# 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)
Characterizations of dichotomies of evolution equations on the half-line. (English) Zbl 0995.34038
The authors characterize (ordinary and exponential) dichotomies of an evolution family $U(t,s)$, $t \geq s$, of bounded linear operators on a Banach space $X$ in terms of properties of certain operators $I_0$ and $I_X$ which are defined on subspaces of $L_p(\bbfR_+,X)$ using the integral equation $$u(t) = U(t,s) u(s) + \int_s^t U(t,\xi) f(\xi) d\xi .$$

##### MSC:
 34D09 Dichotomy, trichotomy 34G10 Linear ODE in abstract spaces 47D03 (Semi)groups of linear operators
Full Text:
##### References:
 [1] Aulbach, B.; Minh, N. V.: Semigroups and exponential stability of nonautonomous linear differential on the half-line. Dynamical systems and application, 45-61 (1995) · Zbl 0842.34059 [2] Daleckii, Ju.L.; Krein, M. G.: Stability of solution of differential equation in Banach spaces. Trans. amer. Math. soc. (1974) [3] Datko, R.: Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM J. Math. anal. 3, 428-445 (1972) · Zbl 0241.34071 [4] Levitan, B. M.; Zhikov, V. V.: Almost periodic functions and differential equations. (1978) · Zbl 0414.43008 [5] Y. Latushkin, S. Montgomery, Smith, and, T. Randolph, Evolutionary semigroups and robust stability of evolution operators on Banach spaces, preprint. · Zbl 0881.47020 [6] Massera, J. J.; Schäffer, J. J.: Linear differential equations and function spaces. (1966) · Zbl 0243.34107 [7] Minh, N. V.: On the proof of characterization of the exponential dichotomy. Proc. amer. Math. soc. 127, 779-782 (1999) · Zbl 0911.34054 [8] Minh, N. V.; Rabiger, F.; Schnaubelt, R.: Exponential dichotomy exponential expansiveness and exponential dichotomy of evolution equation on the half line. Integral equations operator theory 32, 332-353 (1998) · Zbl 0977.34056 [9] Murakami, S.; Naito, T.; Minh, N. V.: Evolution semigroups and sums of commuting operators: A new approach to the admissibility of function spaces. J. differential equations 164, 240-285 (2000) · Zbl 0966.34049 [10] Nagel, R.: One-parameter semigroups of positive operators. Lecture notes in math. 1184 (1986) · Zbl 0585.47030 [11] Naito, T.; Minh, N. V.: Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations. J. differential equations 152, 358-376 (1999) · Zbl 0924.34038 [12] Van Neerven, J.: The asymptotic behaviour of semigroups of linear operator. Operator theory, advances and applications 88 (1996) · Zbl 0905.47001 [13] Pazy, A.: Semigroup of linear operators and application to partial differential equations. (1983) · Zbl 0516.47023 [14] Perron, O.: Die stabilitätsfrage bei differentialgleichungen. Math. Z. 32, 703-728 (1930) · Zbl 56.1040.01 [15] Zhang, W.: The fredhom alternative and exponential dichotomies for parabolic equations. J. math. Anal. appl. 191, 180-201 (1995) · Zbl 0832.34050