On Polynomial Stability of Variational Nonautonomous Difference Equations in Banach Spaces

Abstract

Our goal in this paper is to give characterizations for some concepts of polynomial stability for variational nonautonomous difference equations. The obtained results can be considered generalizations for the case of variational nonautonomous difference equations of some theorems proved by Barbashin (1967), Datko (1973), and Lyapunov (1992), for evolution operators.

1. Introduction

In this paper we define and characterize two types of polynomial stability: (nonuniform) polynomial stability and strong polynomial stability for variational nonautonomous difference equations. These concepts are different from the concept of exponential stability studied for variational nonautonomous difference equations in [1], as shown in this paper.

In the case of evolution operators, the concept of nonuniform polynomial stability was studied by Barreira and Valls [2]. Moreover, characterizations for polynomial stability of evolution operators have been given in [3].

The variational nonautonomous difference equations considered in this paper generate discrete evolution cocycle over a discrete evolution semiflow. The concept of evolution cocycle was introduced by Megan and Stoica in [4].

We will consider the sets Δ={(𝑚,𝑛)∈ℕ²,with𝑚≥𝑛} and 𝑇={(𝑚,𝑛,𝑝)∈ℕ³,with𝑚≥𝑛≥𝑝}, a metric space (𝑋,𝑑) and 𝑉 a real or complex Banach space. The norm on 𝑉 and on ℬ(𝑉) (the Banach algebra of all bounded linear operators on 𝑉) will be denoted by ‖⋅‖.

Definition 1. A mapping 𝜑∶Δ×𝑋→𝑋 is called a discrete evolution semiflow on 𝑋 if the following conditions hold:(𝑠₁)  𝜑(𝑛,𝑛,𝑥)=𝑥, for all (𝑛,𝑥)∈ℕ×𝑋; (𝑠₂)  𝜑(𝑚,𝑛,𝜑(𝑛,𝑝,𝑥))=𝜑(𝑚,𝑝,𝑥), for all (𝑚,𝑛,𝑝,𝑥)∈𝑇×𝑋.

Given a sequence (𝐴_𝑚)_𝑚∈ℕ with 𝐴_𝑚∶𝑋→ℬ(𝑉) and a discrete evolution semiflow 𝜑∶Δ×𝑋→𝑋, we consider the problem of existence of a sequence (𝑣_𝑚)_𝑚∈ℕ with 𝑣_𝑚∶ℕ×𝑋→𝑋 such that 𝑣_𝑚+1(𝑛,𝑥)=𝐴_𝑚𝜑(𝑚,𝑛,𝑥)𝑣_𝑚(𝑛,𝑥) (1) for all (𝑚,𝑛,𝑥)∈Δ×𝑋. We will denote this problem by (𝐴,𝜑) and we say that (𝐴,𝜑) is a variational (nonautonomous) discrete-time system.

For (𝑚,𝑛)∈Δ we define the application Φ^𝑛_𝑚∶𝑋→ℬ(𝑉) by Φ^𝑛_𝑚(𝑥)𝑣=𝐴_𝑚−1𝜑(𝑚−1,𝑛,𝑥)⋯𝐴_𝑛+1𝜑(𝑛+1,𝑛,𝑥)𝐴_𝑛(𝑥)𝑣,if𝑚>𝑛𝑣,if𝑚=𝑛. (2)

Remark 2. From the definitions of 𝑣_𝑚 and Φ^𝑛_𝑚 it follows that (𝑐₁)  Φ^𝑚_𝑚(𝑥)𝑣=𝑣, for all (𝑚,𝑥,𝑣)∈ℕ×𝑋×𝑉; (𝑐₂)  Φ^𝑝_𝑚(𝑥)=Φ^𝑛_𝑚(𝜑(𝑛,𝑝,𝑥))Φ^𝑝_𝑛(𝑥), for all (𝑚,𝑛,𝑝,𝑥)∈𝑇×𝑋; (𝑐₃)  𝑣_𝑚(𝑛,𝑥)=Φ^𝑛_𝑚(𝑥)𝑣_𝑛(𝑛,𝑥), for all (𝑚,𝑛,𝑥)∈Δ×𝑋.

