# On Polynomial Stability of Variational Nonautonomous Difference Equations in Banach Spaces

Authors: Mihail Megan; Traian Ceauşu; Mihaela Aurelia Tomescu
Journal: International Journal of Analysis
 (2013)

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## 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.


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## Body

## 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 setsΔ={(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 mappingφ: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(Am)m∈ℕ with Am:X→ℬ(V) and a discrete evolution semiflow φ:Δ×X→X, we consider the problem of existence of a sequence (vm)m∈ℕ with vm:ℕ×X→X such that
(1)vm+1(n,x)=Am(φ(m,n,x))vm(n,x)
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 Φmn:X→ℬ(V) by
(2)Φmn(x)v={Am-1(φ(m-1,n,x))⋯An+1(φ(n+1,n,x))An(x)v,ifm>nv,ifm=n.Remark 2.
From the definitions ofvm and Φmn it follows that(c1)
Φmm(x)v=v, for all (m,x,v)∈ℕ×X×V;(c2)
Φmp(x)=Φmn(φ(n,p,x))Φnp(x), for all (m,n,p,x)∈T×X;(c3)
vm(n,x)=Φmn(x)vn(n,x), for all (m,n,x)∈Δ×X.Definition 3.
A mappingΦ: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.
IfΦ 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 Remark2 it results that the mapping
(3)Φ:Δ×X→ℬ(V),Φ(m,n,x)v=Φmn(x)v
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=(Am), where Am: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 constantsN≥1, α>0 and β≥0, such that
(4)eα(m-n)∥Φmn(x)v∥≤Neβn∥v∥for all(m,n,x,v)∈Δ×X×V;(ii)
polynomially stable (and denoted as p.s.) if there exist the constantsN≥1, α>0 and β≥0 such that
(5)(m+1)α∥Φmn(x)v∥≤N(n+1)α+β∥v∥for all(m,n,x,v)∈Δ×X×V.Remark 6.
The system(A,φ) is(i)
exponentially stable if and only if there areN≥1, α>0, and β≥0 with
(6)eα(m-n)∥Φmp(x)v∥≤Neβn∥Φnp(x)v∥for all(m,n,p,x,v)∈T×X×V;(ii)
polynomially stable if and only if there existN≥1, α>0, and β≥0 with
(7)(m+1)α∥Φmp(x)v∥≤N(n+1)α+β∥Φnp(x)v∥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(8)e.s.⇒p.s.The following example shows that the converse implication is not valid.Example 8.
Let𝒞=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)=supt∈ℝ+|x1(t)-x2(t)|. Let f:ℝ+→(0,∞) be a bounded decreasing function with the property that there exists limt→∞f(t)=l>0. We denote by X the closure in 𝒞 of the set {ft,t∈ℝ+}, where ft(s)=f(t+s) for all s∈ℝ+. The mapping φ:Δ×X→X defined by φ(m,n,x)=xm-n is a discrete evolution semiflow. Let us consider the Banach space V and let the sequence of mappings Am:X→ℬ(V), defined by
(9)Am(x)v=u(m)x(τ)u(m+1)x(τ+1)v
for all (m,x,v)∈ℕ×X×V, where the sequence u:ℕ→ℝ is given by
(10)u(m)=(m+1)(m+1-mcosmπ2)2.
Then
(11)Φmn(x)v=(n+1)(n+1-ncos(nπ/2))2x(τ)(m+1)(m+1-mcos(mπ/2))2x(m-n+τ)v,
and it results that
(12)∥Φmn(x)v∥≤(2n+1)2(n+1)x(0)(m+1)l∥v∥≤Nn+1m+1(n+1)2∥v∥
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
(13)eα(m-n)∥Φmn(x)v∥≤Neβn∥v∥
for all (m,n,x,v)∈Δ×X×V. The previous inequality for the considered system becomes
(14)eα(m-n)×(n+1)(n+1-ncos(nπ/2))2x(τ)(m+1)(m+1-mcos(mπ/2))2x(m-n+τ)≤Neβn
for all (m,n,x)∈Δ×X. If we take m=4k2+4 and n=4k+2, k∈ℕ, then
(15)e4αk2+k(α-β-2)+2(4k+3)(8k+5)24k2+5×x(τ)x(4k2-4k+2+τ)≤Ne2β.
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
