A Perron-type Theorem for Nonautonomous Difference Equations with Nonuniform Behavior

We show that if the Lyapunov exponents of a linear difference equation are limits, then the same happens with the Lyapunov exponents of the solutions of the nonlinear equations for any sufficiently small nonlinear perturbations. We consider the case with a very general nonuniform behavior, which is called nonuniform (h, k, µ, ν) behavior.

For example, e am , m a + b, m a e bm , m a log(b + m) with a, b > 0 are growth rates.Given a growth rate h, we can define h-Lyapunov exponent λ : C n → R ∪ {−∞} associated with the linear difference equation where x ∈ C n , (A m ) m∈N is a sequence of n × n invertible matrices with complex entries such that sup m∈N A m < +∞, B Corresponding author.Email: hai-long-zhu@163.com and A is the cocycle produced by (A m ) m∈N , that is In this paper, we show that if all h-Lyapunov exponents of (1.1) are limits, the asymptotic behavior of (1.1) persists under sufficiently small perturbations for the nonlinear equation where the perturbation f m : C m → C m is continuous and small enough.More precisely, if the sequence (1.3) is not eventually zero, the limit exists and coincides with an h-Lyapunov exponent of (1.1).The required smallness of the perturbation is that or simply for some δ 1 , δ 2 > 0, where µ, ν are two given growth rates.When µ(m) = ν(m) = e m , we recover the result in [9] and (1.f m (x) x < +∞, for some δ > 0.
In the literature, the results related to the above problems are called "Perron-type theorems".For the case A m = A being constant, the results were proved by Coffman [13].A related result for perturbations of a differential equation x = Ax with constant coefficient can be found in the book [14].More results can be found in [15-19, 22, 23].Recently, Barreira and Valls established the Perron-type theorems for nonautonomous differential equations [8] and nonautonomous difference equations [7,9,10], based on Lyapunov's theory of regularity.Especially, they considered the cases with nonuniform exponential behavior.In this paper, we will follow the ideas of Barreira and Valls.Such problems are also very close to the theory of nonuniform exponential dichotomies, which was inspired both by the classical notion of exponential dichotomy and by the notion of nonuniformly hyperbolic trajectory introduced by Barreira and Pesin (see [3]), and have been developed in a systematic way by Barreira and Valls (see [4][5][6] and the references therein) during the last several years.As explained by Barreira and Valls, in comparison to the notion of exponential dichotomies introduced by Perron in [21], nonuniform exponential dichotomy is a useful and weaker notion.A very general type of nonuniform exponential dichotomy, the so-called (µ, ν) exponential dichotomy, has been considered in [1,2,11,12].
Compared with those results in the literature, the novelty of this work is that we establish the Perron-type theorem for nonautonomous difference equations with different growth rates in the uniform parts and nonuniform parts.More precisely, we consider the (h, k, µ, ν) nonuniform behavior and this creates additional complications in the analysis.We refer the reader to [20] for some results on the so-called (h, k)-dichotomies, which were introduced by Pinto.

Preliminaries
Given a growth rate h and consider a sequence (A m ) m∈N of invertible n × n matrices with complex entries such that lim sup The h-Lyapunov exponent λ : ) is defined by the formula (1.2), with the convention that log 0 = −∞ to illustrate the value λ(0) = −∞.It follows from (2.1) that λ never takes the value +∞.By the general theory of Lyapunov exponents (see [3] for details), we know that the Lyapunov exponent λ can take on only finitely many distinct values −∞ ≤ λ 1 < • • • < λ p , where p ≤ n.Furthermore, for each 1 ≤ i ≤ p, we define as a linear subspace over C n (with the convention that E 0 = {0}).Obviously, We set Now we describe the assumptions in the paper.
(H1) There exist decompositions Thus for a given number b ∈ R which is not a h-Lyapunov exponent, there exist a decomposition ) , where (H2) Take a < b < c such that the interval [a, c] contains no Lyapunov exponent and a given constant ε > 0, there exists a constant and in which P l and Q l are the projections associated with the decomposition (2.2) and h, k, µ, ν are growth rates.
(H3) The growth rates h, k, µ, ν satisfy and h, k satisfy the compensation condition: there exists a constant 0 4), we can obtain Moreover, for every m, l ∈ N, we have The compensation condition in H3 is very important in our analysis.For the uniform (h, k) behavior, [20, Section VI] illustrate importance of "h and k are compensated".
In Section 4, we will give two explicit examples of sequences (A m ) m∈N which satisfy assumptions (H1)-(H3).

Main results
The following is our main result.It claims that under sufficiently small perturbations, the Lyapunov exponent of (1.3) coincides with some Lyapunov exponent of the unperturbed difference equation (1.1).Theorem 3.1.Let (x m ) m∈N be a sequence satisfying (1.3) and where the sequence γ m satisfies for some δ 1 , δ 2 ≥ ε > 0 and two growth rates µ, ν are given in (H2).Assume that conditions (H1)-(H3) are satisfied.Then one of the following alternatives hold: (1) x m = 0 for all sufficiently large m; (2) the limit exists and coincides with a Lyapunov exponent of (1.1).
Before presenting the proof of Theorem 3.1, we prove several lemmas.