Definition 3. A mapping Φ∶Δ×𝑋→ℬ(𝑉) is called a discrete evolution cocycle over the discrete evolution semiflow 𝜑∶Δ×𝑋→𝑋 if the following properties hold: (𝑐₁)  Φ(𝑛,𝑛,𝑥)=𝐼 (the identity operator on 𝑉), for all (𝑛,𝑥)∈ℕ×𝑋, (𝑐₂)  Φ(𝑚,𝑝,𝑥)=Φ(𝑚,𝑛,(𝜑(𝑛,𝑝,𝑥))Φ(𝑛,𝑝,𝑥), for all (𝑚,𝑛,𝑝,𝑥)∈𝑇×𝑋.
If Φ is a discrete evolution cocycle over the discrete evolution semiflow 𝜑, then the pair 𝑆=(Φ,𝜑) is called a discrete skew-evolution semiflow on 𝑋.

Remark 4. From Remark 2 it results that the mapping Φ∶Δ×𝑋→ℬ(𝑉),Φ(𝑚,𝑛,𝑥)𝑣=Φ^𝑛_𝑚(𝑥)𝑣 (3) is a discrete evolution cocycle over discrete evolution semiflow 𝜑.

2. Polynomial Stability

Let (𝐴,𝜑) be a discrete variational system associated with the discrete evolution semiflow 𝜑∶Δ×𝑋→𝑋 and with the sequence of mappings 𝐴=(𝐴_𝑚), where 𝐴_𝑚∶𝑋→ℬ(𝑉), for all 𝑚∈ℕ.

Definition 5. The system (𝐴,𝜑) is said to be (i)  exponentially stable (and denoted as e.s.) if there exist the constants 𝑁≥1, 𝛼>0 and 𝛽≥0, such that 𝑒^{𝛼(𝑚−𝑛)}Φ^𝑛_𝑚(𝑥)𝑣≤𝑁𝑒^𝛽𝑛‖𝑣‖ (4) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉; (ii)  polynomially stable (and denoted as p.s.) if there exist the constants 𝑁≥1, 𝛼>0 and 𝛽≥0 such that (𝑚+1)^𝛼Φ^𝑛_𝑚(𝑥)𝑣≤𝑁(𝑛+1)^𝛼+𝛽‖𝑣‖ (5) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Remark 6. The system (𝐴,𝜑) is (i)  exponentially stable if and only if there are 𝑁≥1, 𝛼>0, and 𝛽≥0 with 𝑒^{𝛼(𝑚−𝑛)}Φ^𝑝_𝑚(𝑥)𝑣≤𝑁𝑒^𝛽𝑛Φ^𝑝_𝑛(𝑥)𝑣 (6) for all (𝑚,𝑛,𝑝,𝑥,𝑣)∈𝑇×𝑋×𝑉; (ii)  polynomially stable if and only if there exist 𝑁≥1, 𝛼>0, and 𝛽≥0 with (𝑚+1)^𝛼Φ^𝑝_𝑚(𝑥)𝑣≤𝑁(𝑛+1)^𝛼+𝛽Φ^𝑝_𝑛(𝑥)𝑣 (7) for all (𝑚,𝑛,𝑝,𝑥,𝑣)∈𝑇×𝑋×𝑉.

The connection between the two concepts of stability defined previously is established in the following.

Remark 7. It is obvious that e.s.⇒p.s. (8)

The following example shows that the converse implication is not valid.

