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

ADIVAR, MURAT

Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 463-480

Résumé

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

Détail

DOI : 10.1017/S0017089511000073

Mots-clés : 34N05, 45D05, 39A13, 45J05

Bibliographie

[1] 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. Google Scholar | DOI

[2] 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] 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] 4.Bohner, M. and Peterson, A., Dynamic equations on time scales, in An introduction with applications (Birkhäuser Boston, Boston, MA, 2001). Google Scholar

[5] 5.Bohner, M. and Peterson, A., Advances in dynamic equations on time scales (Birkhäuser Boston, Boston, MA, 2003). Google Scholar | DOI

[6] 6.Burton, T. A., Stability and periodic solutions of ordinary and functional differential equations (Dover Publications, Mineola, NY, 2005). Google Scholar

[7] 7.Burton, T. A., Fixed points, Volterra equations, and Becker's resolvent, Acta Math. Hungar. 108 (3) (2005), 261–281. Google Scholar | DOI

[8] 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] 9.Elaydi, S., Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl. 181 (1994), 483–492. Google Scholar | DOI

[10] 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] 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. Google Scholar | DOI

[12] 12.Graves, L. M., The theory of functions of real variables (McGraw-Hill, New York, NY, 1946). Google Scholar

[13] 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] 14.Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56. Google Scholar | DOI

[15] 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. Google Scholar | DOI

[16] 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] 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. Google Scholar | DOI

[18] 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. Google Scholar | DOI

[19] 19.Zafer, A., The exponential of a constant matrix on time scales, ANZIAM J. 48 (2006), 99–106. Google Scholar | DOI

[20] 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