Abstract
For a sequence of bounded linear operators on a Banach space X, we investigate the characterization of exponential dichotomy of the difference equations vn+1 = A n v n . We characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations vn+1 = A n v n + f n in l p spaces (1 ≤ p < ∞). Then we apply the results to study the robustness of exponential dichotomy of difference equations.
Highlights
Introduction and preliminariesWe consider the difference equation xn+1 = Anxn, 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
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 = Anvn
We characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations vn+1 = Anvn + fn in lp spaces (1 ≤ p < ∞)
Summary
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 = Anvn. 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]). We would like to note that if one considers the difference equation (1.1) defined on Z, 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 = { fn}n∈Z, or, in other words, to the invertibility of the operator T on suitable sequence spaces defined on Z.
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