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  • Research Article
  • Open Access

Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity

Advances in Difference Equations20072007:065012

  • Received: 25 December 2006
  • Accepted: 8 May 2007
  • Published:


Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation


Authors’ Affiliations

Dipartimento di Matematica e Informatica, Universita di Salerno, Fisciano, Sa, 84084, Italy
Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev 163-a, Donetsk, 83015, Ukraine


  1. Blizorukov MG: On the construction of solutions of linear difference systems with continuous time. Differentsial'nye Uravneniya 1996,32(1):127–128. translation in Differential Equations, vol. 32, no. 1, pp. 133–134, 1996MathSciNetGoogle Scholar
  2. Korenevskiĭ DG: Criteria for the stability of systems of linear deterministic and stochastic difference equations with continuous time and with delay. Matematicheskie Zametki 2001,70(2):213–229. translation in Mathematical Notes, vol. 70, no. 2, pp. 192–205, 2001View ArticleGoogle Scholar
  3. Luo J, Shaikhet L: Stability in probability of nonlinear stochastic Volterra difference equations with continuous variable. Stochastic Analysis and Applications 2007.,25(3):Google Scholar
  4. Sharkovsky AN, Maĭstrenko YuL: Difference equations with continuous time as mathematical models of the structure emergences. In Dynamical Systems and Environmental Models (Eisenach, 1986), Math. Ecol.. Akademie, Berlin, Germany; 1987:40–49.Google Scholar
  5. Péics H: Representation of solutions of difference equations with continuous time. In Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999), 2000, Szeged, Hungary, Proc. Colloq. Qual. Theory Differ. Equ.. Volume 21. Electronic Journal of Qualitative Theory of Differential Equations; 1–8.Google Scholar
  6. Pelyukh GP: Representation of solutions of difference equations with a continuous argument. Differentsial'nye Uravneniya 1996,32(2):256–264. translation in Differential Equations, vol. 32, no. 2, pp. 260–268, 1996MathSciNetGoogle Scholar
  7. Philos ChG, Purnaras IK: An asymptotic result for some delay difference equations with continuous variable. Advances in Difference Equations 2004,2004(1):1–10. 10.1155/S1687183904310058MATHMathSciNetView ArticleGoogle Scholar
  8. Shaikhet L: Lyapunov functionals construction for stochastic difference second-kind Volterra equations with continuous time. Advances in Difference Equations 2004,2004(1):67–91. 10.1155/S1687183904308022MATHMathSciNetView ArticleGoogle Scholar
  9. Kolmanovskiĭ VB: On the stability of some discrete-time Volterra equations. Journal of Applied Mathematics and Mechanics 1999,63(4):537–543. 10.1016/S0021-8928(99)00068-4MathSciNetView ArticleGoogle Scholar
  10. Paternoster B, Shaikhet L: Application of the general method of Lyapunov functionals construction for difference Volterra equations. Computers & Mathematics with Applications 2004,47(8–9):1165–1176. 10.1016/S0898-1221(04)90111-3MATHMathSciNetView ArticleGoogle Scholar
  11. Shaikhet L, Roberts JA: Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations. Advances in Difference Equations 2006, 2006: 22 pages.MathSciNetView ArticleGoogle Scholar
  12. Volterra V: Lesons sur la theorie mathematique de la lutte pour la vie. Gauthier-Villars, Paris, France; 1931.Google Scholar
  13. Kolmanovskiĭ VB, Shaikhet L: New results in stability theory for stochastic functional-differential equations (SFDEs) and their applications. In Proceedings of Dynamic Systems and Applications, Vol. 1 (Atlanta, GA, 1993), 1994, Atlanta, Ga, USA. Dynamic; 167–171.Google Scholar
  14. Kolmanovskiĭ VB, Shaikhet L: General method of Lyapunov functionals construction for stability investigation of stochastic difference equations. In Dynamical Systems and Applications, World Sci. Ser. Appl. Anal.. Volume 4. World Scientific, River Edge, NJ, USA; 1995:397–439.View ArticleGoogle Scholar
  15. Kolmanovskiĭ VB, Shaikhet L: A method for constructing Lyapunov functionals for stochastic differential equations of neutral type. Differentsial'nye Uravneniya 1995,31(11):1851–1857, 1941. translation in Differential Equations, vol. 31, no. 11, pp. 1819–1825 (1996), 1995Google Scholar
  16. Kolmanovskiĭ VB, Shaikhet L: Some peculiarities of the general method of Lyapunov functionals construction. Applied Mathematics Letters 2002,15(3):355–360. 10.1016/S0893-9659(01)00143-4MATHMathSciNetView ArticleGoogle Scholar
  17. Kolmanovskiĭ VB, Shaikhet L: Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results. Mathematical and Computer Modelling 2002,36(6):691–716. 10.1016/S0895-7177(02)00168-1MATHMathSciNetView ArticleGoogle Scholar
  18. Kolmanovskiĭ VB, Shaikhet L: About one application of the general method of Lyapunov functionals construction. International Journal of Robust and Nonlinear Control 2003,13(9):805–818. special issue on Time-delay systems 10.1002/rnc.846MATHMathSciNetView ArticleGoogle Scholar
  19. Shaikhet L: Stability in probability of nonlinear stochastic hereditary systems. Dynamic Systems and Applications 1995,4(2):199–204.MATHMathSciNetGoogle Scholar
  20. Shaikhet L: Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems. Theory of Stochastic Processes 1996,18(1–2):248–259.MathSciNetGoogle Scholar
  21. Shaikhet L: Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations. Applied Mathematics Letters 1997,10(3):111–115. 10.1016/S0893-9659(97)00045-1MATHMathSciNetView ArticleGoogle Scholar


© B. Paternoster and L. Shaikhet. 2007

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