A New Approach on Datko–Zabczyk Method for Nonuniform Exponential Stability

Abstract

We provide a sequence of projections on the linear space of all sequences and connect the existence of nonuniform exponential stability to the restrictions of these projections on a class of Banach sequence spaces defined by a discrete dynamics. As a consequence, we obtain a Datko–Zabczyk type characterization of nonuniform exponential stability. We develop our analysis without any assumption on the invertibility of the dynamics, thus our results are applicable to a large class of difference equations.

1. Introduction

A wide variety of real world phenomena and processes are described by dynamical systems and, in particular, by difference equations. There is an analogy between difference equations and differential equations, via discretization method. More precisely, a difference equation in an infinite-dimensional Banach space typically occurs as a discretization of a compact semigroup (see [1], p. 12). Therefore, it is natural to develop a theory without any assumption on the invertibility of the dynamics. On the other hand, it is possible to apply some results in the discrete case to the continuous situation (see, for instance, [2], ([3], pp. 229–230), [4]). For interesting and useful details on the theory of discrete dynamical systems we refer the reader to the monographs of Elaydi [5] and Pötzsche [1].

One of the most remarkable results in the stability theory of dynamical systems and control theory has been obtained by Datko in [6] for $C_0$-semigroups and in [7] for evolution families. In fact, Datko established the connections between uniform exponential stability of a $C_0$-semigroup/an evolution family and p-integrability of the associated trajectories. The first result related to Datko’s theorem in the case of discrete-time systems are due to Zabczyk [8], who proved that a $C_0$-semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ on a Banach space X is uniformly exponentially stable, i.e., there exist $M,\alpha >0$ such that

$$\parallel T\left(t\right)\parallel \le M{e}^{-\alpha t},\mathrm{for}\mathrm{all}t\ge 0,$$

if and only if there is a continuous strictly increasing convex function $N:[0,\infty )\to [0,\infty )$ with $N\left(0\right)=0$ such that for each $x\in X$ there exits $c_x>0$ such that

$$\sum _{n=0}^{\infty }N(c_x\parallel T\left(n\right)x\parallel )<\infty .$$

This result was extended by Przyłuski and Rolewicz [9] to the case of discrete nonautonomous systems. For more recent contributions in this direction of the research we refer the reader in particular to [2,10,11,12,13].

The aim of this paper is to obtain a Datko–Zabczyk type characterization of nonuniform exponential stability for a discrete dynamics defined by a sequence of bounded linear operators on a Banach space, which is a special case of nonuniform hyperbolicity (while nonuniform hyperbolicity is related to the non-vanishing of Lyapunov exponents, nonuniform exponential stability corresponds to negativity of all Lyapunov exponents). The importance of nonuniform behavior lies in the ergodic theory. We refer to monographs [14,15] for detailed expositions of this type of behavior. We also mention [16] for a more recent and valuable paper devoted to nonuniform behavior.

We point out that our approach is new even in the particular case of uniform behavior (the case where the growth sequence ${\left(M_n\right)}_{n\in \mathbb{N}}$ in Definition 1 is bounded). Indeed, the major task accomplished in this paper was to establish a version of Datko–Zabczyk theorem on a class of Banach sequence spaces, specific to nonuniform behavior. Recently, Lupa and Popescu [17] developed similar results to those in this paper for continuous time dynamics. Although the approach in this paper is quite similar to that in [17], we emphasize that it requires nontrivial changes. For example, the first step in the proof of Theorem 1, which is the main result of this paper, is based on Faulhaber’s formula, while its analogue in the continuous case follows from a simple integral computation. Furthermore, in comparison with [17], we consider a general class of Banach sequence spaces $\mathcal{S}$, which contains ${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ as a particular element (see Corrolay 3).

