Exponential dichotomy of difference equations and applications to evolution equations on the half-line. (English) Zbl 1016.39007

This article deals with homogeneous and inhomogeneous difference equations \[ x_{n+1}= A_nx_n(n=0,1,2, \dots)\text{ and }x_{n+1}= A_n x_n+f_n (n=0,1,2, \dots), \] where \(A_n\), \(n=1,2, \dots\), is a sequence of uniformly bounded linear operators on a given Banach space \(X\). The main result is the equivalence of the exponential dichotomy property for homogeneous equation and the following properties of inhomogeneous equation: (a) the operator \[ T(u_1,\dots, u_n,\dots)= (u_1-A_0u_0, \dots, u_{n+1}-A_nu_n, \dots): \ell_\infty \to \ell_\infty \] is surjective; (b) the subspace \(X_0(0)= \{x \in X:\sup_{n\geq 0}\|A_{n-1}A_{n-2}\dots A_0 x\|<\infty\}\) is complemented in \(X\). As application of this theorem the authors present a similar result for strongly continuous and exponential bounded evolution families \(U(t,s)\), \(0\leq s\leq t<\infty\).


39A11 Stability of difference equations (MSC2000)
47D06 One-parameter semigroups and linear evolution equations
46B20 Geometry and structure of normed linear spaces
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
39A12 Discrete version of topics in analysis
Full Text: DOI


[1] Li, T., Die Stabilitätsfrage bei Differenzegleichungen, Acta Math., 63, 99-141 (1934) · JFM 60.0397.03
[2] Coffman, C. V.; Schäffer, J. J., Dichotomies for linear difference equations, Math. Anal., 172, 139-166 (1967) · Zbl 0189.40303
[3] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer: Springer Berlin, Lecture Notes in Mathematics, No. 840 · Zbl 0456.35001
[4] Sljusarchuk, V. E., On exponential dichotomy of solutions of discrete systems, Ukrain. Mat. Zh., 35, 109-115 (1983)
[5] Aulbach, B.; Minh, N. V., The concept of spectral dichotomy for difference equations. II, Journal of Difference Equations and Applications, 2, 3, 251-262 (1996) · Zbl 0880.39009
[6] Minh, N. V.; Rabiger, F.; Schnaubelt, R., Exponential expansiveness and exponential dichotomy of evolution equation on the half line, Integral Eq. and Oper. Theory, 32, 332-353 (1998) · Zbl 0977.34056
[7] Pazy, A., Semigroup of Linear Operators and Application to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0516.47023
[8] Sacker, R.; Sell, G., Dichotomies for linear evolutionary equations in Banach spaces, J. Diff. Eq., 113, 17-67 (1994) · Zbl 0815.34049
[9] Aulbach, B.; Minh, N. V., Semigroups and exponential stability of nonautonomous linear differential on the half-line, (Agarwal, R. P., Dynamical Systems and Application (1995), World Scientific: World Scientific Singapore), 45-61 · Zbl 0842.34059
[10] Coffman, C. V.; Schäffer, J. J., Linear differential equations with delays: Admissibility and conditional exponential stability, J. Diff. Eq., 9, 521-535 (1971) · Zbl 0256.34078
[11] Chow, S. N.; Leiva, H., Existence and roughness of the exponential dichotomy for skew-product semiflows in Banach spaces, J. Diff. Eq., 120, 429-477 (1995) · Zbl 0831.34067
[12] Daleckii, L. Ju.; Krein, M. G., Stability of solution of differential equation in Banach spaces, Trans. Amer. Math. Soc. (1974)
[13] Datko, R., Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3, 428-445 (1972) · Zbl 0241.34071
[14] Levitan, B. M.; Zhikov, V. V., Almost Periodic Functions and Differential Equations (1982), Univ. Publ. House: Univ. Publ. House Moscow, English translation by Cambridge University Press · Zbl 0499.43005
[15] Massera, J. J.; Schäffer, J. J., Linear Differential Equations and Function Spaces (1966), Academic Press: Academic Press New York · Zbl 0202.14701
[16] Murakami, S.; Naito, T.; Minh, N. V., Evolution semigroups and sums of commuting operators: A new approach to the admissibility of function spaces, J. Diff. Equations, 164, 240-285 (2000) · Zbl 0966.34049
[17] Naito, T.; Minh, N. V., Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Diff. Eq., 152, 358-376 (1999) · Zbl 0924.34038
[18] van Neerven, J., The asymptotic behaviour of semigroups of linear operator, Operator Theory, Advances and Applications, 88 (1996), Birkhäuser: Birkhäuser Basel · Zbl 0905.47001
[19] Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32, 703-728 (1930) · JFM 56.1040.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.