#### 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 and , a metric space and a real or complex Banach space. The norm on and on (the Banach algebra of all bounded linear operators on ) will be denoted by .

Definition 1. A mapping is called a discrete evolution semiflow on if the following conditions hold:  , for all ;  , for all .

Given a sequence with and a discrete evolution semiflow , we consider the problem of existence of a sequence with such that for all . We will denote this problem by and we say that is a variational (nonautonomous) discrete-time system.

For we define the application by

Remark 2. From the definitions of and it follows that  , for all ;  , for all ;  , for all .

Definition 3. A mapping is called a discrete evolution cocycle over the discrete evolution semiflow if the following properties hold:   (the identity operator on ), for all ,  , for all .
If is a discrete evolution cocycle over the discrete evolution semiflow , then the pair is called a discrete skew-evolution semiflow on .

Remark 4. From Remark 2 it results that the mapping is a discrete evolution cocycle over discrete evolution semiflow .

#### 2. Polynomial Stability

Let be a discrete variational system associated with the discrete evolution semiflow and with the sequence of mappings , where , for all .

Definition 5. The system is said to be  exponentially stable (and denoted as e.s.) if there exist the constants , and , such that for all ;  polynomially stable (and denoted as p.s.) if there exist the constants , and such that for all .

Remark 6. The system is  exponentially stable if and only if there are , , and with for all ;  polynomially stable if and only if there exist , , and with for all .

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

Remark 7. It is obvious that

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

Example 8. Let be the metric space of all bounded continuous functions , with the topology of uniform convergence. is metrizable with respect to the metric . Let be a bounded decreasing function with the property that there exists . We denote by the closure in of the set , where for all . The mapping defined by is a discrete evolution semiflow. Let us consider the Banach space and let the sequence of mappings , defined by for all , where the sequence is given by Then and it results that for all , where . Hence is p.s. Assume by a contradiction that is e.s. According to Definition 5, there are , , and such that for all . The previous inequality for the considered system becomes for all . If we take and , , then Passing to the limit for we obtain a contradiction. We have shown that is not e.s.

Lemma 9. The system is polynomially stable if and only if there are and such that for all .

Proof. Necessity. If is p.s., then there are , , and such that for all . Hence inequality (16) holds for and .
Sufficiency. From the hypothesis it results that relation (5) of Definition 5 holds for and .

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

Theorem 10. If the system is polynomially stable, then there are , , , and such that for all and for all .

Proof. Let , , and as in Definition 5. Then, for every we have that for all , where and . In addition for all .

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

Theorem 11. If there are and such that for all and for all , then the system is polynomially stable.

Proof. From the hypothesis it results that for all . We suppose that and we denote by . Then Hence, for all , where . If , then for all thus we have proved that 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 is polynomially stable if and only if there are and such that for all .

Proof. Necessity. Let , , and as in Definition 5. Then, for every with we have that for all , where and .
Sufficiency. From the hypothesis we have for all . Hence and relation (16) from Lemma 9 holds for .

Definition 13. An application is called a Lyapunov polynomial stability function for the system if there exists such that for all , with .

The constant is called the order of the Lyapunov function .

Theorem 14. If the system is polynomially stable, then there are a Lyapunov polynomial stability function for the system and constants , and such that for all .

Proof. From Theorem 10 we have that there are , , , and such that for all . We define the application by for all . Then, for all , with we have In addition, from Theorem 10 we have that Consequently, relations (33) are satisfied for , and .

Theorem 15. If there exist a Lyapunov polynomial stability function with the order for the system and the constants , , and with such that for all , then the system is polynomially stable.

Proof. From the hypothesis and Definition 13 we have that for all , with . Passing to the limit for we obtain that for all . Now, from Theorem 11 the conclusion follows.

#### 3. Strong Polynomial Stability

Definition 16. The system is said to be strongly polynomially stable (and denoted as s.p.s.) if there are three constants and such that for all .

Remark 17. It is easy to see that is strongly polynomially stable if and only if there are and with for all .

Remark 18. It is obvious that

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

Example 19. Let be the metric space, let be a Banach space, and let be the evolution semiflow given as in Example 8. We define the sequence of mappings by for all , where the sequence is given by Then and it follows that for all , where . Hence we have proved that is p.s. Let us suppose now that the system is s.p.s. According to Definition 16, there exist and such that for all . If we take and , , we have that It follows that , , and which implies that , contradicting the fact that . This proves that is not s.p.s.

Lemma 20. The system is strongly polynomially stable if and only if there are and such that for all .

Proof . It is similar to the proof of Lemma 9 with the condition in the case of the necessity and with for the sufficiency.

Theorem 21. If the system is strongly polynomially stable, then there are , , , and such that for all , for all .

Proof. It results from the proof of Theorem 10 with for the first inequality and with for the second inequality.

Theorem 22. If there are and with such that for all and for all , then the system 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 is strongly polynomially stable, then there exist a Lyapunov polynomial stability function for the system and constants and such that for all .

Proof. Using the technique from the proof of Theorem 14 for we obtain the conclusion.

Theorem 24. If there are a Lyapunov polynomial stability function with the order for the system and the constants , and with and such that: for all , then the system is strongly polynomially stable.

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

Remark 25. If the system is strongly polynomially stable, then there are and such that for all .

Remark 26. If there are ,  , and , with such that for all , then the system is strongly polynomially stable.