2. Preliminaries

Throughout this paper X will be a real or complex Banach space and $\mathcal{B}\left(X\right)$ will denote the Banach algebra of all bounded linear operators on X. The norms on X and on $\mathcal{B}\left(X\right)$ will be denoted by $\parallel ·\parallel$. As usually $\mathbb{N}=\{0,1,\dots \}$ denotes the set of all natural numbers. Let $\Delta =\left\{\right(m,n)\in \mathbb{N}\times \mathbb{N}:m\ge n\}$ and ${\Delta }_n=\{m\in \mathbb{N}:m\ge n\}$ for any fixed $n\in \mathbb{N}$. Furthermore, let $X^{\mathbb{N}}$ be the linear space of all sequences $u={\left(u_n\right)}_{n\in \mathbb{N}}$ in X, we write $l^{\infty }(\mathbb{N},X)$ for the Banach space of all bounded sequences in X, equipped with the norm

$${\parallel u\parallel }_{\infty }=\underset{n\in \mathbb{N}}{sup}\parallel u_n\parallel ,u={\left(u_n\right)}_{n\in \mathbb{N}}\in l^{\infty }(\mathbb{N},X),$$

and $c_{00}(\mathbb{N},X)$ denotes the space of all sequences in X with only a finite number of non-zero terms.

Given a sequence $\mathcal{A}={\left(A_n\right)}_{n\in \mathbb{N}}$ of operators in $\mathcal{B}\left(X\right)$, for any $(m,n)\in \Delta$ we set

$$A(m,n)=\left\{\begin{array}{cc}{A}_{m-1}\cdots A_n,\hfill & m>n,\hfill \\ \mathrm{Id},\hfill & m=n.\hfill \end{array}\right.$$

The family of operators ${\left\{A(m,n)\right\}}_{(m,n)\in \Delta }$ is called the discrete evolution family associated to the operator sequence $\mathcal{A}$.

Definition 1.

A real number $\alpha \in \mathbb{R}$ is called an admissible exponent for the operator sequence $\mathcal{A}={\left(A_n\right)}_{n\in \mathbb{N}}$ if there exists a positive real sequence ${\left(M_n\right)}_{n\in \mathbb{N}}$ such that

$$\parallel A(m,n)\parallel \le M_n{e}^{\alpha (m-n)},\mathrm{for}\mathrm{all}(m,n)\in \Delta ,$$

and we denote ${\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ the set of all admissible exponents of $\mathcal{A}$.

If the previous inequality holds for some $\alpha <0$, then $\mathcal{A}$ is called α-nonuniformly exponentially stable.

Evidently, if $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ and $\beta \ge \alpha$, then $\beta \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$, thus the set ${\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ is either a semi-infinite closed interval, or empty. From now on we assume that ${\mathcal{E}}_{ad}\left(\mathcal{A}\right)\ne \varnothing$. In particular, if ${\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ contains negative admissible exponents, $\mathcal{A}$ is said to be nonuniformly exponentially stable.

For any fixed $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$, $k\in \mathbb{N}$ and $u={\left(u_n\right)}_{n\in \mathbb{N}}\in X^{\mathbb{N}}$, we set

$${\phi }_{\alpha }(k,u)=\underset{i\in {\Delta }_k}{sup}{e}^{-\alpha (i-k)}\parallel A(i,k)u_k\parallel .$$

One can easily check that

$$\parallel u_k\parallel \le {\phi }_{\alpha }(k,u)\le M_k\parallel u_k\parallel .$$

(1)

For each $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$, we denote

$${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)=\left\{u={\left(u_n\right)}_{n\in \mathbb{N}}\in X^{\mathbb{N}}:\underset{n\in \mathbb{N}}{sup}{\phi }_{\alpha }(n,u)<\infty \right\},$$

which is a Banach space equipped with the norm ${\parallel u\parallel }_{\alpha }=\underset{n\in \mathbb{N}}{sup}{\phi }_{\alpha }(n,u).$

Remark 1.

${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ contains the space $c_{00}(\mathbb{N},X)$.

