Non-almost periodic solutions of limit periodic and almost periodic homogeneous

Non-almost periodic solutions of limit periodic and almost periodic homogeneous

linear difference systems

Martin Chvátal

Masaryk University, Faculty of Science, Department of Mathematics and Statistics, Kotláˇrská 2, CZ-611 37 Brno, Czech Republic

Received 14 October 2014, appeared 15 January 2015 Communicated by Stevo Stevi´c

Abstract. We study limit periodic and almost periodic homogeneous linear difference systems. The coefficient matrices of the considered systems are taken from a given commutative group. We mention a condition on the group which ensures that, by arbitrarily small changes, the considered systems can be transformed to new systems, which do not possess any almost periodic solution other than the trivial one. The elements of the coefficient matrices are taken from an infinite field with an absolute value.

Keywords: limit periodicity, almost periodicity, almost periodic sequences, almost pe- riodic solutions, limit periodic sequences, linear difference systems.

2010 Mathematics Subject Classification: 39A06, 39A10, 39A24, 42A75.

1 Introduction

In this paper, for a commutative group X of square matrices over a field, we analyse the homogeneous linear difference systems

x_k+1= A_kx_k, k∈_Z, (1.1)

where {A_k}_k∈Z ⊆ X. We consider the case, when the sequence{A_k}_k∈Z is limit periodic or almost periodic. We continue in the research based on the results of papers [8,9,18,22,24].

In [18] (see also [16]), the unitary systems of the form (1.1) are considered. One of the main results of [18] says that the systems with non-almost periodic solutions form a dense subset of the space of all unitary systems. If one is interested in orthogonal difference systems and skew-Hermitian and skew-symmetric differential systems, the corresponding result can be found in [19], [21], and [23], respectively. Concerning almost periodic solutions of these systems, we refer to [12,13,17] as well.

In [8,22], general almost periodic systems (1.1) are examined. There are found groups of matrices such that the homogeneous linear difference systems without any non-trivial almost

BEmail: chvatal.m@mail.muni.cz

periodic solution form a dense subset of the set of all considered systems. Transformable and strongly transformable groups of matrices are introduced. Based on this concept, the above-mentioned result of [18] is generalized for other matrix groups.

In papers [9,24], the limit periodic systems of the form (1.1) are investigated, where ma- trices A_k are taken from a commutative group or from a bounded group. It is shown that any of the systems can be transformed to a new system, which does not possess any non-zero (asymptotically) almost periodic solution. Our goal is to improve the results of [9,24] about systems of the form (1.1) with regard to their non-almost periodic solutions. Furthermore, we recall the corresponding Cauchy problem. Note that the presented results are new even for complex matrix groups.

The fundamental properties of limit periodic and almost periodic sequences or functions have been studied closely. One can easily find many relevant monographs. Here we point out only the books [3, 6,15]. Concerning almost periodic solutions of linear almost periodic difference systems, we can refer to [4,5,25] (see also [7,10,26]). Other properties of (complex) almost periodic systems can be found in [1,11, 14]. The properties of limit periodic homoge- neous linear difference systems with respect to their almost periodic solutions are mentioned, e.g., in [9,24].

This paper is divided into five sections as follows. First, in the next section, the definitions of limit and almost periodicity are recalled. In Section3, we introduce the used notations. In Section4, we collect auxiliary results, which we use in the proof of the main result. Finally, in Section5, we formulate and prove our main result.

2 Limit and almost periodicity

In this section, we recall the definitions of limit periodic and almost periodic sequences in a metric space(M,ρ).

Definition 2.1. We say that a sequence {ϕ_k}_k∈Z is limit periodic if there exists a sequence of periodic sequences{ϕⁿ_k}_k∈Z ⊆ M, n ∈ N, such that limn→∞ϕⁿ_k = ϕ_k and the convergence is uniform with respect tok∈_Z.

Remark 2.2. The limit periodicity can be introduced in another equivalent way (see [2]).

Definition 2.3. A sequence{ϕ_k}_k∈Z ⊆M is called almost periodic if for anyε>0 there exists r(ε)∈_Nsuch that any set consisting ofr(ε)consecutive integers contains at least one number lsatisfying

ρ(ϕ_k+l,ϕ_k)< ε, k∈_Z.