Example 8. Let 𝒞=𝒞(ℝ₊,ℝ) be the metric space of all bounded continuous functions 𝑥∶ℝ₊→ℝ, with the topology of uniform convergence. 𝒞 is metrizable with respect to the metric 𝑑(𝑥₁,𝑥₂)=sup_𝑡∈ℝ+|𝑥₁(𝑡)−𝑥₂(𝑡)|. Let 𝑓∶ℝ₊→(0,∞) be a bounded decreasing function with the property that there exists lim_𝑡→∞𝑓(𝑡)=𝑙>0. We denote by 𝑋 the closure in 𝒞 of the set {𝑓_𝑡,𝑡∈ℝ₊}, where 𝑓_𝑡(𝑠)=𝑓(𝑡+𝑠) for all 𝑠∈ℝ₊. The mapping 𝜑∶Δ×𝑋→𝑋 defined by 𝜑(𝑚,𝑛,𝑥)=𝑥_𝑚−𝑛 is a discrete evolution semiflow. Let us consider the Banach space 𝑉 and let the sequence of mappings 𝐴_𝑚∶𝑋→ℬ(𝑉), defined by 𝐴_𝑚(𝑥)𝑢(𝑚)𝑥(𝜏)𝑣=𝑢()𝑥()𝑣𝑚+1𝜏+1 (9) for all (𝑚,𝑥,𝑣)∈ℕ×𝑋×𝑉, where the sequence 𝑢∶ℕ→ℝ is given by 𝑢(𝑚)=(𝑚+1)𝑚+1−𝑚cos𝑚𝜋2². (10) Then Φ^𝑛_𝑚(𝑥)((𝑣=𝑛+1)(𝑛+1−𝑛cos𝑛𝜋/2))²𝑥(𝜏)(𝑚+1)(𝑚+1−𝑚cos(𝑚𝜋/2))²𝑥(𝑚−𝑛+𝜏)𝑣, (11) and it results that Φ^𝑛_𝑚(𝑥)𝑣≤(2𝑛+1)²(𝑛+1)𝑥(0)()𝑙‖𝑣‖𝑚+1≤𝑁𝑛+1()𝑚+1𝑛+1²‖𝑣‖ (12) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, where 𝑁=4𝑥(0)/𝑙. Hence (𝐴,𝜑) is p.s. Assume by a contradiction that (𝐴,𝜑) is e.s. According to Definition 5, there are 𝑁≥1, 𝛼>0, and 𝛽≥0 such that 𝑒^{𝛼(𝑚−𝑛)}Φ^𝑛_𝑚(𝑥)𝑣≤𝑁𝑒^𝛽𝑛‖𝑣‖ (13) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉. The previous inequality for the considered system becomes 𝑒^{𝛼(𝑚−𝑛)}×(𝑛+1)(𝑛+1−𝑛cos(𝑛𝜋/2))²𝑥(𝜏)(𝑚+1)(𝑚+1−𝑚cos(𝑚𝜋/2))²𝑥(𝑚−𝑛+𝜏)≤𝑁𝑒^𝛽𝑛 (14) for all (𝑚,𝑛,𝑥)∈Δ×𝑋. If we take 𝑚=4𝑘²+4 and 𝑛=4𝑘+2, 𝑘∈ℕ, then 𝑒^{4𝛼𝑘2+𝑘(𝛼−𝛽−2)+2}(4𝑘+3)(8𝑘+5)²4𝑘²×𝑥(𝜏)+5𝑥4𝑘²−4𝑘+2+𝜏≤𝑁𝑒^2𝛽. (15) Passing to the limit for 𝑘→∞ we obtain a contradiction. We have shown that (𝐴,𝜑) is not e.s.

Lemma 9. The system (𝐴,𝜑) is polynomially stable if and only if there are 𝑁≥1 and 0<𝑐≤𝑑 such that (𝑚+1)^𝑐Φ^𝑛_𝑚(𝑥)𝑣≤𝑁(𝑛+1)^𝑑‖𝑣‖ (16) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Proof. Necessity. If (𝐴,𝜑) is p.s., then there are 𝑁≥1, 𝛼>0, and 𝛽≥0 such that (𝑚+1)^𝛼Φ^𝑛_𝑚(𝑥)𝑣≤𝑁(𝑛+1)^𝛼+𝛽‖𝑣‖ (17) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉. Hence inequality (16) holds for 𝑐=𝛼 and 𝑑=𝛼+𝛽.
Sufficiency. From the hypothesis it results that relation (5) of Definition 5 holds for 𝛼=𝑐 and 𝛽=𝑑−𝑐.

A necessary condition for the polynomial stability property is presented by the following theorem.