Indeed, for $u={\left(u_n\right)}_{n\in \mathbb{N}}\in c_{00}(\mathbb{N},X)$, without loss of generality, we may assume that there exists $k\in \mathbb{N}$ such that $u_n=0$ for all $n\in {\Delta }_{k+1}$. Hence, by (1) we get

$$\underset{n\in \mathbb{N}}{sup}{\phi }_{\alpha }(n,u)=sup\left\{{\phi }_{\alpha }(n,u):n=0,1,\dots ,k\right\}\le max\left\{M_n{\parallel u\parallel }_n:n=0,1,\dots ,k\right\}<\infty ,$$

and thus $u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right).$

Following closely the line of Proposition 5.1 in [18] one can easily show that

$${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\subseteq l^{\infty }(\mathbb{N},X),\mathrm{for}\mathrm{each}\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right),$$

with equality if and only if there exists a positive constant $M>0$ such that

$$\parallel A(m,n)\parallel \le M{e}^{\alpha (m-n)},\mathrm{for}\mathrm{all}(m,n)\in \Delta ,$$

(2)

which corresponds to the uniform asymptotic behavior. Thus, in the uniform case we have that

$${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)=l^{\infty }(\mathbb{N},X),$$

for each admissible exponent $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$. If $\alpha <0$ in the inequality (2), then $\mathcal{A}$ is called uniformly exponentially stable. In the nonuniform setting the situation is completely different. More precisely, one can find examples of difference equations admitting (infinitely) many Banach sequence spaces ${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ (see, for instance, [18] (Example 5.5)).

For any fixed $n\in \mathbb{N}$ we consider an operator ${\mathsf{\Phi }}_n:X^{\mathbb{N}}\to X^{\mathbb{N}}$, defined by

$${\left({\mathsf{\Phi }}_{n}u\right)}_i=\left\{\begin{array}{cc}A(i,n)u_n,\hfill & i\in {\Delta }_{n+1},\hfill \\ u_i,\hfill & i\in \{0,1,\dots ,n\}.\hfill \end{array}\right.$$

In the following we present some auxiliary results needed in the subsequent part of the paper.

Lemma 1.

The operator ${\mathsf{\Phi }}_n$ is a projection on $X^{\mathbb{N}}$ for each $n\in \mathbb{N}$. Furthermore,

$${\mathsf{\Phi }}_m{\mathsf{\Phi }}_n={\mathsf{\Phi }}_n,\mathrm{for}\mathrm{all}(m,n)\in \Delta .$$

(3)

Proof. 

Setting $m=n\in \mathbb{N}$ in (3), we get ${\mathsf{\Phi }}_n^2={\mathsf{\Phi }}_n$, i.e., ${\mathsf{\Phi }}_n$ is a projection on $X^{\mathbb{N}}$, and thus it suffices to prove (3). Let $(m,n)\in \Delta$. For $i\in \{0,1,\dots ,m\}$ we have ${\left({\mathsf{\Phi }}_m\left({\mathsf{\Phi }}_{n}u\right)\right)}_i={\left({\mathsf{\Phi }}_{n}u\right)}_i,$ and for $i\in {\Delta }_{m+1}$ we get

$${\left({\mathsf{\Phi }}_m\left({\mathsf{\Phi }}_{n}u\right)\right)}_i=A(i,m){\left({\mathsf{\Phi }}_{n}u\right)}_m=A(i,m)A(m,n)u_n=A(i,n)u_n={\left({\mathsf{\Phi }}_{n}u\right)}_i,$$

therefore (3) holds. □

Lemma 2.

Let $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$. For all $u\in X^{\mathbb{N}}$ and $n\in \mathbb{N}$ one gets:

(i)

${\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)={\phi }_{\alpha }(m,u),$ for $m\in \{0,1,\dots ,n\};$

(ii)

${\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)\le e^{\alpha (m-n)}{\phi }_{\alpha }(n,u),$ for $m\in {\Delta }_{n+1}.$

Proof. 

Let $u\in X^{\mathbb{N}}$ and $n\in \mathbb{N}$. If $m\in \{0,1,\dots ,n\}$, then

$${\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)=\underset{i\in {\Delta }_m}{sup}{e}^{-\alpha (i-m)}\parallel A(i,m)u_m\parallel ={\phi }_{\alpha }(m,u).$$

