The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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BOUNDEDNESS IN THE PERTURBED DIFFERENTIAL SYSTEMS

  • Received : 2013.06.03
  • Accepted : 2013.08.07
  • Published : 2013.08.31
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Abstract

Alexseev's formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In recent years M. Pinto introduced the notion of $h$-stability. S.K. Choi et al. investigated $h$-stability for the nonlinear differential systems using the notion of $t_{\infty}$-similarity. Applying these two notions, we study bounds for solutions of the perturbed differential systems.

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References

  1. V.M. Alexseev: An estimate for the perturbations of the solutions of ordinary differential equations Vestn. Mosk. Univ. Ser. I. Math. Mekh. 2 (1961), 28-36(Russian).
  2. S.K. Choi & N.J. Koo: h-stability for nonlinear perturbed systems. Ann. of Diff. Eqs. 11 (1995), 1-9.
  3. S.K. Choi & H.S. Ryu: h-stability in differential systems. Bull. Inst. Math. Acad. Sinica 21 (1993), 245-262.
  4. S.K. Choi, N.J. Koo & H.S. Ryu: h-stability of differential systems via $t_{\infty}$-similarity. Bull. Korean. Math. Soc. 34 (1997), 371-383.
  5. S.K. Choi, N.J. Koo & S.M. Song: Lipschitz stability for nonlinear functional differential systems. Far East J. Math. Sci(FJMS)I 5 (1999), 689-708.
  6. R. Conti: Sulla $t_{\infty}$-similitudine tra matricie l'equivalenza asintotica dei sistemi differenziali lineari. Rivista di Mat. Univ. Parma 8 (1957), 43-47.
  7. S. Elaydi & R.R.M. Rao: Lipschitz stability for nonlinear Volterra integro-differential systems. Appl. Math. Computations 27 (1988), 191-199. https://doi.org/10.1016/0096-3003(88)90001-X
  8. Y. H. Goo, D.G. Park & D.H. Ryu: Boundedness in perturbed differential systems. J. Appl. Math. and Informatics 30 (2012), 279-287.
  9. Y.H. Goo & S.B. Yang: h-stability of nonlinear perturbed differential systems via $t_{\infty}$-similarity. J. Korean Soc. Math. Educ. Ser.B: Pure Appl. Math. 19 (2012), 171-177. https://doi.org/10.7468/jksmeb.2012.19.2.171
  10. V. Lakshmikantham & S. Leela: Differential and Integral Inequalities: Theory and Applications Vol. I. Academic Press, New York and London, 1969.
  11. M. Pinto: Perturbations of asymptotically stable differential systems. Analysis 4 (1984), 161-175.
  12. M. Pinto: Stability of nonlinear differential systems. Applicable Analysis 43 (1992), 1-20. https://doi.org/10.1080/00036819208840049

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  1. UNIFORM LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED DIFFERENTIAL SYSTEMS vol.26, pp.4, 2013, https://doi.org/10.14403/jcms.2013.26.4.831
  2. BOUNDEDNESS IN THE PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS vol.27, pp.3, 2014, https://doi.org/10.14403/jcms.2014.27.3.479
  3. LIPSCHITZ AND ASYMPTOTIC STABILITY FOR NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS vol.27, pp.4, 2014, https://doi.org/10.14403/jcms.2014.27.4.591
  4. LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS vol.21, pp.1, 2014, https://doi.org/10.7468/jksmeb.2014.21.1.11