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Article Dans Une Revue Nonlinear functional analysis and applications Année : 2014

Criterion For The Exponential Stability of Discrete Evolution Family Over Banach Spaces

Agarwal R. P.
  • Fonction : Auteur
Zada Akbar
  • Fonction : Auteur
Ahmad Nisar
  • Fonction : Auteur

Résumé

Let $T(1)$ be the algebraic generator of the discrete semigroup $\textbf{T}=\{T(n)\}_{n\geq 0}$. We prove that the system $x_{n+1}=T(1)x_n$ is uniformly exponentially stable if and only if for each real number $\mu$ and each $q$-periodic sequence $z(n)$ with $z(0)=0$ the unique solution of the Cauchy Problem \begin{equation*} \left\{ \begin{split} % \nonumber to remove numbering (before each equation) x_{n+1} &= T(1)x_{n}+e^{i\mu(n+1)}z(n+1), \\ x_0&= 0 \end{split} \right.\eqno{(T(1), \mu,0)} \end{equation*} is bounded. We also extend the above result to $q$-periodic system $y_{n+1} = A_{n}y_{n}$ i.e. we proved that the system $y_{n+1} = A_{n}y_{n}$ is uniformly exponentially stable if and only if for each real number $\mu$ and each $q$-periodic sequence $z(n)$, with $z(0)=0$ the unique solution of the Cauchy Problem \begin{equation*} \left\{ \begin{split} % \nonumber to remove numbering (before each equation) y_{n+1} &= A_{n}y_{n}+e^{i\mu(n+1)}z(n+1), \\ y_0&= 0 \end{split} \right.\eqno{(A_{n}, \mu,0)} \end{equation*} is bounded. Here, $A_{n}$ is a sequence of bounded linear operators on Banach space $\mathcal{X}$.
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Dates et versions

hal-01099024 , version 1 (30-12-2014)

Identifiants

  • HAL Id : hal-01099024 , version 1

Citer

Dhaou Lassoued, Agarwal R. P., Zada Akbar, Ahmad Nisar. Criterion For The Exponential Stability of Discrete Evolution Family Over Banach Spaces. Nonlinear functional analysis and applications , 2014, 19 (4), pp.547-561. ⟨hal-01099024⟩
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