On the other hand, for $m\in {\Delta }_{n+1}$ we have

$$\begin{array}{cc}\hfill {\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)& =\underset{i\in {\Delta }_m}{sup}{e}^{-\alpha (i-m)}\parallel A(i,m){\left({\mathsf{\Phi }}_{n}u\right)}_m\parallel \hfill \\ \hfill & =e^{\alpha (m-n)}\underset{i\in {\Delta }_m}{sup}{e}^{-\alpha (i-n)}\parallel A(i,n)u_n\parallel \hfill \\ \hfill & \le e^{\alpha (m-n)}\underset{i\in {\Delta }_n}{sup}{e}^{-\alpha (i-n)}\parallel A(i,n)u_n\parallel \hfill \\ \hfill & =e^{\alpha (m-n)}{\phi }_{\alpha }(n,u).\hfill \end{array}$$

 □

3. The Main Results

In this section we give a complete characterization of the concept of nonuniform exponential stability for a discrete dynamics using the projections ${\mathsf{\Phi }}_n$ defined above. On the other hand, since $0\in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ is a necessary condition for the existence of nonuniform exponential stability, we first characterize the case where zero is an admissible exponent.

Proposition 1.

If $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ and $\alpha \le 0$, then

$${\mathsf{\Phi }}_n\left({\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\right)\subseteq {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right),\mathrm{for}\mathrm{each}n\in \mathbb{N}.$$

(4)

Proof. 

Fix $n\in \mathbb{N}$ and let $u={\left(u_k\right)}_{k\in \mathbb{N}}\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$. If $m\in \{0,1,\dots ,n\}$, from Lemma 2 (i) we have

$${\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)={\phi }_{\alpha }(m,u)\le {\parallel u\parallel }_{\alpha }.$$

On the other hand, for $m\in {\Delta }_{n+1}$ we get

$${\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)\le e^{\alpha (m-n)}{\phi }_{\alpha }(n,u)\le {\phi }_{\alpha }(n,u)\le {\parallel u\parallel }_{\alpha }.$$

Thus, ${\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)\le {\parallel u\parallel }_{\alpha }<\infty$, for every $m\in \mathbb{N}$, and consequently ${\mathsf{\Phi }}_{n}u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right).$ □

Remark 2.

The previous result shows that if α is an admissible exponent for $\mathcal{A}$ with $\alpha \le 0$, in particular if $\mathcal{A}$ is α-nonuniformly exponentially stable, then the restriction of the operator ${\mathsf{\Phi }}_n$ on ${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$, denoted by ${\mathsf{\Phi }}_{\alpha ,n}$, is a projection on ${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ for each $n\in \mathbb{N}.$

Proposition 2.

If there exists $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ such that (4) hods, then $0\in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$.

Proof. 

It suffices to assume that $\alpha >0$. For $x\in X$ and $n\in \mathbb{N}$ we consider a sequence $u_{n,x}={\left(u_{n,x}^k\right)}_{k\in \mathbb{N}}$ such that $u_{n,x}^n=x$ and $u_{n,x}^k=0$, for every $k\in \mathbb{N}$, $k\ne n$. Evidently, $u_{n,x}\in c_{00}(\mathbb{N},X)\subset {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ and thus ${\mathsf{\Phi }}_n{u}_{n,x}\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$. For each $m\in {\Delta }_n$ one gets

$${\phi }_{\alpha }(m,{\mathsf{\Phi }}_n{u}_{n,x})=\underset{i\in {\Delta }_m}{sup}{e}^{-\alpha (i-m)}\parallel A(i,n)x\parallel .$$

Therefore,

$$e^{-\alpha (i-m)}\parallel A(i,n)x\parallel \le {\phi }_{\alpha }(m,{\mathsf{\Phi }}_n{u}_{n,x})\le {\parallel {\mathsf{\Phi }}_n{u}_{n,x}\parallel }_{\alpha }<\infty ,\mathrm{for}\mathrm{all}i\ge m\mathrm{in}{\Delta }_n.$$

The Banach–Steinhaus theorem implies

$$K_n\left(\alpha \right)=sup\left\{e^{-\alpha (i-m)}\parallel A(i,n)\parallel :i\ge m\mathrm{in}{\Delta }_n\right\}<\infty .$$

This yields that $\parallel A(m,n)\parallel \le K_n\left(\alpha \right)$, for all $(m,n)\in \Delta$, that is $0\in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$. □

Combining Proposition 1 and Proposition 2 we establish a necessary and sufficient condition for the case where zero is an admissible exponent for $\mathcal{A}$.

