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 ChapterĀ 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].
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References
Agarwal, R., and Pang, P.Y., On a generalized difference system, Nonlinear Anal., TM and Appl., 30(1997), 365ā376.
Crisci, M.R., Kolmanovskii, V.B., and Vecchio. A., Boundedness of discrete Volterra equations, J. Math. Analy. Appl. 211(1997), 106ā130.
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) 10ā12, 77ā97.
Crisci, M.R., Kolmanovskii, V.B., and Vecchio. A., On the exponential stability of discrete Volterra systems, J. Differ. Equations Appl. 6 (2000) 6, 667ā680.
DiblĆk, 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), 9310ā9320 (2012).
Elaydi, S.E., Periodicity and stability of linear Volterra difference systems, J. Math Anal. Appl. 181(1994), 483ā492.
Elaydi, S.E., stability and asymptotocity of Volterra difference equations: A progress report, J. Compu. and Appl. Math. 228 (2009) 504ā513.
Elaydi, S.E., and Murakami, S., Uniform asymptotic stability in linear Volterra difference equations, Journal of Difference Equations 3(1998), 203ā218.
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. 1033ā1039.
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) 1645ā1653.
Islam, M., and Raffoul, Y., Uniform asymptotic stability in linear Volterra difference equations, PanAmerican Mathematical Journal, 11 (2001), No. 1, pp. 61ā73.
Islam, M., and Raffoul, Y., Exponential stability in nonlinear difference equations, Journal of Difference Equations and Applications, (2003), Vol. 9, No. 9, pp. 819ā825.
Matsunaga, H., and Hajiri, C.,Exact stability sets for a linear difference systems with diagonal delay, J. Math. Anal. 369 (2010) 616ā622.
Medina, R., The asymptotic behavior of the solutions of a Volterra difference equations, Comput. Math. Appl., 181 (1994), no. 1, pp. 19ā26.
Medina, R., Stability results for nonlinear difference equations, Nonlinear Studies, Vol. 6, No. 1, 1999.
Medina, R., Asymptotic equivalence of Volterra difference systems, Intl. Jou. of Diff. Eqns. and Appl. Vol. 1 No.1(2000), 53ā64.
Medina, R., Asymptotic behavior of Volterra difference equations, Computers and Mathematics with Applications, 41, (2001) 679ā687.
Raffoul, Y., Boundedness and Periodicity of Volterra Systems of Difference Equations, Journal of Difference Equations and Applications, 1998, Vol. 4, pp. 381ā393.
Raffoul, Y., General theorems for stability and boundedness for nonlinear functional discrete systems, J. Math. Analy. Appl., 279 (2003), pp. 639ā650.
Raffoul, Y., Periodicity in General Delay Nonlinear Difference Equations Using fixed point Theory, Journal of Difference Equations and Applications, (2004). Vol. 10, pp.1229ā1242.
Raffoul, Y., Necessary and sufficient conditions for uniform boundedness In functional difference equations, EPAM, Volume 2, Issue 2, 2016, Pages 171ā180.
Raffoul, Y., Lyapunov-Razumikhin conditions that leads to stability and boundedness of functional difference equations of Volterra difference type, preprint.
Raffoul, Y., Uniform asymptotic stability and boundedness in functional finite delays difference equations, preprint.
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, 1ā13.
Zhang, S., Stability of infinite delay difference systems, Nonlinear Analysis, Method & Applications, (1994) Vol. 22, No. 9, pp. 1121ā1129.
Zhang, S., and Chen, M.P., A new Razumikhin theorem for delay difference equations, Computers Math. Applic. (1998) Vol. 36, No. 10ā12, pp. 405ā412.
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Raffoul, Y.N. (2018). Functional Difference Equations. In: Qualitative Theory of Volterra Difference Equations. Springer, Cham. https://doi.org/10.1007/978-3-319-97190-2_2
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