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Asymptotic stability and uniform boundedness with respect to parameters for discrete non-autonomous periodic systems. (English) Zbl 1258.39008
Let \(X\) be a complex Banach space, and denote by \(\mathcal{L}(X)\) the Banach algebra of all bounded linear operators acting on \(X\). In the sequel, \(\mathbb{Z}_{+}\) is the set of non-negative integers, \(q\in\mathbb{Z}_{+}\) with \(q>1\), \((A_n)_n\) is a \(q\)-periodic sequence of operators in \(\mathcal{L}(X)\), \((z_n)_n\) is a \(q\)-periodic sequence in \(X\) with \(z_0=0\), \(\mu\) is a real parameter and \(b\in X\). The authors consider the difference equation \((A_n): x_{n+1}=A_n x_n\), \(n\in\mathbb{Z}_{+}\), and the discrete Cauchy problems \[ (A_n,\mu,b,0):\, y_{n+1}=A_n y_n+e^{i\mu n}b, \, n\in\mathbb{Z}_{+}, \, y_0=0, \] \[ (A_n,\mu,z_n,0):\, w_{n+1}=A_n w_n+e^{i\mu n}z_n, \, n\in\mathbb{Z}_{+}, \, w_0=0. \] For \(n\geq j\geq 0\) define \(U(n,j)=A_{n-1}A_{n-2}\cdots A_j\) if \(j\leq n-1\) and \(U(n,n)=\mathrm{Id}\). The system \((A_n)\) is said to be uniformly asymptotically stable (u.a.s. in short) whenever there exist two positive constants \(N\) and \(\nu\) such that \(\|U(n,k)\|\leq Ne^{\nu(n-k)}\) for all \(n\geq k\geq 0.\)
In the case \(X=\mathbb{C}^m\), with \(m\in\mathbb{Z}_{+}\), \(m\geq 1\), the authors prove that the u.a.s. of \((A_n)\) is equivalent to: (i) the solution of \((A_n,\mu,z_n,0)\) is bounded for each \(\mu\) and each \((z_n)\); (ii) for each \(b\) the solution of \((A_n,\mu,b,0)\) is uniformly bounded with respect to \(\mu\); (iii) for each \(\mu\) and each \(b\) the solution of \((A_n,\mu,b,0)\) is bounded and the operator \(\sum_{\nu=1}^{q}e^{i\mu\nu}U(q,\nu)\) is an invertible one.
To prove the result, the main tool is the development in Fourier series of smooth \(q\)-periodic functions \(f:\mathbb{R}\rightarrow \mathbb{C}^m\) and some previous results in the literature are employed (the equivalence between u.a.s. and (i) was established by C. Buşe et al. [J. Difference Equ. Appl. 11, No. 12, 1081–1088 (2005; Zbl 1094.47040)], and the equivalence with (iii) was obtained by S. Arshad et al. [Electron. J. Qual. Theory Differ. Equ. 2011, Paper No. 16, 12 p. (2011; Zbl 1281.39009)]).
Using the same strategy of Fourier series, the authors even show that in the general case of an arbitrary complex Banach space \(X\) the solution of \((A_n,\mu,b,0)\) is bounded uniformly with respect to the parameter \(\mu\) if and only if the Poincaré map \(U(q,0)\) is stable , that is, its spectral radius is less than one, which in turn is equivalent to the u.a.s. of \((A_n).\) As a corollary, the authors mention a Barbashin-type theorem.

MSC:
39A30 Stability theory for difference equations
39A70 Difference operators
39A22 Growth, boundedness, comparison of solutions to difference equations
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34A30 Linear ordinary differential equations and systems, general
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[1] S. Arshad, C. Buşe, A. Nosheen, and A. Zada, Connections between the stability of a Poincare map and boundedness of certain associate sequences, E. J. Qualitative Theory of Ditt. Equ. 16 (2011), pp. 1–12 · Zbl 1281.39009
[2] Barbashin E.A., Introduction in the Theory of Stability (1967) · Zbl 0155.13501
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[5] DOI: 10.1080/10236190500331271 · Zbl 1094.47040 · doi:10.1080/10236190500331271
[6] DOI: 10.1007/s00020-007-1533-x · Zbl 1160.47036 · doi:10.1007/s00020-007-1533-x
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[9] DOI: 10.1007/BF02574130 · Zbl 0892.47040 · doi:10.1007/BF02574130
[10] Vu Quoc P., Proc. Am. Math. Soc. 129 (10) pp 2871– (2002)
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