zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Lipschitz stability of nonlinear systems of differential equations. (English) Zbl 0595.34054
The authors define a new notion of stability - uniform Lipschitz stability (ULS) for nonlinear systems of differential equations (1) $\dot x=f(t,x)$, where $f\in C[J\times R\sp n,R\sp n]$ and $f\sb x=\partial f/\partial x$ exists and is continuous on $J\times R\sp n$, $J=[t\sb 0,\infty)$, $t\sb 0\ge 0$, $f(t,0)=0$, and $x(t,t\sb 0,x\sb 0)$ is the solution of (1) with $x(t\sb 0,t\sb 0,x\sb 0)=x\sb 0$. Definition. The zero solution of (1) is said to be (ULS) if there exists $M>0$ and $\delta >0$ such that $\vert x(t,t\sb 0,x\sb 0)\vert \le M\vert x\sb 0\vert,$ whenever $\vert x\sb 0\vert \le \delta$ and $t\ge t\sb 0\ge 0$. Several criteria for the (ULS) are obtained. It is shown by examples that the (ULS) coincides with the uniform stability in the linear case, which means that the (ULS) is a nonlinear phenomenon. The relationship between (ULS) and various types of stability notions is illustrated by a diagram.

MSC:
34D10Stability perturbations of ODE
WorldCat.org
Full Text: DOI
References:
[1] Alekseev, V. M.: An estimate for the perturbations of the solutions of ordianry differential equations. Vestnik moskov. Univ. ser. I. math. Mekh 2, 28-36 (1961)
[2] Athanassov, Z. S.: Perturbation theorems for nonlinear systems of ordinary differential equations. J. math. Anal. appl. 86, 194-207 (1982) · Zbl 0508.34036
[3] Bihari, I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta math. Hangar. 7, 71-94 (1956) · Zbl 0070.08201
[4] Birkhoff, G.; Rota, G. -C: Ordinary differential equations. (1978) · Zbl 0377.34001
[5] Brauer, F.: Perturbations of nonlinear systems of differential equations. J. math. Anal. appl. 14, 198-206 (1967) · Zbl 0156.09805
[6] Brauer, F.: Perturbations of nonlinear systems of differential equations, II. J. math. Anal. appl. 17, 418-434 (1967) · Zbl 0238.34083
[7] Brauer, F.; Strauss, A.: Perturbations of nonlinear systems of differential equations, III. J. math. Anal. appl. 31, 37-48 (1970) · Zbl 0206.37802
[8] Brauer, F.: Perturbations of nonlinear systems of differential equations, IV. J. math. Anal. appl. 37, 214-222 (1972) · Zbl 0195.10002
[9] Conley, C. C.; Miller, R. K.: Asymptotic stability without uniform stability: almost periodic coefficients. J. differential equations 1, 333-336 (1965) · Zbl 0145.11401
[10] Coppel, W. A.: Stability and asymptotic behaviour of differential equations. (1965) · Zbl 0154.09301
[11] F. M. Dannan, Gronwall-Bellman inequalities and perturbations of nonlinear systems of differential equations, preprint. · Zbl 0586.26008
[12] Lozinskii, S. M.: Error estimates for the numerical integration of ordinary differential equations, I. Izv. vyssh. Uchebn. zaved. Mat. 5, No. No. 6, 52-90 (1958)
[13] Miller, R. K.; Michel, A. N.: Ordinary differential equations. (1982) · Zbl 0552.34001
[14] Patchpatte, B. G.: Stability and asymptotic behaviour of perturbed nonlinear systems. J. differential equations 16, 14-25 (1974)
[15] Patchpatte, B. G.: Perturbations of nonlinear systems of differential equations. J. math. Anal. appl. 51, 550-556 (1975) · Zbl 0313.34047