Exponential dichotomy of difference equations in l p -phase spaces on the half-line
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For a sequence of bounded linear operators {An }∞ n=0 on a Banach space X, we investigate the characterization of exponential dichotomy of the difference equations vn+1 = An vn . We characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations vn+1 = An vn + fn in l p spaces (1 ≤ p < ∞). Then we apply the results to study the robustness of exponential dichotomy of difference equations. Copyright © 2006 N. T. Huy and V. T. N. Ha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and preliminaries We consider the difference equation xn+1 = An xn ,
n ∈ N,
(1.1)
where An , n = 0,1,2,..., is a sequence of bounded linear operators on a given Banach space X, xn ∈ X for n ∈ N. One of the central interests in the asymptotic behavior of solutions to (1.1) is to find conditions for solutions to (1.1) to be stable, unstable, and especially to have an exponential dichotomy (see, e.g., [1, 5, 7, 12, 16–20] and the references therein for more details on the history of this problem). One can also use the results on exponential dichotomy of difference equations to obtain characterization of exponential dichotomy of evolution equations through the discretizing processes (see, e.g., [4, 7, 9, 18]). One can easily see that if An = A for all n ∈ N, then the asymptotic behavior of solutions to (1.1) can be determined by the spectra of the operator A. However, the situation becomes more complicated if {An }n∈N is not a constant sequence because, in this case, the spectra of each operator An cannot determine the asymptotic behavior of the solutions to (1.1). Therefore, in order to find the conditions for (1.1) to have an exponential dichotomy, one tries to relate the exponential dichotomy of (1.1) to the solvability of the Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 58453, Pages 1–14 DOI 10.1155/ADE/2006/58453
2
Exponential dichotomy
following inhomogeneous equation: xn+1 = An xn + fn ,
n ∈ N,
(1.2)
in some certain sequence spaces for each given f = { fn }. In other words, one wants to relate the exponential dichotomy of (1.1) to the surjectiveness of the operator T defined by (Tx)n := xn+1 − An xn
for x = xn belonging to a relevant sequence space.
(1.3)
In the infinite-dimensional case, in order to characterize the exponential dichotomy of (1.1) defined on N, beside the surjectiveness of the operator T, one needs a priori condition that the stable space is complemented (see, e.g., [5]). In our recent paper, we have replaced this condition by the spectral conditions of related operators (see [9, Corollary 3.3]). At this point, we would like to note that if one considers the difference equation (1.1) defined on Z, then the existence of exponential dichotomy of (1.1) is equivalent to the existence and uniqueness of the solution of (1.2) for a given f =
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