# Uniform Exponential Stability of Discrete Evolution Families on Space ofp-Periodic Sequences

Authors: Yongfang Wang; Akbar Zada; Nisar Ahmad; Dhaou Lassoued; Tongxing Li
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We prove that the discrete systemζn+1=Anζn is uniformly exponentially stable if and only if the unique solution of the Cauchy problem ζn+1=Anζn+eiθn+1zn+1, n∈Z+, ζ0=0, is bounded for any real number θ and any p-periodic sequence z(n) with z(0)=0. Here, An is a sequence of bounded linear operators on Banach space X.


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## Body

## 1. Introduction

The investigation of difference equationsζn+1=Anζn or ζn+1=Anζn+fn leads to the idea of discrete evolution family. The main interest is the asymptotic behavior of the solutions and different types of stabilities in the study of such systems. There are a number of spectral criteria for the characterizations of stability of such systems.New difficulties appear in the study of nonautonomous systems, especially because the part of the solution generated by the forced term(fn), that is, ∑k=νn‍U(n,k)fk, is not a convolution in the classical sense. These difficulties may be passed by using the so-called evolution semigroups. The evolution semigroups were exhaustively studied in [1]. Clark et al. [2] developed this efficient method to the study of continuous case. So far, there are few related results regarding the investigation of discrete systems. Recently, the discrete versions of [3, 4] were obtained in [5, 6], respectively.Buşe et al. [7] considered the uniform exponential stability of discrete nonautonomous systems on the space of sequences denoted by C00(Z+,X). The objective of this paper is to extend results obtained in [7] to the space of p-periodic sequences denoted by P0p(Z+,X). The results reported in this paper develop the theory of discrete evolution families on a space of bounded sequences. Similar results of this type for the continuous case may be found in the paper by Buşe and Jitianu [8] and the references cited therein.


## 2. Notation and Preliminaries

LetX be a real or complex Banach space and let B(X) be the Banach algebra of all linear and bounded operators acting on X.We denote by∥·∥ the norms of operators and vectors. Denote by R the set of real numbers and by Z+ the set of all nonnegative integers.LetB(Z+,X) be the space of X-valued bounded sequences with the supremum norm, and let P0p(Z+,X) be the space of p-periodic (with p∈{2,3,…}) sequences z(n) with z(0)=0. Then, P0p(Z+,X) is a closed subspace of B(Z+,X).Throughout this paper,A∈B(X), σ(A) denotes the spectrum of A, and r(A):=sup⁡{|λ|:λ∈σ(A)} denotes the spectral radius of A. It is well known that r(A)=lim⁡n→∞∥An∥1/n. The resolvent set of A is defined as ρ(A)≔C∖σ(A), that is, the set of all λ∈C for which A-λI is an invertible operator in B(X).We give some results in the framework of general Banach space and spaces of sequences as defined above.Recall thatA is power bounded if there exists a positive constant M such that ∥An∥≤M for all n∈Z+.We need the following auxiliary lemma.Lemma 1 (Buşe et al. [3]).
AssumeV∈B(X) and let
(1)sup⁡θ∈Rsup⁡n∈Z+∥∑k=0n‍eiθkVk∥<∞.
Then, r(V)<1.