The definition mentioned above is the so-called Bohr definition of almost periodicity. The almost periodicity can be defined in another equivalent way (see the next theorem). This concept is the so-called Bochner definition.

Theorem 2.4. Let {ϕ_k}_k∈Z ⊆ M be given. The sequence {ϕ_k}_k∈Z is almost periodic if and only if any sequence{ln}_n∈N ⊆ _Zhas a subsequence {l^¯n}_n∈N ⊆ {ln}_n∈N such that, for any ε > 0, there exists K(ε)∈_Nsatisfying

ρ

ϕ_k+l¯i,ϕ_k+l¯j

<ε, i,j>K(ε), k∈_Z. (2.1)

Proof. See, e.g., [20, Theorem 2.3].

3 Preliminaries

In the whole paper, we will consider an infinite field F with an absolute value | · |: F → _R.

Let m ∈ _Nbe arbitrarily given. We denote the set of all m×m matrices with elements in F by the symbol Matm(F). The absolute value gives the norms k · kon F^m andMatm(F)as the sum of the absolute values of elements. The absolute value and norms induce metrics on F and F^m, Matm(F), respectively. We denoteδ-neighbourhoods by symbolOδ in all considered metric spaces.

Let X ⊆ Mat_m(F). We repeat that X is a commutative group. The set of all limit peri- odic and almost periodic sequences with values in X will be denoted by LP(X)and AP(X), respectively. In these sets, we consider the metric

σ({A_k}_k∈Z,{B_k}_k∈Z):=sup

k∈Z

kA_k−B_kk.

For the reader’s convenience, we also denote δ-neighbourhoods of sequences in LP(X) and AP(X)by symbolOδ.

Instead of{Z_k}_k∈Z, we will shortly write {Z_k}. If index kwill be taken from another set, we will specify it at the corresponding place. The identity matrix will be denoted by I. The zero vector will be denoted by 0. Symbolε stands for a positive real number.

In the definitions given below, we recall (and slightly generalize) the propertyPfrom [9].

Definition 3.1. We say that group X has propertyP if there exists ζ > 0 such that for every δ > 0 there exists l ∈ _N such that for every u ∈ F^m fulfilling kuk ≥ 1 there exist matrices M₁,M2, . . . ,M_l ∈ Xwith the property that

M_i ∈O_δ(I), i∈ {1, . . . ,l}, kM_l· · ·M₁u−uk>ζ.

Definition 3.2. Let u ∈ F^m be an arbitrary non-zero vector. We say that group X has pro- perty P with respect to u if there exists ζ > 0 such that for everyδ > 0 there exist matrices M₁,M2, . . . ,M_l ∈ Xwith the property that

M_i ∈O_δ(I), i∈ {1, . . . ,l}, kM_l· · ·M₁u−uk>ζ.

4 Auxiliary results

Definition3.1 can be apprehended in a little larger sense using the following lemma.

Lemma 4.1. For any a > 0, there exists f(a) ∈ F such that kf(a)·vk ≥ 1 for every v ∈ F^m satisfyingkvk ≥a.

Proof. Let a > 0 be arbitrarily given. For v ∈ F^m, kvk ≥ a, kvk ≥ 1, we put f(a) = e, where e denotes the identity element of F. If there exists v ∈ F^m, kvk ≥ a, kvk < 1, then there exists element g ∈ F such that |g| ≤ kvk < 1. Thus, lim_n→_∞|gⁿ| = lim_n→_∞|g|ⁿ = 0 and, consequently, we have that lim_n→_∞|g⁻ⁿ| = lim_n→_∞|g⁻¹|ⁿ = _∞. It means that there exists N∈_Nsuch that|g^−N| ≥1/a. It suffices to put f(a) =g^−N.

Remark 4.2. According to Lemma4.1, Definition3.1can be used to vectors satisfyingkuk ≥a in the following way. Instead of u, we apply the definition for vector f(a)·u. Then the cor- responding inequality takes the formkM_l· · ·M₁f(a)·u−f(a)·uk>ζwhich can be rewritten to kM_l· · ·M₁u−uk> _ζ/|f(a)|_:=ω(a)>_0.