Theorem 10. If the system (𝐴,𝜑) is polynomially stable, then there are 𝐷≥1, 𝑑>0, 𝜔>0, and 𝛾≥0 such that ^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷(𝑛+1)^𝛾‖𝑣‖ (18) for all (𝑛,𝑥,𝑣)∈ℕ×𝑋×𝑉 and Φ^𝑛_𝑚(𝑥)𝑣≤𝐷(𝑚+1)^𝜔(𝑛+1)^𝛾−𝜔‖𝑣‖ (19) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Proof. Let 𝑁≥1, 𝛼>0, and 𝛽≥0 as in Definition 5. Then, for every 𝑑∈(0,𝛼) we have that ^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝑁(𝑛+1)^{𝛼+𝛽−𝑑}‖𝑣‖^∞_𝑘=𝑛1(𝑘+1)^{𝛼+1−𝑑}=𝑁‖𝑣‖(𝑛+1)^{𝛼+𝛽−𝑑}1(𝑛+1)^{𝛼+1−𝑑}+^∞_𝑘=𝑛+11(𝑘+1)^{𝛼+1−𝑑}≤𝑁‖𝑣‖(𝑛+1)^{𝛼+𝛽−𝑑}1(𝑛+1)^{𝛼+1−𝑑}+11𝛼−𝑑(𝑛+1)^𝛼−𝑑≤𝑁‖𝑣‖(𝑛+1)^{𝛼+𝛽−𝑑}11+1𝛼−𝑑(𝑛+1)^𝛼−𝑑=𝑁(1+𝛼−𝑑)‖𝑣)𝛼−𝑑‖(𝑛+1^𝛽≤𝐷‖𝑣‖(𝑛+1)^𝛾 (20) for all (𝑛,𝑥,𝑣)∈ℕ×𝑋×𝑉, where 𝐷=𝑁(1+𝛼−𝑑)/(𝛼−𝑑) and 𝛾=𝛽. In addition Φ^𝑛_𝑚(𝑥)𝑣()≤𝑁𝑛+1^𝛼+𝛽(𝑚+1)^𝛼‖𝑣‖=𝑁𝑚+1𝑛+1^𝜔𝑛+1𝑚+1^𝛼+𝜔(𝑛+1)^𝛽‖𝑣‖≤𝐷𝑚+1𝑛+1^𝜔(𝑛+1)^𝛾‖𝑣‖ (21) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Next, a sufficient condition for the polynomial stability property is presented by.

Theorem 11. If there are 𝐷≥1 and 𝑑>𝜔>𝛾≥0 such that ^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷(𝑛+1)^𝛾‖𝑣‖ (22) for all (𝑛,𝑥,𝑣)∈ℕ×𝑋×𝑉 and Φ^𝑛_𝑚(𝑥)𝑣≤𝐷(𝑚+1)^𝜔(𝑛+1)^𝛾−𝜔‖𝑣‖ (23) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, then the system (𝐴,𝜑) is polynomially stable.