Corollary 1.

$0\in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ if and only if there exists $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ such that

$${\mathsf{\Phi }}_n\left({\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\right)\subseteq {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right),\mathrm{for}\mathrm{every}n\in \mathbb{N}.$$

For any fixed $k\in \mathbb{N}$ we consider the equation

$$u_m=A(m,n)u_n+\sum _{i=n}^{m}A(m,i)v_i,m\ge n\mathrm{in}{\Delta }_k.$$

(5)

Lemma 3.

The operators

$$A_{\alpha ,k}:D\left(A_{\alpha ,k}\right)\subseteq {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\to {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right),A_{\alpha ,k}u=v,$$

where $D\left(A_{\alpha ,k}\right)$ is the set of all sequences $u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ such that there exists $v\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ with $v_i=0$ for $i\in \mathbb{N}\setminus {\Delta }_k$ and $u,v$ satisfy (5), are well-defined and closed in the topology of ${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$.

Proof. 

Let $v^1,v^2\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ with $v_i^1=v_i^2=0$ for $i\in \{0,1,\dots ,k-1\}$ and

$$u_m=A(m,n)u_n+\sum _{i=n}^{m}A(m,i)v_i^{\kappa },\mathrm{for}\mathrm{all}m\ge n\mathrm{in}{\Delta }_k,\kappa =1,2.$$

Setting $m=n\in {\Delta }_k$, we get $v_m^1=v_m^2$ for all $m\in {\Delta }_k$, which implies $v_1=v_2$ and thus $A_{\alpha ,k}$ is well-defined. We now prove that $A_{\alpha ,k}$ is closed. Assume that ${\left(u^j\right)}_{j\in \mathbb{N}}$ is a sequence in $D\left(A_{\alpha ,k}\right)$ converging to $u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ such that the sequence ${\left(A_{\alpha ,k}{u}^j\right)}_{j\in \mathbb{N}}$ converges to some $v\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$, that is

$$\parallel u^j{-u\parallel }_{\alpha }\to 0\mathrm{and}{\parallel A_{\alpha ,k}{u}^j-v\parallel }_{\alpha }\to 0,\mathrm{as}j\to \infty .$$

From (1) we deduce that

$$\underset{j\to \infty }{lim}{u}_i^j=u_i\mathrm{and}\underset{j\to \infty }{lim}{\left(A_{\alpha ,k}{u}^j\right)}_i=v_i\mathrm{in}X,\mathrm{for}\mathrm{each}\mathrm{fixed}i\in \mathbb{N}.$$

Since $u^j\in D\left(A_{\alpha ,k}\right)$, we get that ${\left(A_{\alpha ,k}{u}^j\right)}_i=0$ for every $i\in \{0,1,\dots ,k-1\}$ and

$$u_m^j=A(m,n)u_n^j+\sum _{i=n}^{m}A(m,i){\left(A_{\alpha ,k}{u}^j\right)}_i,m\ge n\mathrm{in}{\Delta }_k.$$

Therefore, $v_i=0$ for every $i\in \{0,1,\dots ,k-1\}$ and

$$u_m=A(m,n)u_n+\sum _{i=n}^{m}A(m,i)v_i,m\ge n\mathrm{in}{\Delta }_k,$$

which implies that $u\in D\left(A_{\alpha ,k}\right)$ and $A_{\alpha ,k}u=v.$ □

Let us notice that the operator $A_{\alpha ,k}$ defined above is the correspondent of the operator $I_X$ in [19] for the case of a discrete nonuniform behavior.

Proposition 3.

$0\in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ if and only if there exists $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ such that

$$\mathrm{Ker}\left(A_{\alpha ,k}\right)={\mathsf{\Phi }}_k\left({\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\right),\mathrm{for}\mathrm{every}k\in \mathbb{N}.$$

Proof. 

