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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:
34D09Dichotomy, trichotomy
34G10Linear ODE in abstract spaces
47D03(Semi)groups of linear operators
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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