Asymptotic stability and uniform boundedness with respect to parameters for discrete non-autonomous periodic systems.

*(English)*Zbl 1258.39008Let \(X\) 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.

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.

Reviewer: Antonio Linero Bas (Murcia)

##### 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 |

##### Keywords:

stable operators; boundedness; discrete Cauchy problems; evolution family; complex Banach space; bounded linear operators; Fourier series; Barbashin’s type theorems; uniform asymptotic stability; difference equation
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\textit{D. Barbu} and \textit{C. Buşe}, J. Difference Equ. Appl. 18, No. 9, 1435--1441 (2012; Zbl 1258.39008)

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##### References:

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