For any fixed $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ and $k\in \mathbb{N}$, the following equivalences hold:

$$\begin{array}{cc}\hfill A_{\alpha ,k}u=0& \iff u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\mathrm{and}{u}_m=A(m,n)u_n,\mathrm{for}\mathrm{all}m\ge n\mathrm{in}{\Delta }_k\hfill \\ \hfill & \iff u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\mathrm{and}{u}_i=A(i,k)u_k,\mathrm{for}\mathrm{all}i\in {\Delta }_k\hfill \\ \hfill & \iff u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\mathrm{and}{\mathsf{\Phi }}_{k}u=u.\hfill \end{array}$$

Since ${\mathsf{\Phi }}_k$ is a projection on $X^{\mathbb{N}}$ we deduce that

$$\mathrm{Ker}\left(A_{\alpha ,k}\right)={\mathsf{\Phi }}_k\left({\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\right)\cap {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right).$$

(6)

The conclusion holds now from Corollary 1. □

From (6), one can easily see that if $\mathcal{A}$ is $\alpha$-nonuniformly exponentially stable, then

$$\mathrm{Ker}\left(A_{\alpha ,n}\right)=\mathrm{Range}\left({\mathsf{\Phi }}_{\alpha ,n}\right),\mathrm{for}\mathrm{every}n\in \mathbb{N},$$

which gives a necessary condition for the existence of $\alpha$-nonuniform exponential stability.

Proposition 4.

If $\mathcal{A}={\left(A_k\right)}_{k\in \mathbb{N}}$ is α-nonuniformly exponentially stable, then for each $p\in (0,\infty )$ there exists a positive real constant $C>0$ such that

$$\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,{\mathsf{\Phi }}_{n}u)\le C{\phi }_{\alpha }^p(n,u),\mathrm{for}\mathrm{all}n\in \mathbb{N}\mathrm{and}u\in X^{\mathbb{N}}.$$

(7)

Proof. 

Let $n\in \mathbb{N}$ and $u\in X^{\mathbb{N}}$. From Lemma 2 we get

$$\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,{\mathsf{\Phi }}_{n}u)\le \sum _{k=n}^{\infty }{e}^{\alpha p(k-n)}{\phi }_{\alpha }^p(n,u)=\left(\sum _{i=0}^{\infty }{e}^{\alpha pi}\right){\phi }_{\alpha }^p(n,u)=\frac{1}{1-e^{\alpha p}}{\phi }_{\alpha }^p(n,u),$$

and thus (7) holds for $C=\frac{1}{1-e^{\alpha p}}$. □

Since ${\mathsf{\Phi }}_{\alpha ,n}$ is a projection on ${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ whenever $\mathcal{A}$ is $\alpha$-nonuniformly exponentially stable, we obtain a simple necessary condition for the existence of $\alpha$-nonuniform exponential stability.

Corollary 2.

Let $p\in (0,\infty )$. If $\mathcal{A}$ is α-nonuniformly exponentially stable, then

$$\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,u)<\infty ,\mathrm{for}\mathrm{all}n\in \mathbb{N}\mathrm{and}u\in \mathrm{Range}\left({\mathsf{\Phi }}_{\alpha ,n}\right).$$

Proof. 

Fix $p\in (0,\infty )$ and $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ with $\alpha <0$. For any $n\in \mathbb{N}$ and $u\in \mathrm{Range}\left({\mathsf{\Phi }}_{\alpha ,n}\right)\subseteq {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$, by Remark 2 and Proposition 4, we get

$$\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,u)=\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,{\mathsf{\Phi }}_{\alpha ,n}u)\le C{\phi }_{\alpha }^p(n,u)\le C{\parallel u\parallel }_{\alpha }^p<\infty ,$$

where $C>0$ is given by Proposition 4. □

Theorem 1.

If there exist $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ and $\mathcal{S}\subseteq X^{\mathbb{N}}$ with $c_{00}(\mathbb{N},X)\subseteq \mathcal{S}$ and

$${\mathsf{\Phi }}_n\mathcal{S}\subseteq \mathcal{S},\mathrm{for}\mathrm{every}n\in \mathbb{N},$$

(8)

such that there exist $p,C>0$ satisfying

$$\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,{\mathsf{\Phi }}_{n}u)\le C{\phi }_{\alpha }^p(n,u),\mathrm{for}\mathrm{all}n\in \mathbb{N}\mathrm{and}u\in \mathcal{S},$$

(9)

then $\mathcal{A}$ is nonuniformly exponentially stable.