Remark 4.3. We can assume that ω(a) is a non-decreasing function of a. It follows from Remark4.2 and from the proof of Lemma4.1.

Remark 4.4. Let a > 0. Let v ∈ F^m satisfy kM_l· · ·M₁v−vk > ω(a). Then, it is valid that kM_l· · ·M₁v−wk>ω(a)/2 ifkw−vk<ω(a)/2 holds. Indeed, it follows directly from

ω(_a)<kM_l· · ·M1v−v+_w−wk ≤ kM_l· · ·M1v−wk+kw−vk.

Now we recall two known lemmas, which we need to prove the main result of this paper.

Lemma 4.5. Let {A_k} ∈ AP(X). For any non-zero almost periodic solution {x_k} of the system x_k+1 = A_kx_k,it holds thatinf_k∈Zkx_kk>0.

Proof. See [22, Lemma 3.10].

Lemma 4.6. Let{A_k} ∈ LP(X)andε > 0be arbitrarily given. Let {δn}_n∈N ⊂ _Rbe a decreasing sequence satisfying

nlim→_∞δ_n=0 and let{Sⁿ_k} ⊂X be periodic sequences for n∈_Nsuch that

Sⁿ_k ∈O_δn(I), k∈_Z, n∈_N, (4.1) S_k^j = I or Sⁱ_k = I, k ∈_Z, i6= j, i,j∈_N. (4.2) If one puts

S_k := A_k·S¹_k·S²_k· · ·S_kⁿ· · · , k∈_Z, then{S_k} ∈LP(X). In addition, if

δ₁< ^ε sup

k∈_Z

kA_kk^{, (4.3)}

then{S_k} ∈O_ε({A_k}).

Proof. See [9, Lemma 5.1.] (and also [20, Theorem 3.5]).

Remark 4.7. Lemma4.6 remains true if LP(X)is replaced by AP(X)in the statement of this lemma, which gives [9, Lemma 5.8.].

5 Results

Now, we can prove the main result. We repeat thatX⊆Mat_m(F)is a commutative group.

Theorem 5.1. Let X have property P and ε > 0 be arbitrary. Then, for every {A_k} ∈ LP(X)and every sequence{un}_n∈Nof non-zero vectors un ∈ F^m, there exists {S_k} ∈Oε({A_k})∩LP(X)such that the solution of

x_k+1 =S_kx_k, x₀ =u_n (5.1)

is not almost periodic for any n∈_N.

Proof. Let ε > 0 be arbitrary. Let ζ be taken from Definition3.1. We use the following con- struction.

In the first step of the construction, let us consider the initial problem

x_k+1 = A_kx_k, x₀ =u₁. (5.2)

Let{x⁽_k^1,1,1)}be its solution and letj(1, 1, 1):=0. For δ₁:= ¹

1+ε · ^ε sup

k∈_Z

kA_kk^{, (5.3)}

there exists l(δ₁)≥ 2 (see Definition3.1). Denote4₁ := 2·l(δ₁). Then, for vector x⁽_j(^1,1,1_1,1,1⁾₎₊₄

1, there exist matricesM⁽₁^1,1,1),M₂^(1,1,1), . . . ,M_l⁽₍^1,1,1_δ ⁾

1) ∈Oδ₁(I)given by propertyPin Definition3.1.

We define periodic sequence{S⁽_k^1,1,1)}with periodp(1, 1, 1):=4₁as follows. Denotea(_1,1,1) :=

x⁽_j(^1,1,1_1,1,1⁾₎

. Ifa_(1,1,1)≤1, then we put

S⁽₀^1,1,1) =· · ·=S⁽_p^1,1,1_(1,1,1⁾₎₋₁= I.

Ifa_(1,1,1) >1 and

x⁽_j(^1,1,1_1,1,1⁾₎₊₄

1−x⁽_j(^1,1,1_1,1,1⁾₎

>ω(a_(1,1,1))/2, then S⁽₀^1,1,1) =· · ·=S⁽_p^1,1,1_(1,1,1⁾₎₋₁= I.

