Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T21:40:07.548Z Has data issue: false hasContentIssue false
  • English
  • Français

PRINCIPAL MATRIX SOLUTIONS AND VARIATION OF PARAMETERS FOR VOLTERRA INTEGRO-DYNAMIC EQUATIONS ON TIME SCALES

Published online by Cambridge University Press:  10 March 2011

MURAT ADIVAR
MURAT ADIVAR*
Affiliation:
Department of Mathematics, Izmir University of Economics35330, Balcova, Izmir, Turkey e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation and prove that it is the unique matrix solution of We also show that the solution of is unique and given by the variation of parameters formula

Keywords

34N0545D0539A1345J05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 463 - 480
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Adıvar, M. and Raffoul, Y. N., Existence of resolvent for Volterra integral equations on time scales, Bull. Aust. Math. Soc. 82 (1) (2010), 139–155.CrossRefGoogle Scholar
2.Becker, L. C., Principal matrix solution and variation of parameters for a Volterra integro-differential equation and its adjoint, E. J. Qual. Theory Differ. Equ. 14 (2006), 1–22.Google Scholar
3.Bohner, M., Some oscillation criteria for first-order delay dynamic equations, Far East J. Appl. Math. 18 (3) (2005), 289–304.Google Scholar
4.Bohner, M. and Peterson, A., Dynamic equations on time scales, in An introduction with applications (Birkhäuser Boston, Boston, MA, 2001).Google Scholar
5.Bohner, M. and Peterson, A., Advances in dynamic equations on time scales (Birkhäuser Boston, Boston, MA, 2003).CrossRefGoogle Scholar
6.Burton, T. A., Stability and periodic solutions of ordinary and functional differential equations (Dover Publications, Mineola, NY, 2005).Google Scholar
7.Burton, T. A., Fixed points, Volterra equations, and Becker's resolvent, Acta Math. Hungar. 108 (3) (2005), 261–281.CrossRefGoogle Scholar
8.Burton, T. A., Integral equations, Volterra equations, and the remarkable resolvent: Contractions, E. J. Qualitative Theory Differ. Equ. 2 (3) (2006), 1–17.Google Scholar
9.Elaydi, S., Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl. 181 (1994), 483–492.CrossRefGoogle Scholar
10.Elaydi, S. and Kocic, V., Global stability of a nonlinear Volterra difference equations, Differ. Equ. Dyn. Syst. 2 (1994), 337–345.Google Scholar
11.Eloe, P., Islam, M. and Raffoul, Y., Uniform asymptotic stability in nonlinear Volterra discrete systems, Comput. Math. Appl. (2002, Special issue advances in difference equations IV) 45 (6–9) (2003), 1033–1039.CrossRefGoogle Scholar
12.Graves, L. M., The theory of functions of real variables (McGraw-Hill, New York, NY, 1946).Google Scholar
13.Grossman, S. I. and Miller, R. K., Perturbation theory for Volterra integro-differential systems, J. Differ. Equ. 8 (1970), 457–474.Google Scholar
14.Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56.CrossRefGoogle Scholar
15.Khandaker, T. M. and Raffoul, Y. N., Stability properties of linear Volterra discrete systems with nonlinear perturbation, J. Differ. Equ. Appl. 8 (10) (2002), 857–874.CrossRefGoogle Scholar
16.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. Differ. Equ. 3 (1) (2008), 103–133.Google Scholar
17.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. 68 (11) (2008), 3504–3524.CrossRefGoogle Scholar
18.Raffoul, Y. N., Stability in neutral nonlinear differential equations with functional delays using fixed point theory, Math. Comput. Modell. 2008 (7–8) (2004), 691–700.CrossRefGoogle Scholar
19.Zafer, A., The exponential of a constant matrix on time scales, ANZIAM J. 48 (2006), 99–106.CrossRefGoogle Scholar
20.Zhang, B., Asymptotic stability criteria and integrability properties of the resolvent of Volterra and functional equations, Funkcialaj Ekvacioj 40, (1997), 335–351.Google Scholar