zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
A dynamical systems result on asymptotic integration of linear differential systems. (English) Zbl 1037.34042
The author studies, under a dynamical systems point of view, the asymptotic integration of linear differential systems of the form $x^{\prime }=[\Lambda (t)+R(t)]x$, where $\Lambda $ is diagonal and $R\in L^{p}[t_{0},\infty )$ for $p\in \lbrack 1,2]$. By using the theory of linear skew-product flows, the paper presents a dichotomy condition in terms of the spectrum over the omega-limit set $w_{\Lambda }$. The theory is illustrated by two examples that are not covered by the Hartman-Winter theorem.

34D09Dichotomy, trichotomy
34A30Linear ODE and systems, general
34L05General spectral theory for OD operators
Full Text: DOI
[1] Behncke, H.; Remling, C.: Asymptotic integration of linear differential equations. J. math. Anal. appl. 210, 585-597 (1997) · Zbl 0883.47041
[2] Benzaid, Z.; Lutz, D. A.: Asymptotic representation of solutions of perturbed systems of linear difference equations. Stud. appl. Math. 77, 195-221 (1987) · Zbl 0628.39002
[3] Bodine, S.; Sacker, R. J.: A new approach to asymptotic diagonalization of linear differential systems. J. dynam. Differential equations 12, 229-245 (2000) · Zbl 0957.34010
[4] Bylov, B. F.: Almost reducible systems. Siberian math. J. 7, 600-625 (1966) · Zbl 0161.05902
[5] Bylov, B. F.: On the reduction of a system of linear equations to diagonal form. Amer. math. Soc. transl. 89, No. 2, 51-59 (1970) · Zbl 0208.11302
[6] Coppel, W. A.: Dichotomies and reducibility. J. differential equations 3, 500-521 (1967) · Zbl 0162.39104
[7] Coppel, W. A.: Dichotomies in stability theory. Lecture notes in mathematics 629 (1978) · Zbl 0376.34001
[8] Eastham, M. S. P.: The asymptotic solution of linear differential systems. (1989) · Zbl 0674.34045
[9] Folland, G. B.: Real analysis. (1984) · Zbl 0549.28001
[10] Gingold, H.: Almost diagonal systems in asymptotic integration. Proc. Edinburgh math. Soc. 28, No. 2, 143-158 (1985) · Zbl 0554.34034
[11] Gingold, H.; Hsieh, P. F.; Sibuya, Y.: Globally analytic simplification and the Levinson theorem. J. math. Anal. appl. 182, 269-286 (1994) · Zbl 0890.34005
[12] Harris, W. A.; Lutz, D. A.: On the asymptotic integration of linear differential systems. J. math. Anal. appl. 48, 1-16 (1974) · Zbl 0304.34043
[13] Harris, W. A.; Lutz, D. A.: A unified theory of asymptotic integration. J. math. Anal. appl. 57, 571-586 (1977) · Zbl 0398.34012
[14] Hartman, P.; Wintner, A.: Asymptotic integration of linear differential equations. Amer. J. Math. 77, 45-86404932 (1955) · Zbl 0064.08703
[15] Hsieh, P. F.; Xie, F.: On asymptotic diagonalization of linear ordinary differential equations. Dyn. contin. Discrete impuls. Systems 4, 351-377 (1998) · Zbl 0919.34033
[16] Johnson, R. A.: Analyticity of spectral subbundles. J. differential equations 35, 366-387 (1980) · Zbl 0458.34017
[17] Johnson, R. A.; Sell, G. R.: Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems. J. differential equations 41, 262-288 (1981) · Zbl 0443.34037
[18] Levinson, N.: The asymptotic nature of solutions of linear systems of differential equations. Duke math. J. 15, 111-126 (1948) · Zbl 0040.19402
[19] Medina, R.; Pinto, M.: Linear differential systems with conditionally integrable coefficients. J. math. Anal. appl. 166, 52-64 (1992) · Zbl 0755.34011
[20] Okikiolu, G. O.: Aspects of the theory of bounded integral operators in lp-spaces. (1971) · Zbl 0219.44002
[21] Palmer, K. J.: Exponential dichotomy, integral separation and diagonalizability of linear systems of ordinary differential equations. J. differential equations 43, 184-203 (1982) · Zbl 0443.34007
[22] Palmer, K. J.: Exponential dichotomies and transversal homoclinic points. J. differential equations 55, 225-256 (1984) · Zbl 0508.58035
[23] Sacker, R. J.; Sell, G. R.: Existence of dichotomies and invariant splittings for linear differential systems, I. J. differential equations 15, 429-458 (1974) · Zbl 0294.58008
[24] Sacker, R. J.; Sell, G. R.: Existence of dichotomies and invariant splittings for linear differential systems, II. J. differential equations 22, 478-496 (1976) · Zbl 0339.58013
[25] Sacker, R. J.; Sell, G. R.: A spectral theory for linear differential systems. J. differential equations 27, 320-358 (1978) · Zbl 0372.34027
[26] Sell, G. R.: Compact sets of nonlinear operators. Funkcial. ekvac. 11, 131-138 (1968) · Zbl 0165.49001
[27] Sell, G. R.: Topological dynamics and ordinary differential equations. (1971) · Zbl 0212.29202