## 3. Uniform Exponential Stability of Discrete Evolution Family on SpaceP0p(Z+,X)

The familyU:={U(n,m):n,m∈Z+,n≥m} of bounded linear operators is called p-periodic discrete evolution family, for a fixed integer p∈{2,3,…}, if it satisfies the following properties:(i)
U(n,n)=I, for all n∈Z+;(ii)
U(n,m)U(m,r)=U(n,r), for all n≥m≥r, n,m,r∈Z+;(iii)
U(n+p,m+p)=U(n,m), for all n≥m, n,m∈Z+;(iv)
the map(n,m)↦U(n,m): {(n,m):n,m∈Z+,n≥m}→X is continuous, for all n≥m.It is well known that anyp-periodic discrete evolution family U is exponentially bounded; that is, there exist ω∈R and an Mω≥0 such that
(2)∥U(n,m)∥≤Mωeω(n-m),∀n≥m∈Z+.
When a family U is exponentially bounded, its growth bound ω0(U) is the infimum of all ω∈R for which there exists an Mω≥1 such that inequality (2) is fulfilled. It is known that
(3)ω0(U)=lim⁡n→∞ln⁡∥U(n,0)∥n=1pln⁡(r(U(p,0))).
As a matter of fact,
(4)ω0(U)=lim⁡n→∞ln⁡∥U(n,0)∥n=lim⁡n→∞ln⁡∥U(np,0)∥np=1plim⁡n→∞ln⁡∥Un(p,0)∥1/n=1pln⁡lim⁡n→∞∥Un(p,0)∥1/n=1pln⁡(r(U(p,0))).
A family U is termed uniformly exponentially stable if ω0(U) is negative or, equivalently, there exist an M>0 and ω>0 such that ∥U(n,m)∥≤Me-ω(n-m), for all n≥m∈Z+. Therefore, we have the following lemma.Lemma 2.
The discrete evolution familyU is uniformly exponentially stable if and only if r(U(p,0))<1.The mapU(p,0) is also called the Poincare map of the evolution family U.Consider the following discrete Cauchy problem:(An,θ,0)ζn+1=Anζn+eiθ(n+1)z(n+1),n∈Z+,ζ0=0,
where the sequence (An) is p-periodic; that is, A(n+p)=An for all n∈Z+ and a fixed p∈{2,3,…}.Let(5)U(n,k)≔{An-1An-2⋯Ak,ifk≤n-1,I,ifk=n.
Then, the family {U(n,k)}n≥k≥0 is a discrete p-periodic evolution family. Over finite dimensional spaces, the uniform exponential stability of the Cauchy problem (An,θ,0) in discrete and continuous autonomous cases has been studied in [9, 10].Let us dividen by p, that is, n=lp+r for some l∈Z+, where r∈{0,1,…,p-1}. We consider the following sets which will be useful along this work:
(6)Aj≔{1+jp,2+jp,…,(j+1)p-1},∀j∈Z+.
If r∈{1,2,…,p-1}, then define
(7)Bl≔{lp+1,lp+2,…,lp+r},C≔{0,p,2p,…,lp}.
It is clear that
(8)⋃j=0l-1‍Aj⋃‍Bl⋃‍C={0,1,2,…,n}.
With the help of partition (8), we construct the space W which consists of all the sequences of the form
(9)z(k)≔{(k-jp)[(1+j)p-k]U(k-jp,0),ifk∈Aj,k(p-k)U(k,0),ifk∈Bl,0,ifk∈C.
That is,
(10)W≔{z(n):z(n)hastheproperty(9)}.
Obviously, W is the subspace of P0p(Z+,X).Our result is stated as follows.Theorem 3.
LetU≔{U(n,m):n,m∈Z+,n≥m} be a discrete evolution family on X. If the sequence
(11)ζn=∑k=0n‍eiθkU(n,k)z(k)
is bounded for each real number θ and each p-periodic sequence z(n)∈W, then U is uniformly exponentially stable.Proof.
Letz(n)∈W. By virtue of (9),
(12)ζn=∑k=1n‍eiθkU(n,k)z(k)=∑k=1lp+r‍eiθkU(lp+r,k)z(k)=∑k∈∪j=0l-1Aj∪Bl∪C‍eiθkU(lp+r,k)z(k)=∑k∈∪j=0l-1Aj‍eiθkU(lp+r,k)z(k)+∑k∈Bl‍eiθkU(lp+r,k)z(k)+∑k∈C‍eiθkU(lp+r,k)z(k)=∑j=0l-1‍∑k=1+jpp-1+jp‍eiθkU(lp+r,k)z(k)+∑k=lp+1lp+r‍eiθkU(lp+r,k)z(k)+∑k∈C‍eiθkU(lp+r,k)z(k).
By virtue of the definition of z(k), we have
(13)ζn=∑k=1n‍eiθkU(n,k)z(k)=∑j=0l-1‍∑k=1+jpp-1+jp‍eiθkU(lp+r,k)(k-jp)×[(1+j)p-k]U(k-jp,0)+∑k=lp+1lp+r‍eiθkU(lp+r,k)k(p-k)U(k,0)=∑j=0l-1‍∑k=1+jpp-1+jp‍eiθkU(lp+r,k)(k-jp)×[(1+j)p-k]U(k-jp,0)+∑k=lp+1lp+r‍eiθkU(lp+r,k)k(p-k)U(k,0)=I1+I2,
where
(14)I1≔∑j=0l-1‍∑k=1+jpp-1+jp‍eiθkU(lp+r,k)(k-jp)×[(1+j)p-k]U(k-jp,0),I2≔∑k=lp+1lp+r‍eiθkU(lp+r,k)k(p-k)U(k,0).
Write I1 in the form
(15)I1=∑j=0l-1‍∑k=1+jpp-1+jp‍eiθkU(lp+r,k)(k-jp)×[(1+j)p-k]U(k-jp,0)=∑j=0l-1‍∑k=1+jpp-1+jp‍eiθkU(lp+r,k)(k-jp)×[(1+j)p-k]U(k,jp)=∑j=0l-1‍∑k=1+jpp-1+jp‍eiθkU(lp+r,jp)(k-jp)×[(1+j)p-k]=∑j=0l-1‍U(lp+r,jp)×∑k=1+jpp-1+jp‍eiθk(k-jp)[(1+j)p-k]=∑j=0l-1‍U(lp+r,jp)×∑k=1+jpp-1+jp‍eiθk(k-jp)[p-(k-jp)]=∑j=0l-1‍U(r,0)Ul-j(p,0)eiθjp∑v=1p-1‍eiθvv(p-v)=U(r,0)∑v=1p-1‍eiθvv(p-v)∑j=0l-1‍eiθjpUl-j(p,0)=U(r,0)∑v=1p-1‍eiθvv(p-v)∑α=1l‍eiθp(l-α)Uα(p,0)=U(r,0)∑v=1p-1‍eiθvv(p-v)eiθpl×∑α=1l‍e-iθpαUα(p,0)=G(θ,p)∑α=1l‍e-iθpαUα(p,0),
where G(θ,p):=U(r,0)∑v=1p-1‍eiθvv(p-v)eiθpl≠0. Furthermore,
(16)I2=∑k=lp+1lp+r‍eiθkU(lp+r,k)k(p-k)U(k,0)=∑k=lp+1lp+r‍eiθkU(lp+r,0)k(p-k)=U(lp+r,0)∑k=lp+1lp+r‍eiθkk(p-k).
Therefore,
(17)∑k=0n‍eiθkU(n,k)z(k)=G(θ,p)∑α=1l‍e-iθpαUα(p,0)+U(lp+r,0)∑k=lp+1lp+r‍eiθkk(p-k).
Since ζn is bounded, we conclude that I1 is bounded; that is,
(18)sup⁡l≥0∥∑α=0l‍e-iθpαUα(p,0)∥<∞.
Using Lemma 1, we deduce that r(U(p,0))<1. Hence, by Lemma 2, U is uniformly exponentially stable. This completes the proof.



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*Source: 784289-2014-08-31.xml*