Lemma 3.2.
There exists a constant K > 0 such that for every m, l ∈ N with m ≥ l and d > λ p .In particular, given r ∈ N there exists for all l ≤ sr ≤ m ≤ (s + 1)r.
Proof.For each m ≥ l, (1.3) has a solution x m which can be written as Then by (3.1) and (3.5), we obtain and hence,

One can use induction to show that
Therefore, by using (3.2), we know that property (3.3) holds with In particular, (3.3) implies that property (3.4) holds with This completes the proof of the lemma.
Let b ∈ R be a number that is not an h-Lyapunov exponent.Let also a < b < c be as in Section 2. We consider the norm for each m ∈ N and x ∈ C n .We have and one can easily verify that and Similarly, for m ≥ l we have This completes the proof of the lemma.Now let (x m ) m∈N be a sequence satisfying (1.3).Using the decomposition in (2.2), we can write x m = y m + z m , where Applying P m and Q m to both sides of (3.5) and using (2.6), we obtain, and Using (2.5), (3.4), (3.6), (3.7) and (3.8), it follows from (H3) that for m ≤ (s + 1)r, with with By (3.2), provided that τ is sufficiently small.Under the assumption (H3), it is easy to see that This shows that (3.28) holds.Thus, we show that if (3.23) fails, then (3.24) holds.As a consequence, we have the following two cases.
Case 2. Now we assume that (3.24) holds.We show that (3.13) and (3.14) hold.Given τ > 0, there exists s 0 such that τ 1 s , τ 2 s < τ and y sr sr < z sr sr for s ≥ s 0 .By (3.26), we find that for s ≥ s 0 , z (s+1)r (s+1)r ≥ (α s − 2Dτ) z sr sr , which implies that Together with (3.4), (3.7) and (3.8), this yields that for for s ≥ s 0 and sr ≤ m ≤ (s + 1)r, Under the assumptions (H2), thus we have Since τ is arbitrary and provided that ε is sufficiently small, we obtain This establishes (3.13).Now we prove (3.14).We define Proof of Theorem 3.1.Let (x m ) m∈N be a sequence satisfying the hypotheses of Theorem 3.1.If x k = 0 for some k , then it follows from (3.3) that x k = 0 for all k ≥ k , and hence, the first alternative in the theorem holds.Now we assume that x k = 0 for all k ≥ k .Let λ 1 < • • • < λ p be the Lyapunov exponents of the sequence (A m ) m∈N .On both sides of λ i , take real numbers b j such that

Examples
In this section, we present the following examples which will show the (h, k, µ, ν)-dichotomies.
To show the difference with different values of h, k, µ and ν, we follow the ideas of Naulin and Pinto in [20].In order to make precise statements, we first introduce some notations and concepts for difference equations.Now, we introduce the sequence spaces which equipped with the norm It is easy to see that the spaces (l h , • h ) and (l h,0 , • h ) are Banach spaces.Let V h be the subspace of C m defining by the following property: if ξ ∈ V h , then the solution of the linear difference system (1.1) with initial condition x 0 = ξ belongs to l h .Analogously we introduce the subspace V h,0 of the initial conditions by the following property: if ξ ∈ V h,0 , then the solution of the linear difference system (1.1) with initial condition x 0 = ξ belongs to l h,0 .
Following the ideas of [20] (see also Chapter 2 in [15]), we consider the nonuniform behavior, and then we have the following property: and and (H1) is fulfilled, then where P : C m → C m is a projection such that PP = P. Now two linear difference systems, which admit (h, k, µ, ν)-dichotomies but does not admit (h, k)-dichotomies, will be given to illustrate the relation of V h,0 , V k,0 , P[C m ], V h and V k .
Example 4.1.Now we consider the system and define the projection matrices Besides, it is easy to verify that the nonuniform part can not be removed, see [24] for details.Thus we can list the following (h, k, µ, ν)-dichotomies: D1'.With projection P = P 1 the system (4.

Lemma 3 . 4 . 1 ∑
Let b ∈ R be a number that is not an h-Lyapunov exponent, then one of the following alternatives holds: For m ≥ sr we have y m = A(m, sr)P sr x sr + m−j=sr A(m, j + 1)P j+1 f j (x j ) (3.15)andz m = A(m, sr)Q sr x sr + m−1 ∑ j=sr A(m, j + 1)Q j+1 f j (x j ).(3.16)By (3.8) and (3.10), it follows from (3.16) that for m ≥ sr z m m ≥ A(m, sr)Q kr x sr m − m−1 + 1) ε γ j .

1 .
and b p > λ p .Applying Lemma 3.4 to each number b = b j , we conclude that there exists j ∈ {1, . . ., Considering h(m) = k(m) and letting b j λ j and b j−1 λ j , we find that lim m→+∞ log x m log h(m) = λ j .Now the proof is finished.