Proof. From the hypothesis it results that Φ^𝑛_𝑚(𝑥)𝑣≤𝐷(𝑚+1)^𝜔(𝑘+1)^𝛾−𝜔Φ^𝑛_𝑘(𝑥)𝑣 (24) for all (𝑚,𝑘,𝑛,𝑥,𝑣)∈𝑇×𝑋×𝑉. We suppose that 𝑚≥2𝑛 and we denote by 𝑗=[𝑚/2]. Then 𝑚+22(𝑚+1)^𝑑−𝜔(𝑛+1)^{𝑑−𝜔+𝛾}Φ^𝑛_𝑚(𝑥)𝑣≤^𝑚_𝑘=𝑗(𝑚+1)^𝑑−𝜔(𝑛+1)^{𝑑−𝜔+𝛾}Φ^𝑛_𝑚(𝑥)𝑣≤𝐷^𝑚_𝑘=𝑗(𝑚+1)^𝑑−𝜔(𝑛+1)^{𝑑−𝜔+𝛾}𝑚+1𝑘+1^𝜔×(𝑘+1)^𝛾Φ^𝑛_𝑘(𝑥)𝑣=𝐷^𝑚_𝑘=𝑗1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣×𝑚+1𝑘+1^𝑑𝑛+1𝑘+1^𝜔−𝛾(𝑘+1)≤𝐷4^𝑑(𝑚+1)^𝑚_𝑘=𝑗1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷4^𝑑(𝑚+1)^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷²4^𝑑(𝑚+1)(𝑛+1)^𝛾‖𝑣‖. (25) Hence, 𝑚+1𝑛+1^𝑑−𝜔Φ^𝑛_𝑚(𝑥)𝑣≤𝐷²4^𝑑2(𝑚+1)()𝑚+2𝑛+1^2𝛾‖𝑣‖≤𝐷²2^2𝑑+1(𝑛+1)^2𝛾‖𝑣‖=𝑁(𝑛+1)^2𝛾‖𝑣‖ (26) for all (𝑥,𝑣)∈𝑋×𝑉, where 𝑁=𝐷²2^2𝑑+1. If 𝑛≤𝑚<2𝑛, then 𝑚+1𝑛+1^𝑑−𝜔Φ^𝑛_𝑚(𝑥)𝑣≤𝐷𝑚+1𝑛+1^𝑑−𝜔𝑚+1𝑛+1^𝜔(𝑛+1)^𝛾‖𝑣‖≤𝐷2^𝑑(𝑛+1)^𝛾‖𝑣‖≤𝑁(𝑛+1)^2𝛾‖𝑣‖ (27) for all (𝑥,𝑣)∈𝑋×𝑉 thus we have proved that (𝐴,𝜑) is p.s.

As a generalization of a theorem of Barbashin [5], we give the following characterization of the polynomial stability property.

Theorem 12. The system (𝐴,𝜑) is polynomially stable if and only if there are 𝐵≥1 and 𝑎>𝑏≥0 such that ^𝑚_𝑘=𝑛1𝑘+1𝑚+1𝑘+1^𝑎Φ^𝑘_𝑚(𝑥)𝑣≤𝐵(𝑚+1)^𝑏‖𝑣‖ (28) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Proof. Necessity. Let 𝑁≥1, 𝛼>0, and 𝛽≥0 as in Definition 5. Then, for every 𝑎>0 with 0≤𝛽<𝑎<𝛼+𝛽<𝑎+1 we have that ^𝑚_𝑘=𝑛1𝑘+1𝑚+1𝑘+1^𝑎Φ^𝑘_𝑚(𝑥)𝑣≤𝑁‖𝑣‖^𝑚_𝑘=𝑛1𝑘+1𝑚+1𝑘+1^𝑎(𝑘+1)^𝛼+𝛽(𝑚+1)^𝛼=𝑁‖𝑣‖(𝑚+1)^{𝑚𝑎−𝛼}_𝑘=𝑛1(𝑘+1)^{𝑎+1−𝛼−𝛽}≤𝑁‖𝑣‖(𝑚+1)^𝑎−𝛼1()𝛼+𝛽−𝑎𝑚+1^{𝛼+𝛽−𝑎}≤𝐷(𝑚+1)^𝑏‖𝑣‖ (29) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, where 𝐷=𝑁/(𝛼+𝛽−𝑎) and 𝑏=𝛽.
Sufficiency. From the hypothesis we have 1𝑛+1𝑚+1𝑛+1^𝑎Φ^𝑛_𝑚(𝑥)𝑣≤𝐵(𝑚+1)^𝑏‖𝑣‖ (30) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉. Hence (𝑚+1)^𝑎−𝑏Φ^𝑛_𝑚(𝑥)𝑣≤𝐵(𝑛+1)^𝑎+1‖𝑣‖, (31) and relation (16) from Lemma 9 holds for 0<𝑐=𝑎−𝑏≤𝑎<𝑎+1=𝑑.

Definition 13. An application 𝐿∶Δ×𝑋×𝑉→ℝ₊ is called a Lyapunov polynomial stability function for the system (𝐴,𝜑) if there exists 𝑙>0 such that 𝐿𝑚,𝑝,𝑥,𝑣+^𝑚−1_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑙Φ^𝑝_𝑘(𝑥)𝑣≤𝐿𝑛,𝑝,𝑥,𝑣 (32) for all (𝑚,𝑛,𝑝,𝑥,𝑣)∈𝑇×𝑋×𝑉, with 𝑚>𝑛.

