Open Access

Boundedness in functional dynamic equations on time scales

Advances in Difference Equations20062006:079689

https://doi.org/10.1155/ADE/2006/79689

Received: 1 February 2006

Accepted: 27 March 2006

Published: 28 September 2006

Abstract

Using nonnegative definite Lyapunov functionals, we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales. We apply our obtained results to linear and nonlinear Volterra integro-dynamic equations on time scales by displaying suitable Lyapunov functionals.

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Authors’ Affiliations

(1)
Department of Mathematics and Statistics, University of Missouri-Rolla
(2)
Department of Mathematics, University of Dayton

References

  1. Akın-Bohner E, Bohner M, Akın F: Pachpatte inequalities on time scales. Journal of Inequalities in Pure and Applied Mathematics 2005,6(1):1–23. article 6MathSciNetMATHGoogle Scholar
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  3. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.MATHGoogle Scholar
  4. Bohner M, Raffoul YN: Volterra dynamic equations on time scales. preprintGoogle Scholar
  5. Peterson AC, Raffoul YN: Exponential stability of dynamic equations on time scales. Advances in Difference Equations 2005,2005(2):133–144. 10.1155/ADE.2005.133MathSciNetView ArticleMATHGoogle Scholar
  6. Peterson AC, Tisdell CC: Boundedness and uniqueness of solutions to dynamic equations on time scales. Journal of Difference Equations and Applications 2004,10(13–15):1295–1306.MathSciNetView ArticleMATHGoogle Scholar
  7. Raffoul YN: Boundedness in nonlinear differential equations. Nonlinear Studies 2003,10(4):343–350.MathSciNetMATHGoogle Scholar
  8. Raffoul YN: Boundedness in nonlinear functional differential equations with applications to Volterra integrodifferential equations. Journal of Integral Equations and Applications 2004,16(4):375–388. 10.1216/jiea/1181075297MathSciNetView ArticleMATHGoogle Scholar

Copyright

© E.Akin-Bohner and Y.N. Raffoul. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.