(16)(m+1)c∥Φmn(x)v∥≤N(n+1)d∥v∥
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
(17)(m+1)α∥Φmn(x)v∥≤N(n+1)α+β∥v∥
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
(18)∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D(n+1)γ∥v∥
for all (n,x,v)∈ℕ×X×V and
(19)∥Φmn(x)v∥≤D(m+1)ω(n+1)γ-ω∥v∥
for all (m,n,x,v)∈Δ×X×V.Proof.
LetN≥1, α>0, and β≥0 as in Definition 5. Then, for every d∈(0,α) we have that
(20)∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤N(n+1)α+β-d∥v∥∑k=n∞‍1(k+1)α+1-d=N∥v∥(n+1)α+β-d(1(n+1)α+1-d+∑k=n+1∞‍1(k+1)α+1-d)≤N∥v∥(n+1)α+β-d(1(n+1)α+1-d+1α-d1(n+1)α-d)≤N∥v∥(n+1)α+β-d(1+1α-d)1(n+1)α-d=N(1+α-d)α-d∥v∥(n+1)β≤D∥v∥(n+1)γ
for all (n,x,v)∈ℕ×X×V, where D=N(1+α-d)/(α-d) and γ=β. In addition
(21)∥Φmn(x)v∥≤N(n+1)α+β(m+1)α∥v∥=N(m+1n+1)ω(n+1m+1)α+ω(n+1)β∥v∥≤D(m+1n+1)ω(n+1)γ∥v∥
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 areD≥1 and d>ω>γ≥0 such that
(22)∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D(n+1)γ∥v∥
for all (n,x,v)∈ℕ×X×V and
(23)∥Φmn(x)v∥≤D(m+1)ω(n+1)γ-ω∥v∥
for all (m,n,x,v)∈Δ×X×V, then the system (A,φ) is polynomially stable.Proof.
From the hypothesis it results that(24)∥Φmn(x)v∥≤D(m+1)ω(k+1)γ-ω∥Φkn(x)v∥
for all (m,k,n,x,v)∈T×X×V. We suppose that m≥2n and we denote by j=[m/2]. Then
(25)m+22(m+1)d-ω(n+1)d-ω+γ∥Φmn(x)v∥≤∑k=jm‍(m+1)d-ω(n+1)d-ω+γ∥Φmn(x)v∥≤D∑k=jm‍(m+1)d-ω(n+1)d-ω+γ(m+1k+1)ω×(k+1)γ∥Φkn(x)v∥=D∑k=jm‍1k+1(k+1n+1)d∥Φkn(x)v∥×(m+1k+1)d(n+1k+1)ω-γ(k+1)≤D4d(m+1)∑k=jm‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D4d(m+1)∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D24d(m+1)(n+1)γ∥v∥.
Hence,
(26)(m+1n+1)d-ω∥Φmn(x)v∥≤D24d2(m+1)m+2(n+1)2γ∥v∥≤D222d+1(n+1)2γ∥v∥=N(n+1)2γ∥v∥
for all (x,v)∈X×V, where N=D222d+1. If n≤m<2n, then
(27)(m+1n+1)d-ω∥Φmn(x)v∥≤D(m+1n+1)d-ω(m+1n+1)ω(n+1)γ∥v∥≤D2d(n+1)γ∥v∥≤N(n+1)2γ∥v∥
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
(28)∑k=nm‍1k+1(m+1k+1)a∥Φmk(x)v∥≤B(m+1)b∥v∥
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
(29)∑k=nm‍1k+1(m+1k+1)a∥Φmk(x)v∥≤N∥v∥∑k=nm‍1k+1(m+1k+1)a(k+1)α+β(m+1)α=N∥v∥(m+1)a-α∑k=nm‍1(k+1)a+1-α-β≤N∥v∥(m+1)a-α1α+β-a(m+1)α+β-a≤D(m+1)b∥v∥
for all (m,n,x,v)∈Δ×X×V, where D=N/(α+β-a) and b=β.
Sufficiency. From the hypothesis we have
(30)1n+1(m+1n+1)a∥Φmn(x)v∥≤B(m+1)b∥v∥
for all (m,n,x,v)∈Δ×X×V. Hence
(31)(m+1)a-b∥Φmn(x)v∥≤B(n+1)a+1∥v∥,
and relation (16) from Lemma 9 holds for 0<c=a-b≤a<a+1=d.Definition 13.
An applicationL:Δ×X×V→ℝ+ is called a Lyapunov polynomial stability function for the system (A,φ) if there exists l>0 such that
(32)L(m,p,x,v)+∑k=nm-1‍1k+1(k+1n+1)l∥Φkp(x)v∥≤L(n,p,x,v)
for all (m,n,p,x,v)∈T×X×V, with m>n.The constantl>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
(33)L(m,n,x,v)≤K(n+1)δ∥v∥,∥Φmn(x)v∥≤K(m+1)ν(n+1)δ-ν∥v∥
for all (m,n,x,v)∈Δ×X×V.Proof.
From Theorem10 we have that there are D≥1, d>0, ω>0, and γ≥0 such that
(34)∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D(n+1)γ∥v∥,∥Φmn(x)v∥≤D(m+1)ω(n+1)γ-ω∥v∥
for all (m,n,x,v)∈Δ×X×V. We define the application L:Δ×X×V→ℝ+ by
(35)L(m,p,x,v)=∑k=m∞‍1k+1(k+1m+1)d∥Φkp(x)v∥
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
(36)L(m,p,x,v)+∑k=nm-1‍1k+1(k+1n+1)d∥Φkp(x)v∥=∑k=m∞‍1k+1(k+1m+1)d∥Φkp(x)v∥+∑k=nm-1‍1k+1(k+1n+1)d∥Φkp(x)v∥≤∑k=n∞‍1k+1(k+1n+1)d∥Φkp(x)v∥=L(n,p,x,v).
