On Barreira-Valls Polynomial Stability of Evolution Operators in Banach Spaces

Our main objective is to consider a concept of nonuniform behavior and obtain appropriate versions of the well-known stability due to R. Datko and L. Barbashin. This concept has been considered in the works of L. Barreira and C. Valls. Our approach is based on the extension of techniques for exponential stability to the case of polynomial stability.


Introduction
In the theory of differential equations both in finite-dimensional and infinite-dimensional spaces, there is a very extensive literature concerning uniform exponential stability.
For some of the most relevant early contributions in this area we refer to the books of J.L.Massera and J.J.Schäffer [9] and by J. Daletski and M.G.Krein [7].
In their notable contribution [2], L. Barreira and C. Valls obtain results in the case of a notion of nonuniform exponential dichotomy, which is motivated by ergodic theory.
A principal motivation for weakening the assumption of uniform exponential behavior is that from the point of view of ergodic theory, almost all linear variational equations in a finite-dimensional space admit a nonuniform exponential dichotomy.
In this paper we consider a concept of nonuniform stability for evolution operators in Banach spaces.This concept has been considered in the works [2] and [3] due to L. Barreira and C. Valls.This causes that the stability results discussed in the paper hold for a much larger class of differential equations than in the classical theory of uniform exponential stability.

Evolution operators
In this section we recall some definitions which will be used in what follows.
Let X be a real or complex Banach space and let I be the identity operator on X.The norm on X and on B(X) the Banach algebra of all bounded linear operators on X , will be denoted by We recall that an operator-valued function Φ : ∆ → B(X) is called an evolution operator on the Banach spaces X iff: e 1 )Φ (t, t) = I for every t ≥ 0; e 2 )Φ (t, s) Φ (s, t 0 ) = Φ (t, t 0 ) for all (t, s)and (s, t 0 ) ∈ ∆.

Remark 2.1
In the examples considered in this paper we consider evolution operators on X defined by where u : R + → R * + = (0, ∞).An evolution operator Φ : ∆ → B(X) with the property e 3 ) there exists a nondecreasing function ϕ : R + → [1, ∞) such that: Φ(t, s) ≤ ϕ(t − s) for all (t, s) ∈ ∆ then Φ is called the evolution operator with uniform growth.
The evolution operator Φ : ∆ → B(X) is said to be strongly measurable, iff e 4 ) for all (s, x) ∈ R + × X the mapping defined by t → Φ (t, s) x is measurable on [s, ∞).

Remark 3.2
The evolution operator Φ is : (i) uniformly polynomially stable iff there are N ≥ 1 and α > 0 such that for all (t, s, t 0 , x 0 ) ∈ T × X; (ii) (nonuniformly) polynomially stable iff there exist α > 0 and a nondecreasing function for every (t, s, t 0 , x 0 ) ∈ T × X; (iii) polynomially stable in the sense of Barreira and Valls iff there are N ≥ 1, α > 0 and β ≥ α such that: The converse implications between these stability concepts are not valid.This is proved in the following two examples.
The following example shows an evolution operator that is B.V.p.s which is not u.p.s. .Then x is an evolution operator on X with: If we suppose that Φ is u.p.s then there exist N ≥ 1 and α > 0 such that: for all (t, s) ∈ ∆.
Then for s = n and t = n + 1 n 2 we obtain which for n → ∞ gives a contradiction and hence Φ is not B.V.p.s.
Theorem 3.1 Let Φ : ∆ → B(X) be a strongly measurable evolution operator with uniform growth.If there are D ≥ 1, γ > 0 and δ ≥ 0 such that: for all (t, s, x) ∈ ∆×X, then Φ is polynomially stable in the sense of Barreira and Valls.
Remark 3.4 Theorem 3.1 is a generalization for the case of polynomial stability in the sense of Barreira and Valls of the classic result proved by R.Datko in Theorem 11 of [8] for the case of uniform exponential stability.The case of exponential stability has been considered by Buse in [4].
Remark 3.5 The converse of the preceding theorem is valid in the case when the constant α given by Definition 3.1 (iii) satisfies the condition α > 1 and we consider 0 < γ < α − 1 and δ = β + 1.
For the case when α ∈ (0, 1) and β > 0 the converse of Theorem 3.1 is not valid, result illustrated by Example 3.3 The evolution operator for all t ≥ s ≥ t 0 = 1 and all x ∈ X. Hence Φ is B.V.p.s. with α = 1 3 ∈ (0, 1) and β = 1 2 .We observe that Some immediate characterizations of the polynomial stability in the sense of Barreira and Valls are given by: Proposition 3.1 Let Φ : ∆ → B(X) be an evolution operator.The following statements are equivalent: (i) Φ is polynomially stable in the sense of Barreira and Valls; (ii) there are N ≥ 1 , ν > 0 and β ∈ [0, ν) such that: for all (t, s, x) ∈ ∆ × X; (iii) there are N ≥ 1, a, b > 0 and b ≥ a such that: for all (t, s, x) ∈ ∆ × X.
Corollary 3.1 Let Φ : ∆ → B(X) be an evolution operator.The following statements are equivalent: (i) Φ is polynomially stable in the sense of Barreira and Valls; (ii) there are N ≥ 1 , ν > 0 and β ∈ [0, ν) such that: for all (t, s, x 0 ) ∈ ∆ × X; (iii) there are N ≥ 1, a, b > 0 and b ≥ a such that: for all (t, s, x 0 ) ∈ ∆ × X. Proof where B = 1 + N α−b+1 and δ = β + 1. Sufficiency.If t ≥ s + 1 and s ≥ 0 then: Remark 3.6 Theorem 3.2 is a generalization for the case of polynomial stability in the sense Barreira-Valls of a classic result due to Barbashin [1](see also [5] and [13]) for uniform exponential stability.The case of exponential stability has been considered by Buse in [6].