Asymptotic behavior of discrete evolution families in Banach spaces

Constantin Buşe; Aftab Khan; Lan T. Nguyen; Donal O’Regan; Gul Rahmat

doi:10.1080/00036811.2016.1257122

Asymptotic behavior of discrete evolution families in Banach spaces

Constantin Buşe
, Aftab Khan
, Lan T. Nguyen
, Donal O’Regan
, Gul Rahmat

Mathematics

Politehnica University of Timisoara
Shaheed Benazir Bhutto University
Western Kentucky University
Islamia College University Peshawar

Research output: Contribution to a Journal (Peer & Non Peer) › Article › peer-review

3 Citations (Scopus)

Abstract

Let X be a Banach space, A=(A_n)n∈ℤ₊ be an operator-valued sequence and let u = {∪(n,m) : n ≥ m ∈ ℤ₊} be the discrete evolution family associated to A In this paper we prove that the family u is non-uniformly strongly stable (i.e. for every nonnegative integer m and every (Formula presented.) if and only if it is l¹ ₀(ℤ₊, X) -approximative admissible, i.e. for every sequence f = (f_n) in l¹ ₀(ℤ₊, X) and every positive number Ɛ there exists the sequence g = (g_n) in l¹ ₀(ℤ₊, X) satisfying (Formula presented.) such that the solution of the discrete Cauchy Problem x_n+1 = A_nX_n + g_n+1, n ∈ ℤ₊, x₀= 0, belongs to l¹ ₀(ℤ₊, X). Other types of asymptotic behavior of the family u are also analyzed.

Original language	English
Pages (from-to)	160-178
Number of pages	19
Journal	Applicable Analysis
Volume	97
Issue number	2
DOIs	https://doi.org/10.1080/00036811.2016.1257122
Publication status	Published - 25 Jan 2018

Keywords

Non-autonomous difference equations
boundedness and asymptotic stability
discrete evolution families of bounded linear operators
discrete evolution semigroups

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10.1080/00036811.2016.1257122

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title = "Asymptotic behavior of discrete evolution families in Banach spaces",

abstract = "Let X be a Banach space, A=(An)n∈ℤ+ be an operator-valued sequence and let u = \{∪(n,m) : n ≥ m ∈ ℤ+\} be the discrete evolution family associated to A In this paper we prove that the family u is non-uniformly strongly stable (i.e. for every nonnegative integer m and every (Formula presented.) if and only if it is l1 0(ℤ+, X) -approximative admissible, i.e. for every sequence f = (fn) in l1 0(ℤ+, X) and every positive number Ɛ there exists the sequence g = (gn) in l1 0(ℤ+, X) satisfying (Formula presented.) such that the solution of the discrete Cauchy Problem xn+1 = AnXn + gn+1, n ∈ ℤ+, x0= 0, belongs to l1 0(ℤ+, X). Other types of asymptotic behavior of the family u are also analyzed.",

keywords = "Non-autonomous difference equations, boundedness and asymptotic stability, discrete evolution families of bounded linear operators, discrete evolution semigroups",

author = "Constantin Bu{\c s}e and Aftab Khan and Nguyen, \{Lan T.\} and Donal O{\textquoteright}Regan and Gul Rahmat",

note = "Publisher Copyright: {\textcopyright} 2016 Informa UK Limited, trading as Taylor \& Francis Group.",

year = "2018",

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doi = "10.1080/00036811.2016.1257122",

language = "English",

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T1 - Asymptotic behavior of discrete evolution families in Banach spaces

AU - Buşe, Constantin

AU - Khan, Aftab

AU - Nguyen, Lan T.

AU - O’Regan, Donal

AU - Rahmat, Gul

PY - 2018/1/25

Y1 - 2018/1/25

N2 - Let X be a Banach space, A=(An)n∈ℤ+ be an operator-valued sequence and let u = {∪(n,m) : n ≥ m ∈ ℤ+} be the discrete evolution family associated to A In this paper we prove that the family u is non-uniformly strongly stable (i.e. for every nonnegative integer m and every (Formula presented.) if and only if it is l1 0(ℤ+, X) -approximative admissible, i.e. for every sequence f = (fn) in l1 0(ℤ+, X) and every positive number Ɛ there exists the sequence g = (gn) in l1 0(ℤ+, X) satisfying (Formula presented.) such that the solution of the discrete Cauchy Problem xn+1 = AnXn + gn+1, n ∈ ℤ+, x0= 0, belongs to l1 0(ℤ+, X). Other types of asymptotic behavior of the family u are also analyzed.

AB - Let X be a Banach space, A=(An)n∈ℤ+ be an operator-valued sequence and let u = {∪(n,m) : n ≥ m ∈ ℤ+} be the discrete evolution family associated to A In this paper we prove that the family u is non-uniformly strongly stable (i.e. for every nonnegative integer m and every (Formula presented.) if and only if it is l1 0(ℤ+, X) -approximative admissible, i.e. for every sequence f = (fn) in l1 0(ℤ+, X) and every positive number Ɛ there exists the sequence g = (gn) in l1 0(ℤ+, X) satisfying (Formula presented.) such that the solution of the discrete Cauchy Problem xn+1 = AnXn + gn+1, n ∈ ℤ+, x0= 0, belongs to l1 0(ℤ+, X). Other types of asymptotic behavior of the family u are also analyzed.

KW - Non-autonomous difference equations

KW - boundedness and asymptotic stability

KW - discrete evolution families of bounded linear operators

KW - discrete evolution semigroups

UR - https://www.scopus.com/pages/publications/84994885249

U2 - 10.1080/00036811.2016.1257122

DO - 10.1080/00036811.2016.1257122

M3 - Article

SN - 0003-6811

VL - 97

SP - 160

EP - 178

JO - Applicable Analysis

JF - Applicable Analysis

IS - 2

ER -

Asymptotic behavior of discrete evolution families in Banach spaces

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