaptive control applications, Decision and Control 1991. Proceedings of the 30th IEEE Conference on, pp. 2917–2919 vol.3, 1991.Google Scholar
  32. 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
  33. 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
  34. 138.
    Raffoul, Y., Inequalities that lead to exponential stability and instability in delay difference equations, Journal of Inequalities in Pure and Applied Mathematics, Vol. 10, iss.3, art, 70, 2009.Google Scholar
  35. 143.
    Raffoul, Y., Total and asymptotic stability in linear Volterra integro-differential equations with nonlinear perturbation, preprint.Google Scholar
  36. 160.
    Smith, M. J., and Wisten, M.B. A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium, Annals of Operations Research. (1995). 60 (1): 59–79. doi:10.1007/BF02031940.CrossRefzbMATHGoogle Scholar
  37. 161.
    Song, Y., and Baker, C., Qualitative behavior of numerical approximations to Volterra integro-differential equations, Journal of Computational and Applied Mathematics 172 (2004) 101–115.MathSciNetCrossRefGoogle Scholar
  38. 163.
    Taniguchi, T.,Asymptotic behavior of solutions of nonautonomous difference equations, J. Math. Anal. Appl. 184(2006) 342–347.Google Scholar
  39. 177.
    Zhang, B., Asymptotic criteria and integrability properties of the resolvent of Volterra and functional equations, Funkcialaj Ekvacioj, 40(1997), 355–351.MathSciNetGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

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