Proof. 

For the convenience of the reader we will divide the proof of the theorem into several steps. Evidently, it suffices to assume that $\alpha \ge 0$.

Step 1. First, we will prove that for all $k\in \mathbb{N}$, one has

$${\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{n}u)\frac{{(m-n)}^k}{k!C^k}\le C{\phi }_{\alpha }^p(n,u),\mathrm{for}\mathrm{all}(m,n)\in \Delta \mathrm{and}u\in \mathcal{S},$$

(10)

with the convention $0^0=1$.

Since

$${\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{n}u)\le \sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,{\mathsf{\Phi }}_{n}u)\le C{\phi }_{\alpha }^p(n,u),\mathrm{for}\mathrm{all}(m,n)\in \Delta \mathrm{and}u\in \mathcal{S},$$

(11)

it follows that (10) works for $k=0$. Assume now that (10) holds for some fixed $k\in \mathbb{N}$. Let $(m,n)\in \Delta$ and $u\in \mathcal{S}$. For each $i\in \{n,n+1,\dots ,m\}$ we have

$${\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{i}u)\frac{{(m-i)}^k}{k!C^k}\le C{\phi }_{\alpha }^p(i,u).$$

Replacing u by ${\mathsf{\Phi }}_{n}u\in \mathcal{S}$ in the above inequality and using relation (3) for $(i,n)$, one gets

$${\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{n}u)\frac{{(m-i)}^k}{k!C^k}\le C{\phi }_{\alpha }^p(i,{\mathsf{\Phi }}_{n}u).$$

(12)

From a simple consequence of Faulhaber’s formula (see Appendix 2 in [18]),

$$\frac{{(m-n)}^{k+1}}{k+1}\le \sum _{i=n}^m{(m-i)}^k,$$

and using inequalities (12) and (9), we obtain

$$\begin{array}{cc}\hfill {\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{n}u)\frac{{(m-n)}^{k+1}}{(k+1)!C^{k+1}}& \le \frac{1}{C}\sum _{i=n}^m{\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{n}u)\frac{{(m-i)}^k}{k!C^k}\hfill \\ \hfill & \le \sum _{i=n}^m{\phi }_{\alpha }^p(i,{\mathsf{\Phi }}_{n}u)\hfill \\ \hfill & \le \sum _{i=n}^{\infty }{\phi }_{\alpha }^p(i,{\mathsf{\Phi }}_{n}u)\hfill \\ \hfill & \le C{\phi }_{\alpha }^p(n,u).\hfill \end{array}$$

Therefore, (10) holds for $k+1$. By induction argument it follows that it is valid for all $k\in \mathbb{N}$.

Step 2. We will prove that for any fixed $\delta \in (0,1)$ there exists $N>0$ such that

$$\parallel A(m,n)u_n\parallel \le N{e}^{-\frac{\delta }{pC}(m-n)}{\phi }_{\alpha }(n,u),\mathrm{for}\mathrm{all}(m,n)\in \Delta \mathrm{and}u\in \mathcal{S}.$$

(13)

Let $(m,n)\in \Delta$, $u\in \mathcal{S}$, and pick $\delta \in (0,1)$. Multiplying inequality (10) by ${\delta }^k$ and summing over all $k\in \mathbb{N}$ one gets

$${\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{n}u)\sum _{k=0}^{\infty }\frac{1}{k!}{\left[\frac{\delta (m-n)}{C}\right]}^k\le C{\phi }_{\alpha }^p(n,u)\left(\sum _{k=0}^{\infty }{\delta }^k\right),$$

which is equivalent to

$${\phi }_{\alpha }^p(m,{\mathsf{\Phi }}_{n}u)\le \frac{C}{1-\delta }{e}^{-\frac{\delta }{C}(m-n)}{\phi }_{\alpha }^p(n,u).$$

Since

$$\begin{array}{cc}\hfill \parallel A(m,n)u_n\parallel & \le \underset{i\in {\Delta }_m}{sup}{e}^{-\alpha (i-m)}\parallel A(i,n)u_n\parallel ={\phi }_{\alpha }(m,{\mathsf{\Phi }}_{n}u)\hfill \\ \hfill & \le {\left(\frac{C}{1-\delta }\right)}^{1/p}{e}^{-\frac{\delta }{pC}(m-n)}{\phi }_{\alpha }(n,u),\hfill \end{array}$$

we deduce that (13) works with $N={\left(\frac{C}{1-\delta }\right)}^{1/p}$.

