Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

PERTURBATIONS OF FUNCTIONAL DIFFERENTIAL SYSTEMS

  • Im, Dong Man (Department of Mathematics Education, Cheongju University)
  • Received : 2019.03.02
  • Accepted : 2019.04.22
  • Published : 2019.05.15
Download PDF
⟨ Previous Next ⟩

Abstract

We show the boundedness and uniform Lipschitz stability for the solutions to the functional perturbed differential system $$y^{\prime}=f(t,y)+{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{t_0}}^t}g(s,y(s),\;T_1y(s))ds+h(t,y(t),\;T_2y(t))$$, under perturbations. We impose conditions on the perturbed part ${\int_{t_0}^{t}}g(s,y(s)$, $T_1y(s))ds$, $h(t,y(t)$, $T_2y(t))$, and on the fundamental matrix of the unperturbed system y' = f(t, y) using the notion of h-stability.

Keywords

Acknowledgement

Supported by : Cheongju University

References

  1. V. M. Alekseev, 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. F. Brauer, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 14 (1966), 198-206. https://doi.org/10.1016/0022-247X(66)90021-7
  3. F. Brauer and A. Strauss, Perturbations of nonlinear systems of differential equations, III, J. Math. Anal. Appl. 31 (1970), 37-48. https://doi.org/10.1016/0022-247X(70)90118-6
  4. F. Brauer, Perturbations of nonlinear systems of differential equations, IV, J. Math. Anal. Appl. 37 (1972), 214-222. https://doi.org/10.1016/0022-247X(72)90269-7
  5. S. I. Choi and Y. H. Goo, Lipschitz and asymptotic stability of nonlinear systems of perturbed differential equations, Korean J. Math. 23 (2015), 181-197. https://doi.org/10.11568/kjm.2015.23.1.181
  6. S. I. Choi and Y. H. Goo, Lipschitz stability for perturbed functional differential systems, Far East J. Math. Sci. 96 (2015), 573-591.
  7. S. K. Choi and H. S. Ryu, h−stability in differential systems, Bull. Inst. Math. Acad. Sinica, 21 (1993), 245-262.
  8. S. K. Choi, N. J. Koo and H. S. Ryu, h-stability of differential systems via $t_{\infty}$-similarity, Bull. Korean. Math. Soc. 34 (1997), 371-383.
  9. R. Conti, Sulla $t_{\infty}$-similitudine tra matricie l'equivalenza asintotica dei sistemi differenziali lineari, Rivista di Mat. Univ. Parma 8 (1957), 43-47.
  10. F. M. Dannan and S. Elaydi, Lipschitz stability of nonlinear systems of differential systems, J. Math. Anal. Appl. 113 (1986), 562-577. https://doi.org/10.1016/0022-247X(86)90325-2
  11. Y. H. Goo, Boundedness in nonlinear perturbed functional differential systems, J. Appl. Math. and Informatics, 33 (2015), 447-457. https://doi.org/10.14317/jami.2015.447
  12. Y. H. Goo, h-stability and boundedness in the functional differential systems, Far East J. Math. Sci. 99 (2016), 1215-1231.
  13. Y. H. Goo, Perturbations of nonlinear functional differential systems, Far East J. Math. Sci. 103 (2018), 767-784.
  14. G. A. Hewer, Stability properties of the equation by $t_{\infty}$-similarity, J. Math. Anal. Appl. 41 (1973), 336-344. https://doi.org/10.1016/0022-247X(73)90209-6
  15. D. M. Im, Boundedness for nonlinear perturbed functional differential systems via $t_{\infty}$-similarity, J. Chungcheong Math. Soc. 29 (2016), 585-598. https://doi.org/10.14403/jcms.2016.29.4.585
  16. D. M. Im and Y. H. Goo, Asymptotic property for perturbed nonlinear functional differential systems, J. Appl. Math. and Informatics, 33 (2015), 687-697. https://doi.org/10.14317/jami.2015.687
  17. D. M. Im and Y. H. Goo, Uniformly Lipschitz stability and asymptotic property of perturbed functional differential systems, Korean J. Math. 24 (2016), 1-13. https://doi.org/10.11568/kjm.2016.24.1.1
  18. D. M. Im and Y. H. Goo, Perturbations of nonlinear perturbed differential systems, Far East J. Math. Sci. 101 (2017), 1509-1531.
  19. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications Vol. 1, Academic Press, New York and London, 1969.
  20. B. G. Pachpatte, Stability and asymptotic behavior of perturbed nonlinear systems, J. diff. equations, 16 (1974) 14-25. https://doi.org/10.1016/0022-0396(74)90025-4
  21. B. G. Pachpatte, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 51 (1975), 550-556. https://doi.org/10.1016/0022-247X(75)90106-7
  22. M. Pinto, Perturbations of asymptotically stable differential systems, Analysis 4 (1984), 161-175. https://doi.org/10.1524/anly.1984.4.12.161
  23. M. Pinto, Stability of nonlinear differential systems, Applicable Analysis, 43 (1992), 1-20. https://doi.org/10.1080/00036819208840049