The constant 𝑙>0 is called the order of the Lyapunov function 𝐿.

Theorem 14. If the system (𝐴,𝜑) is polynomially stable, then there are a Lyapunov polynomial stability function for the system (𝐴,𝜑) and constants 𝐾≥1, 𝜈>0 and 𝛿≥0 such that 𝐿(𝑚,𝑛,𝑥,𝑣)≤𝐾(𝑛+1)^𝛿‖𝑣‖,Φ^𝑛_𝑚(𝑥)𝑣≤𝐾(𝑚+1)^𝜈(𝑛+1)^𝛿−𝜈‖𝑣‖ (33) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Proof. From Theorem 10 we have that there are 𝐷≥1, 𝑑>0, 𝜔>0, and 𝛾≥0 such that ^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷(𝑛+1)^𝛾‖𝑣‖,Φ^𝑛_𝑚(𝑥)𝑣≤𝐷(𝑚+1)^𝜔(𝑛+1)^𝛾−𝜔‖𝑣‖ (34) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉. We define the application 𝐿∶Δ×𝑋×𝑉→ℝ₊ by 𝐿𝑚,𝑝,𝑥,𝑣=^∞_𝑘=𝑚1𝑘+1𝑘+1𝑚+1^𝑑Φ^𝑝_𝑘(𝑥)𝑣 (35) for all (𝑚,𝑝,𝑥,𝑣)∈Δ×𝑋×𝑉. Then, for all (𝑚,𝑛,𝑝,x,𝑣)∈𝑇×𝑋×𝑉, with 𝑚>𝑛 we have 𝐿𝑚,𝑝,𝑥,𝑣+^𝑚−1_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑝_𝑘(𝑥)𝑣=^∞_𝑘=𝑚1𝑘+1𝑘+1𝑚+1^𝑑Φ^𝑝_𝑘(𝑥)𝑣+^𝑚−1_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑝_𝑘(𝑥)𝑣≤^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑝_𝑘(𝑥)𝑣=𝐿𝑛,𝑝,𝑥,𝑣. (36) In addition, from Theorem 10 we have that 𝐿(𝑚,𝑛,𝑥,𝑣)=^∞_𝑘=𝑚1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷(𝑛+1)^𝛾‖𝑣‖,Φ^𝑛_𝑚(𝑥)𝑣≤𝐷(𝑚+1)^𝜔(𝑛+1)^𝛾−𝜔‖𝑣‖. (37) Consequently, relations (33) are satisfied for 𝐾=𝐷, 𝛿=𝛾 and 𝜈=𝜔.

Theorem 15. If there exist a Lyapunov polynomial stability function with the order 𝑙>0 for the system (𝐴,𝜑) and the constants 𝐾≥1, 𝜈>0, and 𝛿≥0 with 𝑙>𝜈>𝛿≥0 such that 𝐿(𝑚,𝑛,𝑥,𝑣)≤𝐾(𝑛+1)^𝛿‖𝑣‖,Φ^𝑛_𝑚(𝑥)𝑣≤𝐾(𝑚+1)^𝜈(𝑛+1)^𝛿−𝜈‖𝑣‖ (38) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, then the system (𝐴,𝜑) is polynomially stable.

Proof. From the hypothesis and Definition 13 we have that ^𝑚−1_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑙Φ^𝑛_𝑘(𝑥)𝑣()()()()≤𝐿𝑛,𝑛,𝑥,𝑣−𝐿𝑚,𝑛,𝑥,𝑣≤𝐿𝑛,𝑛,𝑥,𝑣≤𝐾𝑛+1^𝛿‖𝑣‖ (39) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, with 𝑚>𝑛. Passing to the limit for 𝑚→∞ we obtain that ^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑙Φ^𝑛_𝑘(𝑥)𝑣≤𝐾(𝑛+1)^𝛿‖𝑣‖ (40) for all (𝑛,𝑥,𝑣)∈ℕ×𝑋×𝑉. Now, from Theorem 11 the conclusion follows.

