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Exponential stability in functional dynamic equations on time scales. (English) Zbl 1194.34173
Summary: We are interested in the exponential stability of the zero solution of a functional dynamic equation on a time scale, a nonempty closed subset of real numbers. The approach is based on suitable Lyapunov functionals and certain inequalities. We apply our results to obtain exponential stability in Volterra integrodynamic equations on time scales.

34N05Dynamic equations on time scales or measure chains
34K20Stability theory of functional-differential equations
time scales
Full Text: Euclid
[1] M. Ad\ivar and Y. Raffoul, Existence results for periodic solutions of integro-dynamic equations on time scales. Annali di Mathematica, DOI 10.1007/s1023-008-0088-z . · Zbl 1195.34138
[2] S. Bodine and D. A. Lutz, Exponential functions on time scales: Their asymptotic behavior and calculation. Dynam. Systems Appl. 12 (2003), pp 23-43. · Zbl 1053.39029
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[4] E. Ak\in-Bohner, and Y. Raffoul, Boundeness in functional dynamic equations on time scales. Adv. Difference Equ. 2006 2006, pp 1-18. · Zbl 1139.39005 · doi:10.1155/ADE/2006/79689 · eudml:54213
[5] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications , Birkhäuser, Boston, 2001. · Zbl 0978.39001
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[7] M. Bohner and Y. Raffoul, Volterra Dynamic Equations on Time Scales. · Zbl 1139.39005
[8] A. Peterson and Y. Raffoul, Exponential stability of dynamic equations on time scales. Adv. Difference Equ. 2 (2005), pp 133-144. · Zbl 1100.39013 · doi:10.1155/ADE.2005.133 · eudml:52829
[9] A. Peterson and C. C. Tisdell, Boundedness and uniqueness of solutions to dynamic equations on time scales. J. Diff. Equations Appl. 10 No. 13-15 (2004), pp 1295-1306. · Zbl 1072.39017 · doi:10.1080/10236190410001652793
[10] C. Poetzsche, Chain rule and invariance principle on measure chains. Dynamic equations on time scales. J. Comput. Appl. Math., 141, No. 1-2 (2002), pp 249-254. · Zbl 1011.34045 · doi:10.1016/S0377-0427(01)00450-2
[11] Y. Raffoul, Boundedness in nonlinear functional differential equations with applications to volterra integrodifferential. J. Integral Equations Appl. 16, No. 4 , Winter 2004. · Zbl 1090.34056 · doi:10.1216/jiea/1181075297
[12] Y. Raffoul, Boundedness in Nonlinear Differential Equations. Nonlinear Stud. 10 (2003), pp 343-350. · Zbl 1050.34046