Browse > Article
http://dx.doi.org/10.14403/jcms.2017.30.2.273

UNIFORMLY LIPSCHITZ STABILITY OF PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS  

Choi, Sang Il (Department of Mathematics Hanseo University)
Lee, Ji Yeon (Department of Information Security Daejeon University)
Goo, Yoon Hoe (Department of Mathematics Hanseo University)
Publication Information
Journal of the Chungcheong Mathematical Society / v.30, no.2, 2017 , pp. 273-284 More about this Journal
Abstract
In this paper, we study that the solutions to perturbed differential system $$y^{\prime}=f(t,y)+{{\displaystyle\smashmargin{2}{\int\nolimits_{t_0}}^{t}}g(s,y(s),T_1y(s))ds+h(t,y(t),T_2y(t))$$ have uniformly Lipschitz stability by imposing conditions on the perturbed part ${\int_{t0}^{t}}g(s,y(s),T_1y(s))ds,h(t,y(t),T_2y(t))$, and on the fundamental matrix of the unperturbed system y' = f(t, y) using integral inequalities.
Keywords
variational systems; perturbed differential systems; uniformly Lipschitz stability; uniformly Lipschitz stability in variation;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
연도 인용수 순위
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, with English summary).
2 F. Brauer, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 14 (1966), 198-206.   DOI
3 F. Brauer and A. Strauss, Perturbations of nonlinear systems of differential equations, III, J. Math. Anal. Appl. 31 (1970), 37-48.   DOI
4 F. Brauer, Perturbations of nonlinear systems of differential equations, IV, J. Math. Anal. Appl. 37 (1972), 214-222.   DOI
5 S. I. Choi and Y. H. Goo, Uniformly Lipschitz stability and asymptotic behavior of perturbed differential systems, J. Chungcheong Math. Soc. 29 (2016), 429-442.   DOI
6 S. I. Choi and Y. H. Goo, Uniform Lipschitz stability of perturbed differential systems, Far East J. Math. Sci(FJMS) 101 (2017), 721-735.   DOI
7 S. K. Choi, Y. H. Goo, and N. J. Koo, Lipschitz and exponential asymptotic stability for nonlinear functional systems, Dynamic Systems and Applications 6 (1997), 397-410.
8 S. K. Choi, N. J. Koo, and S. M. Song, Lipschitz stability for nonlinear functional differential systems, Far East J. Math. Sci(FJMS) 5 (1999), 689-708.
9 F. M. Dannan and S. Elaydi, Lipschitz stability of nonlinear systems of differential systems, J. Math. Anal. Appl. 113 (1986), 562-577.   DOI
10 Y. H. Goo, Lipschitz and asymptotic stability for perturbed nonlinear differential systems, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 21 (2014), 11-21.
11 Y. H. Goo, Uniform Lipschitz stability and asymptotic behavior for perturbed differential systems, Far East J. Math. Sciences, 99 (2016), 393-412.   DOI
12 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.   DOI
13 V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications Vol. I, Academic Press, New York and London, 1969.
14 B.G. Pachpatte, Stability and asymptotic behavior of perturbed nonlinear systems, J. diff. equations, 16 (1974), 14-25.   DOI
15 B.G. Pachpatte, Perturbations of nonlinear systems of differential equations, J. Math. Anal. Appl. 51 (1975), 550-556.   DOI