3. Strong Polynomial Stability

Definition 16. The system (𝐴,𝜑) is said to be strongly polynomially stable (and denoted as s.p.s.) if there are three constants 𝑁≥1 and 𝛼>𝛽≥0 such that (𝑚+1)^𝛼Φ^𝑛_𝑚(𝑥)𝑣≤𝑁(𝑛+1)^𝛼+𝛽‖𝑣‖ (41) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Remark 17. It is easy to see that (𝐴,𝜑) is strongly polynomially stable if and only if there are 𝑁≥1 and 𝛼>𝛽≥0 with (𝑚+1)^𝛼Φ^𝑝_𝑚(𝑥)𝑣≤𝑁(𝑛+1)^𝛼+𝛽Φ^𝑝_𝑛(𝑥)𝑣 (42) for all (𝑚,𝑛,𝑝,𝑥,𝑣)∈𝑇×𝑋×𝑉.

Remark 18. It is obvious that s.p.s.⇒p.s. (43)

The following example shows that the converse implication is not valid.

Example 19. Let (𝑋,𝑑) be the metric space, let 𝑉 be a Banach space, and let 𝜑 be the evolution semiflow given as in Example 8. We define the sequence of mappings 𝐴_𝑚∶𝑋→ℬ(ℝ) by 𝐴_𝑚(𝑥)𝑢(𝑚)𝑥(𝜏)𝑣=𝑢()𝑥()𝑣𝑚+1𝜏+1 (44) for all (𝑚,𝑥,𝑣)∈ℕ×𝑋×𝑉, where the sequence 𝑢∶ℕ→ℝ is given by 𝑢(𝑚)=(𝑚+1)𝑚+1−𝑚cos𝑚𝜋2. (45) Then Φ^𝑛_𝑚(𝑥)((𝑥(𝜏)𝑣=𝑛+1)(𝑛+1−𝑛cos𝑛𝜋/2))(𝑚+1)(𝑚+1−𝑚cos(𝑚𝜋/2))𝑥(𝑚−𝑛+𝜏)𝑣, (46) and it follows that Φ^𝑛_𝑚(𝑥)𝑣≤()()𝑥(0)2𝑛+1𝑛+1()𝑙‖𝑣‖𝑚+1≤𝑁𝑛+1(𝑣‖𝑚+1𝑛+1)‖ (47) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, where 𝑁=2𝑥(0)/𝑙. Hence we have proved that (𝐴,𝜑) is p.s. Let us suppose now that the system (𝐴,𝜑) is s.p.s. According to Definition 16, there exist 𝑁≥1 and 𝛼>𝛽≥0 such that (𝑛+1)(𝑛+1−𝑛cos(𝑛𝜋/2))𝑥(𝜏)(𝑚+1)(𝑚+1−𝑚cos(𝑚𝜋/2))𝑥(𝑚−𝑛+𝜏)≤𝑁𝑛+1𝑚+1^𝛼(𝑛+1)^𝛽 (48) for all (𝑚,n,𝑥)∈Δ×𝑋. If we take 𝑛=4𝑘+2 and 𝑚=4𝑘²+4, 𝑘∈ℕ, we have that 4𝑘+34𝑘²+5^1−𝛼8𝑘+5𝑥(𝜏)4𝑘+3𝑥4𝑘²()−4𝑘+2+𝜏≤𝑁4𝑘+3^𝛽−1. (49) It follows that 𝛼≤1, 𝛽≥1, and 𝑁≥2 which implies that 0<𝛼≤1≤𝛽, contradicting the fact that 𝛼>𝛽≥0. This proves that (𝐴,𝜑) is not s.p.s.

Lemma 20. The system (𝐴,𝜑) is strongly polynomially stable if and only if there are 𝑁≥1 and 0<𝑐≤𝑑<2𝑐 such that (𝑚+1)^𝑐Φ^𝑛_𝑚(𝑥)𝑣≤𝑁(𝑛+1)^𝑑‖𝑣‖ (50) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Proof . It is similar to the proof of Lemma 9 with the condition 0<𝛼=𝑐≤𝑑=𝛼+𝛽<2𝛼=2𝑐 in the case of the necessity and with 0≤𝛽=𝑑−𝑐<𝑐=𝛼 for the sufficiency.