Ifa_(1,1,1) >1 and

x⁽_j(^1,1,1_1,1,1⁾₎₊₄

1−x⁽_j(^1,1,1_1,1,1⁾₎

≤ω(a_(1,1,1))/2, then

S⁽₀^1,1,1) = I, S₁^(1,1,1)= M₁^(1,1,1), S⁽₂^1,1,1)= I, S⁽₃^1,1,1)= M⁽₂^1,1,1), S⁽₄^1,1,1) = I, . . . ,S⁽_p^1,1,1_(1,1,1⁾₎₋₁= M⁽_l(^1,1,1)

δ1) . We denoteS¹_k =S_k^(1,1,1)andR¹_k = A_kS¹_k fork∈_Z.

In the second step, we consider the Cauchy problem x_k+1 =R¹_kx_k, x₀=u₁.

Let {x⁽_k^2,1,1)}be its solution. Then, there exists a positive integer j(2, 1, 1)divisible by 4 satis- fying j(2, 1, 1)> p(1, 1, 1). For

δ₂:= ¹

4+ε · ^ε sup

k∈_Z

kA_kk^,

there exists l(δ₂) (see Definition3.1). Without loss of generality, we can assume thatl(δ₂) ≥ l(δ₁). Denote4₂ :=64·l(δ₂)·l(δ₁). Forx⁽_j(^2,1,1_2,1,1⁾₎₊₄

2, there exist matrices M₁^(2,1,1),M⁽₂^2,1,1), . . . ,M⁽_l(^2,1,1_δ ⁾

2) ∈O_δ2(I)

taken from Definition3.1. We define the periodic sequence{S⁽_k^2,1,1)}with period p(_{2, 1, 1})_:= [j(_{2, 1, 1}) +4₂]p(_{1, 1, 1})

in the following way. Denotea(_2,1,1) := x⁽_j(^2,1,1_2,1,1⁾₎

. Ifa(_2,1,1)≤1/2, then we define S⁽₀^2,1,1) =· · ·= S⁽_p^2,1,1_(2,1,1⁾₎₋₁= I.

Ifa_(2,1,1) >_{1/2 and}x⁽_j(^2,1,1_2,1,1⁾₎₊₄

2−x⁽_j(^2,1,1_2,1,1⁾₎

>ω(a_(2,1,1))_{/2, then} S⁽₀^2,1,1) =· · ·= S⁽_p^2,1,1_(2,1,1⁾₎₋₁= I. Ifa_(2,1,1) >1/2 and

x⁽_j(^2,1,1_2,1,1⁾₎₊₄

2−x⁽_j(^2,1,1_2,1,1⁾₎

≤ω(a_(2,1,1))/2, then S₀^(2,1,1)= · · ·=S⁽_j(^2,1,1_2,1,1⁾₎₋₁ = I,

S⁽_j(^2,1,1_2,1,1⁾₎ = I, S⁽_j(^2,1,1_2,1,1⁾₎₊₁= I, S⁽_j(^2,1,1_2,1,1⁾₎₊₂= M⁽₁^2,1,1),

S⁽_j(^2,1,1_2,1,1⁾₎₊₃ = I, S⁽_j(^2,1,1_2,1,1⁾₎₊₄= I, S⁽_j(^2,1,1_2,1,1⁾₎₊₅= I, S⁽_j(^2,1,1_2,1,1⁾₎₊₆= M⁽₂^2,1,1), ...

S⁽_j(^2,1,1_2,1,1⁾_)+4(l(δ

2)−1)) = I, S⁽_j(^2,1,1_2,1,1⁾_)+1+4(l(δ

2)−1)) = I, S⁽_j(^2,1,1_2,1,1⁾_)+2+4(l(δ

2)−1)= M_l⁽₍^2,1,1_δ ⁾

2) , S⁽_j(^2,1,1_2,1,1⁾_)+3+4(l(δ

2)−1)=· · · =S⁽_p^2,1,1_(2,1,1⁾₎₋₁ = I.