Step 3. Let us fix $x\in X$ and $(m,n)\in \Delta$. We consider $u_{x,n}={\left(u_{x,n}^k\right)}_{k\in \mathbb{N}}$, defined by $u_{x,n}^n=x$ and $u_{x,n}^k=0$, for all $k\in \mathbb{N}$, $k\ne n$. Since $u_{x,n}\in c_{00}(\mathbb{N},X)$, we have $u_{x,n}\in \mathcal{S}$. Therefore, from Step 2 we get that for any $\delta \in (0,1)$ there exists $N>0$ with

$$\parallel A(m,n)x\parallel \le N{e}^{-\frac{\delta }{pC}(m-n)}{\phi }_{\alpha }(n,u_{x,n})\le N{M}_n{e}^{-\frac{\delta }{pC}(m-n)}\parallel x\parallel ,$$

which proves that $-\frac{\delta }{pC}\in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$, and thus $\mathcal{A}$ is nonuniformly exponentially stable. □

Now, we are able to give a complete characterization of nonuniform exponential stability for a discrete dynamics. This result can be considered a Datko–Zabczyk type theorem for nonuniform exponential stability.

Corollary 3.

$\mathcal{A}$ is nonuniformly exponentially stable if and only if there exist $\alpha \in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$ and $p,C>0$ such that

$$\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,{\mathsf{\Phi }}_{n}u)\le C{\phi }_{\alpha }^p(n,u),\mathrm{for}\mathrm{all}n\in \mathbb{N}\mathrm{and}u\in {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right).$$

Proof. 

Necessity clearly holds from Proposition 4. On the other hand, by Remark 1 we have

$$c_{00}(\mathbb{N},X)\subseteq {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right),$$

and thus to establish sufficiency it suffices to observe that replacing $\mathcal{S}$ by ${\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ in (11) we get

$${\mathsf{\Phi }}_n{\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)\subseteq {\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right),\mathrm{for}\mathrm{every}n\in \mathbb{N},$$

which is equivalent to $0\in {\mathcal{E}}_{ad}\left(\mathcal{A}\right)$. Considering now $\mathcal{S}={\ell }_{\alpha }^{\infty }\left(\mathcal{A}\right)$ in Theorem 1, we deduce that $\mathcal{A}$ is nonuniformly exponentially stable. □

In the particular case of uniform exponential behavior we obtain the following result:

Corollary 4.

$\mathcal{A}$ is uniformly exponentially stable if and only if (2) holds for some $M>0$ and $\alpha \in \mathbb{R}$, and there exist $p,C>0$ such that

$$\sum _{k=n}^{\infty }{\phi }_{\alpha }^p(k,{\mathsf{\Phi }}_{n}u)\le C{\phi }_{\alpha }^p(n,u),\mathrm{for}\mathrm{all}n\in \mathbb{N}\mathrm{and}u\in l^{\infty }(\mathbb{N},X).$$

4. Conclusions

In contrast with previous techniques developed in this topic (see, in particular, [12,13]), this paper brings into attention a new type of growth by means of the notion of admissible exponent (Definition 1), letting a nonuniformity to be satisfied (only) in respect to the growth sequence ${\left(M_n\right)}_{n\in \mathbb{N}}$ (see also [17,18,20]). On the other hand, our approach is based on the Lyapunov type map ${\phi }_{\alpha }$, which is the key of our method. A similar approach (based on a sequence of norms in the Banach space X) is developed in [2], where the author considered the case of constant sequences $u={\left(u_n\right)}_{n\in \mathbb{N}}$ in X.