### Abstract

We give an affirmative answer to a question formulated by Aulbach and Van Minh by showing that the linear difference equation x_{n+1} = A _{n}x_{n}, for n ε ℕ in a Banach space B is exponentially stable if and only if for every f = {f_{n}} _{n=1}^{ε} ∫ l_{p}(ℕ, B), where 1 <p <∞, the solution of the Cauchy problem x_{n+1} = A _{n}x_{n} + f_{n}, for n ε ℕ, x_{1} = 0 is bounded on ℕ.

Original language | English |
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Pages (from-to) | 779-783 |

Number of pages | 5 |

Journal | Applied Mathematics Letters |

Volume | 17 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 2004 |

### Fingerprint

### Keywords

- Difference equations in Banach spaces
- Exponential stability
- L-spaces
- Linear equation

### ASJC Scopus subject areas

- Computational Mechanics
- Control and Systems Engineering
- Applied Mathematics
- Numerical Analysis

### Cite this

**A criterion for the exponential stability of linear difference equations.** / Pituk, M.

Research output: Contribution to journal › Article

*Applied Mathematics Letters*, vol. 17, no. 7, pp. 779-783. https://doi.org/10.1016/j.aml.2004.06.005

}

TY - JOUR

T1 - A criterion for the exponential stability of linear difference equations

AU - Pituk, M.

PY - 2004/7

Y1 - 2004/7

N2 - We give an affirmative answer to a question formulated by Aulbach and Van Minh by showing that the linear difference equation xn+1 = A nxn, for n ε ℕ in a Banach space B is exponentially stable if and only if for every f = {fn} n=1ε ∫ lp(ℕ, B), where 1 n+1 = A nxn + fn, for n ε ℕ, x1 = 0 is bounded on ℕ.

AB - We give an affirmative answer to a question formulated by Aulbach and Van Minh by showing that the linear difference equation xn+1 = A nxn, for n ε ℕ in a Banach space B is exponentially stable if and only if for every f = {fn} n=1ε ∫ lp(ℕ, B), where 1 n+1 = A nxn + fn, for n ε ℕ, x1 = 0 is bounded on ℕ.

KW - Difference equations in Banach spaces

KW - Exponential stability

KW - L-spaces

KW - Linear equation

UR - http://www.scopus.com/inward/record.url?scp=7444246244&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=7444246244&partnerID=8YFLogxK

U2 - 10.1016/j.aml.2004.06.005

DO - 10.1016/j.aml.2004.06.005

M3 - Article

VL - 17

SP - 779

EP - 783

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

IS - 7

ER -