We putR⁽_k^2,1,1)= R¹_kS⁽_k^2,1,1),k∈_Z. Next, we consider

x_k+1 =R⁽_k^2,1,1)x_k, x₀ =u₁

with the solution {x⁽_k^2,2,1)}. There exists j(2, 2, 1) ∈ _N divisible by 8 such that j(2, 2, 1) >

p(_{2, 1, 1})_{. Property}P(see Definition3.1) used for vectorx⁽_j(^2,2,1_2,2,1⁾₎₊₄

2−4₁ gives the matrices M⁽₁^2,2,1),M⁽₂^2,2,1), . . . ,M⁽_l(^2,2,1_δ ⁾

2) ∈Oδ₂(I). Let us define the periodic sequence{S⁽_k^2,2,1)}with period

p(2, 2, 1):= [j(2, 2, 1) +4₂− 4₁]p(2, 1, 1) as follows. Denotea_(2,2,1) := x⁽_j(^2,2,1_2,2,1⁾₎

. Ifa_(2,2,1) ≤1/2, then S⁽₀^2,2,1) =· · ·= S⁽_p^2,2,1_(2,2,1⁾₎₋₁= I. Ifa_(2,2,1) >1/2 and

x⁽_j(^2,2,1_2,2,1⁾₎₊₄

2−4₁ −x⁽_j(^2,2,1_2,2,1⁾₎

> ω(a_(2,2,1))/2, then S⁽₀^2,2,1) =· · ·= S⁽_p^2,2,1_(2,2,1⁾₎₋₁= I. Ifa_(2,2,1) >1/2 and

x⁽_j(^2,2,1_2,2,1⁾₎₊₄

2−4₁ −x⁽_j(^2,2,1_2,2,1⁾₎

≤ ω(a_(2,2,1))/2, then S₀^(2,2,1)= · · ·=S⁽_j(^2,2,1_2,2,1⁾₎₋₁ = I,

S⁽_j(^2,2,1_2,2,1⁾₎ =· · ·= S⁽_j(^2,2,1_2,2,1⁾₎₊₃ = I, S⁽_j(^2,2,1_2,2,1⁾₎₊₄= M⁽₁^2,2,1), S⁽_j(^2,2,1_2,2,1⁾₎₊₅ =· · ·= S⁽_j(^2,2,1_2,2,1⁾₎₊₁₁= I, S⁽_j(^2,2,1_2,2,1⁾₎₊₁₂= M⁽₂^2,2,1),

... S⁽_j(^2,2,1_2,2,1⁾_)+8(l(δ

2)−2)+5= · · ·=S⁽_j(^2,2,1_2,2,1⁾_)+8(l(δ

2)−2)+11= I, S⁽_j(^2,2,1_2,2,1⁾_)+8(l(δ

2)−1)+4 = M_l⁽₍^2,2,1_δ ⁾

2) , S⁽_j(^2,2,1_2,2,1⁾_)+8(l(δ

2)−1)+5=· · · =S⁽_p^2,2,1_(2,2,1⁾₎₋₁ = I.

Again, we denote R⁽_k^2,2,1) =R⁽_k^2,1,1)S⁽_k^2,2,1),k ∈_Z.

Next, we consider the initial problem

x_k+1= R⁽_k^2,2,1)x_k, x₀=u₂.

Let {x_k^(2,1,2)}be its solution. Then there exists a positive integer j(2, 1, 2)divisible by 16 satis- fying j(2, 1, 2)> p(2, 2, 1). For x⁽_j(^2,1,2_2,1,2⁾₎₊₄

2, there exist matrices M₁^(2,1,2),M⁽₂^2,1,2), . . . ,M⁽_l(^2,1,2_δ ⁾

2) ∈O_δ2(I)

taken from Definition3.1. We define the periodic sequence{S⁽_k^2,1,2)}with period p(2, 1, 2):= [j(2, 1, 2) +4₂]p(2, 2, 1)

in the following way. Denotea_(2,1,2):= x⁽_j(^2,1,2_2,1,2⁾₎

. Ifa_(2,1,2)≤1/2, then we put S⁽₀^2,1,2) =· · ·=S⁽_p^2,1,2_(2,1,2⁾₎₋₁= I.

Ifa_(2,1,2) >_{1/2 and}x⁽_j(^2,1,2_2,1,2⁾₎₊₄

2 −x⁽_j(^2,1,2_2,1,2⁾₎

>ω(a_(2,1,2))_{/2, then} S⁽₀^2,1,2) =· · ·=S⁽_p^2,1,2_(2,1,2⁾₎₋₁= I.

