Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

Advar, Murat; Raffoul, Youssef N.

doi:10.2478/auom-2013-0039

Abstract

We consider the system of Volterra integro-dynamic equations

and obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time scales. At the end of the paper, we furnish an example in which we deploy an appropriate Lyapunov functional. In addition to generalization, the results of this paper provides improvements for its counterparts in integro-differential and integro-difference equations which are the most important particular cases of our equation.

References

[1] Adivar, M., Principal matrix solutions and variation of parameters for Volterra integro-dynamic equations on time scales, Glasg. Math. J. 53 (2011) 463-480.10.1017/S0017089511000073
Search in Google Scholar
[2] Adivar, M. and Raffoul, Y. N., Existence of resolvent for Volterra integral equations on time scales, Bull. of Aust. Math. Soc., 82(1), (2010), 139155.10.1017/S0004972709001166
Search in Google Scholar
[3] Adivar, M. and Raffoul, Y. N., Existence results for periodic solutions of integro-dynamic equations on time scales, Ann. Mat. Pura Appl., 188 (4), 2009, 543-559.10.1007/s10231-008-0088-z
Search in Google Scholar
[4] Adivar, M., Function bounds for solutions of Volterra integro-dynamic equations on time scales, Electron. J. Qual. Theory Differ. Equ., No. 7. (2010), pp. 1-22.
Search in Google Scholar
[5] Akin-Bohner, E. and Raffoul Y. N., Boundedness in functional dynamic equations on time scales, Adv. Difference Equ., Volume 2006, Art. ID 79689, Pages 1-18.10.1155/ADE/2006/79689
Search in Google Scholar
[6] Bohner, M. and Peterson, A., Dynamic Equations on Time Scales, An introduction with applications. Birkhauser Boston Inc., Boston, MA, 2001.10.1007/978-1-4612-0201-1
Search in Google Scholar
[7] Bohner, M. and Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser Boston Inc., Boston, MA, 2003.10.1007/978-0-8176-8230-9
Search in Google Scholar
[8] Eloe, P., Islam, M., and Zhang, B., Uniform asymptotic stability in linear Volterra integro-differential equations with applications to delay systems, Dynam. Systems Appl., 9 (2000), 331-344.
Search in Google Scholar
[9] Khandaker T. M., Raffoul Y. N, Stability properties of linear Volterra discrete systems with nonlinear perturbation, J. Difference Equ. Appl. 8 (2002), no. 10, 857-874.
Search in Google Scholar
[10] Kulik, T. and Tisdell, C. C., Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains. Int. J. Difference Equ. 3 (2008), no. 1, 103-133.
Search in Google Scholar
[11] Miller, R. K., Asymptotic stability properties of Volterra integro- differential systems, J. Differential Equations, 10(1971), 485-50610.1016/0022-0396(71)90008-8
Search in Google Scholar
[12] Miller, R. K., Nonlinear Volterra integral equations. W. A. Benjamin, Inc., Menlo Park, Calif., 1971.
Search in Google Scholar
[13] Tisdell, C. C. and Zaidi, A., Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling. Nonlinear Anal., Vol. 68(2008), No. 11,. 3504-3524.
Search in Google Scholar

Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

Abstract

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