Theorem 21. If the system (𝐴,𝜑) is strongly polynomially stable, then there are 𝐷≥1, 𝑑>0, 𝜔>0, and 𝛾≥0 such that ^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷(𝑛+1)^𝛾‖𝑣‖ (51) for all (𝑛,𝑥,𝑣)∈ℕ×𝑋×𝑉, Φ^𝑛_𝑚(𝑥)𝑣≤𝐷(𝑚+1)^𝜔(𝑛+1)^𝛾−𝜔‖𝑣‖ (52) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Proof. It results from the proof of Theorem 10 with 0≤𝛽<𝑑<𝛼 for the first inequality and with 0≤𝛽=𝛾<𝜔<𝑑<𝛼 for the second inequality.

Theorem 22. If there are 𝐷≥1 and 𝑑>𝜔>𝛾≥0 with 2𝛾+𝜔<𝑑 such that ^∞_𝑘=𝑛1𝑘+1𝑘+1𝑛+1^𝑑Φ^𝑛_𝑘(𝑥)𝑣≤𝐷(𝑛+1)^𝛾‖𝑣‖ (53) for all (𝑛,𝑥,𝑣)∈ℕ×𝑋×𝑉 and Φ^𝑛_𝑚(𝑥)𝑣≤𝐷(𝑚+1)^𝜔(𝑛+1)^𝛾−𝜔‖𝑣‖ (54) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, then the system (𝐴,𝜑) is strongly polynomially stable.

Proof. It is analogous to the proof of Theorem 11.

Next, we present a generalization of a theorem due to Lyapunov for the case of strong polynomial stability of discrete variational systems.

Theorem 23. If the system (𝐴,𝜑) is strongly polynomially stable, then there exist a Lyapunov polynomial stability function for the system (𝐴,𝜑) and constants 𝐾≥1 and 0≤𝛿<𝜈 such that 𝐿(𝑚,𝑛,𝑥,𝑣)≤𝐾(𝑛+1)^𝛿‖𝑣‖,Φ^𝑛_𝑚(𝑥)𝑣≤𝐾(𝑚+1)^𝜈(𝑛+1)^𝛿−𝜈‖𝑣‖ (55) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Proof. Using the technique from the proof of Theorem 14 for 0≤𝛾<𝜔<𝑑 we obtain the conclusion.

Theorem 24. If there are a Lyapunov polynomial stability function with the order 𝑙>0 for the system (𝐴,𝜑) and the constants 𝐾≥1, 𝜈>0 and 𝛿≥0 with 𝑙>𝜈>𝛿≥0 and 2𝛿+𝜈<𝑙 such that: 𝐿(m,𝑛,𝑥,𝑣)≤𝐾(𝑛+1)^𝛿‖𝑣‖,Φ^𝑛_𝑚(𝑥)𝑣≤𝐾(𝑚+1)^𝜈(𝑛+1)^𝛿−𝜈‖𝑣‖ (56) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, then the system (𝐴,𝜑) is strongly polynomially stable.

Proof. It is the same as the proof of Theorem 15.

Remark 25. If the system (𝐴,𝜑) is strongly polynomially stable, then there are 𝐵≥1 and 𝑎>𝑏≥0 such that ^𝑚_𝑘=𝑛1𝑘+1𝑚+1𝑘+1^𝑎Φ^𝑘_𝑚(𝑥)𝑣≤𝐵(𝑚+1)^𝑏‖𝑣‖ (57) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉.

Remark 26. If there are 𝐵≥1,  𝑎>0, and 𝑏≥0, with 𝑎>2𝑏+1 such that ^𝑚_𝑘=𝑛1𝑘+1𝑚+1𝑘+1^𝑎Φ^𝑘_𝑚(𝑥)𝑣≤𝐵(𝑚+1)^𝑏‖𝑣‖ (58) for all (𝑚,𝑛,𝑥,𝑣)∈Δ×𝑋×𝑉, then the system (𝐴,𝜑) is strongly polynomially stable.