The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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A REFINEMENT OF LYAPUNOV-TYPE INEQUALITY FOR A CLASS OF NONLINEAR SYSTEMS

  • Received : 2011.07.26
  • Accepted : 2011.10.20
  • Published : 2011.11.30
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Abstract

Some new Lyapunov-type inequalities for a class of nonlinear differential systems, which are natural refinements and generalizations of the well-known Lyapunov inequality for linear second order differential equations, are given. The results of this paper cover some previous results on this topic.

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References

  1. A.M. Lyapunov: Sur une serie relative a la theorie des equations differentielles lineaires avec coeficient periodiques. C. R. Math. Acad. Sci., Paris 123 (1896), 1248- 1252.
  2. S.S. Cheng: Lyapunov inequalities for differential and difference equations. Fasc. Math. 23 (1991), 25-41.
  3. S.B. Eliason: A Lyapunov type inequality for certain nonlinear differential equation. J. London Math. Soc. 3 (1970), 461-466.
  4. G. Guseinov & B. Kaymakcalan: Lyapunov inequalities for discrete linear Hamiltonian system. Comp. Math. Appl. 45 (2003), 1399-1416. https://doi.org/10.1016/S0898-1221(03)00095-6
  5. P. Hartman: Ordinary Differential Equations, second ed. Birkhauser, Boston, 1982.
  6. C. Lee, C. Yeh, C. Hong & R.P. Agarwal: Lyapunov and Wirtinger inequalities. Appl. Math. Lett. 17 (2004), 847-853. https://doi.org/10.1016/j.aml.2004.06.016
  7. W. Leighton: On Lyapunov's inequality. Proc. Amer. Math. Soc. 33 (1972), 627-628.
  8. Z. Nehari: On an inequality of Lyapunov, In "Studies in Mathematical Analysis and Related Topics". pp. 256-261, Stanford Univ. Press, Stanford, CA, 1962.
  9. B.G. Pachpatte: On Lyapunov-type inequalities for certain higher order differential equations. J. Math. Anal. Appl. 195 (1995), 527-536. https://doi.org/10.1006/jmaa.1995.1372
  10. A. Tiryaki, M. Unal & D. Cakmak: Lyapunov-type inequalities for nonlinear systems. J. Math. Anal. Appl. 332 (2007), 497-511. https://doi.org/10.1016/j.jmaa.2006.10.010
  11. X. Wang: Lyapunov type inequalities for second order half-linear differential equations. submitted to J. Math. Anal. Appl.
  12. X. Yang: On inequalities of Lyapunov type. Appl.Math. Computation. 134 (2003), 293-300. https://doi.org/10.1016/S0096-3003(01)00283-1
  13. X. Yang & K. Lo: Lyapunov-type inequality for a class of even-order differential equations. Appl. Math. Computation 245 (2010), 3884-3890.
  14. X. Yang, Y. Kim & K. Lo: Lyapunov-type inequality for a class of odd-order differential equations. J. Comput. Appl. Math. 234 (2010), 2962-2969. https://doi.org/10.1016/j.cam.2010.04.008