In addition, from Theorem 10 we have that
(37)L(m,n,x,v)=∑k=m∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D(n+1)γ∥v∥,∥Φmn(x)v∥≤D(m+1)ω(n+1)γ-ω∥v∥.
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 orderl>0 for the system (A,φ) and the constants K≥1, ν>0, and δ≥0 with l>ν>δ≥0 such that
(38)L(m,n,x,v)≤K(n+1)δ∥v∥,∥Φmn(x)v∥≤K(m+1)ν(n+1)δ-ν∥v∥
for all (m,n,x,v)∈Δ×X×V, then the system (A,φ) is polynomially stable.Proof.
From the hypothesis and Definition13 we have that
(39)∑k=nm-1‍1k+1(k+1n+1)l∥Φkn(x)v∥≤L(n,n,x,v)-L(m,n,x,v)≤L(n,n,x,v)≤K(n+1)δ∥v∥
for all (m,n,x,v)∈Δ×X×V, with m>n. Passing to the limit for m→∞ we obtain that
(40)∑k=n∞‍1k+1(k+1n+1)l∥Φkn(x)v∥≤K(n+1)δ∥v∥
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
(41)(m+1)α∥Φmn(x)v∥≤N(n+1)α+β∥v∥
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
(42)(m+1)α∥Φmp(x)v∥≤N(n+1)α+β∥Φnp(x)v∥
for all (m,n,p,x,v)∈T×X×V.Remark 18.
It is obvious that(43)s.p.s.⇒p.s.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 Am:X→ℬ(ℝ) by
(44)Am(x)v=u(m)x(τ)u(m+1)x(τ+1)v
for all (m,x,v)∈ℕ×X×V, where the sequence u:ℕ→ℝ is given by
(45)u(m)=(m+1)(m+1-mcosmπ2).
Then
(46)Φmn(x)v=(n+1)(n+1-ncos(nπ/2))x(τ)(m+1)(m+1-mcos(mπ/2))x(m-n+τ)v,
and it follows that
(47)∥Φmn(x)v∥≤(2n+1)(n+1)x(0)(m+1)l∥v∥≤Nn+1m+1(n+1)∥v∥
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
(48)(n+1)(n+1-ncos(nπ/2))x(τ)(m+1)(m+1-mcos(mπ/2))x(m-n+τ)≤N(n+1m+1)α(n+1)β
for all (m,n,x)∈Δ×X. If we take n=4k+2 and m=4k2+4, k∈ℕ, we have that
(49)(4k+34k2+5)1-α8k+54k+3x(τ)x(4k2-4k+2+τ)≤N(4k+3)β-1.
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
(50)(m+1)c∥Φmn(x)v∥≤N(n+1)d∥v∥
for all (m,n,x,v)∈Δ×X×V.Proof .
It is similar to the proof of Lemma9 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
(51)∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D(n+1)γ∥v∥
for all (n,x,v)∈ℕ×X×V,
(52)∥Φmn(x)v∥≤D(m+1)ω(n+1)γ-ω∥v∥
for all (m,n,x,v)∈Δ×X×V.Proof.
It results from the proof of Theorem10 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 areD≥1 and d>ω>γ≥0 with 2γ+ω<d such that
(53)∑k=n∞‍1k+1(k+1n+1)d∥Φkn(x)v∥≤D(n+1)γ∥v∥
for all (n,x,v)∈ℕ×X×V and
(54)∥Φmn(x)v∥≤D(m+1)ω(n+1)γ-ω∥v∥
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 Theorem11.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
(55)L(m,n,x,v)≤K(n+1)δ∥v∥,∥Φmn(x)v∥≤K(m+1)ν(n+1)δ-ν∥v∥
for all (m,n,x,v)∈Δ×X×V.Proof.
Using the technique from the proof of Theorem14 for 0≤γ<ω<d we obtain the conclusion.Theorem 24.
If there are a Lyapunov polynomial stability function with the orderl>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:
(56)L(m,n,x,v)≤K(n+1)δ∥v∥,∥Φmn(x)v∥≤K(m+1)ν(n+1)δ-ν∥v∥
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 Theorem15.Remark 25.
If the system(A,φ) is strongly polynomially stable, then there are B≥1 and a>b≥0 such that
(57)∑k=nm‍1k+1(m+1k+1)a∥Φmk(x)v∥≤B(m+1)b∥v∥
for all (m,n,x,v)∈Δ×X×V.Remark 26.
If there areB≥1,  a>0, and b≥0, with a>2b+1 such that
(58)∑k=nm‍1k+1(m+1k+1)a∥Φmk(x)v∥≤B(m+1)b∥v∥
for all (m,n,x,v)∈Δ×X×V, then the system (A,φ) is strongly polynomially stable.



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*Source: 407958-2013-03-27.xml*