바로가기메뉴

본문 바로가기 주메뉴 바로가기

logo

  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

Article Contents

Prev Next
e-Submission

Vol.21 No.1

Citation Share

LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2014, v.21 no.1, pp.11-21
https://doi.org/10.7468/jksmeb.2014.21.1.11
Goo, Yoon Hoe
Goo,, Y. H. (2014). LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 21(1), 11-21, https://doi.org/10.7468/jksmeb.2014.21.1.11

Abstract

The present paper is concerned with the notions of Lipschitz and asymptotic stability for perturbed nonlinear differential system knowing the corresponding stability of nonlinear differential system. We investigate Lipschitz and asymtotic stability for perturbed nonlinear differential systems. The main tool used is integral inequalities of the Bihari-type, in special some consequences of an extension of Bihari's result to Pinto and Pachpatte, and all that sort of things.

keywords
uniformly Lipschitz stability, uniformly Lipschitz stability in variation, exponentially asymptotic stability, exponentially asymptotic stability in variation

Reference

1.

V.M. Alekseev. (1961). An estimate for the perturbations of the solutions of ordinary differential equations. Vestn. Mosk. Univ. Ser. I. Math. Mekh.(Russian), 2, 28-36.

2.

F. Brauer. (1967). Perturbations of nonlinear systems of differential equations, II. J. Math. Anal. Appl., 17, 418-434. 10.1016/0022-247X(67)90132-1.

3.

F. Brauer & A. Strauss. (1970). Perturbations of nonlinear systems of differential equations, III. J. Math. Anal. Appl., 31, 37-48. 10.1016/0022-247X(70)90118-6.

4.

F. Brauer. (1972). Perturbations of nonlinear systems of differential equations, IV. J. Math. Anal. Appl., 37, 214-222. 10.1016/0022-247X(72)90269-7.

5.

S.K. Choi & N.J. Koo. (1995). h-stability for nonlinear perturbed systems. Ann. Diff. Eqs., 11, 1-9.

6.

S.K. Choi, Y.H. Goo & N.J. Koo. (1997). Lipschitz and exponential asymptotic stability for nonlinear functional systems. Dynamic Systems and Applications, 6, 397-410.

7.

S.K. Choi, N.J. Koo & S.M. Song. (1999). Lipschitz stability for nonlinear functional differential systems. Far East J. Math. Sci(FJMS)I, 5, 689-708.

8.

F.M. Dannan & S. Elaydi. (1986). Lipschitz stability of nonlinear systems of differential systems. J. Math. Anal. Appl., 113, 562-577. 10.1016/0022-247X(86)90325-2.

9.

Goo, Yoon Hoe;. (2013). BOUNDEDNESS IN THE PERTURBED DIFFERENTIAL SYSTEMS. The Pure and Applied Mathematics, 20(3), 223-232. 10.7468/jksmeb.2013.20.3.223.

10.

Goo, Yoon-Hoe;Yang, Seung-Bum;. (2012). h-STABILITY OF NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS VIA t<sub>∞</sub>-SIMILARITY. The Pure and Applied Mathematics, 19(2), 171-177. 10.7468/jksmeb.2012.19.2.171.

11.

Goo, Yoon-Hoe;Yang, Seung-Bum;. (2012). h-STABILITY OF NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS VIA t<sub>∞</sub>-SIMILARITY. The Pure and Applied Mathematics, 19(2), 171-177. 10.7468/jksmeb.2012.19.2.171.

12.

Goo, Yoon Hoe;. (2013). BOUNDEDNESS IN PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS. Journal of the Chungcheong Mathematical Society, 26(3), 605-613. 10.14403/jcms.2013.26.3.605.

13.

V. Lakshmikantham & S. Leela. Differential and Integral Inequalities: Theory and Applications Vol.I.

14.

B.G. Pachpatte. (1973). A note on Gronwall-Bellman inequality. J. Math. Anal. Appl., 44, 758-762. 10.1016/0022-247X(73)90014-0.

15.

P. Gonzalez & M. Pinto. (1994). Stability properties of the solutions of the nonlinear functional differential systems. J. Math. Anal. Appl., 181, 562-573. 10.1006/jmaa.1994.1044.

16.

M. Pinto. (1984). Perturbations of asymptotically stable differential systems. Analysis, 4, 161-175.

17.

M. Pinto. (1990). Integral inequalities of Bihari-type and applications. Funkcial. Ekvac., 33, 387-404.

18.

M. Pinto. (1990). Variationally stable differential system. J. Math. Anal. Appl., 151, 254-260. 10.1016/0022-247X(90)90255-E.

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

e-Submission OA Policy