Ifa_(2,1,2) >1/2 and

x⁽_j(^2,1,2_2,1,2⁾₎₊₄

2 −x⁽_j(^2,1,2_2,1,2⁾₎

≤ω(a_(2,1,2))/2, then S⁽₀^2,1,2)=· · · =S⁽_j(^2,1,2_2,1,2⁾₎₋₁ = I,

S⁽_j(^2,1,2_2,1,2⁾₎ =· · ·= S⁽_j(^2,1,2_2,1,2⁾₎₊₇ = I, S⁽_j(^2,1,2_2,1,2⁾₎₊₈= M⁽₁^2,1,2), S⁽_j(^2,1,2_2,1,2⁾₎₊₉ =· · ·= S⁽_j(^2,1,2_2,1,2⁾₎₊₂₃= I, S⁽_j(^2,1,2_2,1,2⁾₎₊₂₄= M⁽₂^2,1,2),

... S⁽_j(^2,1,2_2,1,2⁾_)+16(l(

δ2)−2)+9= · · ·=S⁽_j(^2,1,2_2,1,2⁾_)+16(l(

δ2)−2)+23= I, S⁽_j(^2,1,2_2,1,2⁾_)+16(l(

δ2)−1)+8 = M_l⁽₍^2,1,2)

δ2) , S⁽_j(^2,1,2_2,1,2⁾_)+16(l(

δ2)−1)+9 =· · · =S⁽_p^2,1,2_(2,1,2⁾₎₋₁= I.

We put R⁽_k^2,1,2) =R⁽_k^2,2,1)S⁽_k^2,1,2),k∈_Z.

We consider the system with the initial value

x_k+1= R⁽_k^2,1,2)x_k, x₀=u₂.

Let{x⁽_k^2,2,2)} be its solution. There exists a positive integer j(2, 2, 2)divisible by 32 such that j(2, 2, 2)> p(2, 1, 2). Again, forx⁽_j(^2,2,2_2,2,2⁾₎₊₄

2−4₁, there exist matrices (see Definition3.1) M⁽₁^2,2,2),M⁽₂^2,2,2), . . . ,M⁽_l(^2,2,2_δ ⁾

2) ∈O_δ2(I). Let us define the periodic sequence{S⁽_k^2,2,2)}with period

p(2, 2, 2):= [j(2, 2, 2) +4₂− 4₁]p(2, 1, 2) as follows. Denotea_(2,2,2) := x⁽_j(^2,2,2_2,2,2⁾₎

. Ifa_(2,2,2) ≤1/2, then we define S⁽₀^2,2,2) =· · ·= S⁽_p^2,2,2_(2,2,2⁾₎₋₁= I.

Ifa_(2,2,2) >1/2 and

x⁽_j(^2,2,2_2,2,2⁾₎₊₄

2−4₁ −x⁽_j(^2,2,2_2,2,2⁾₎

> ω(a_(2,2,2))/2, then S⁽₀^2,2,2) =· · ·= S⁽_p^2,2,2_(2,2,2⁾₎₋₁= I. Ifa_(2,2,2) >1/2 and

x⁽_j(^2,2,2_2,2,2⁾₎₊₄

2−4₁ −x⁽_j(^2,2,2_2,2,2⁾₎

≤ ω(a_(2,2,2))/2, then S₀^(2,2,2)= · · ·=S⁽_j(^2,2,2_2,2,2⁾₎₋₁ = I,

S⁽_j(^2,2,2_2,2,2⁾₎=· · · =S⁽_j(^2,2,2_2,2,2⁾₎₊₁₅ = I, S⁽_j(^2,2,2_2,2,2⁾₎₊₁₆ = M₁^(2,2,2), S⁽_j(^2,2,2_2,2,2⁾₎₊₁₇=· · · =S⁽_j(^2,2,2_2,2,2⁾₎₊₄₇ = I, S⁽_j(^2,2,2_2,2,2⁾₎₊₄₈ = M⁽₂^2,2,2),

... S⁽_j(^2,2,2_2,2,2⁾_)+32(l(

δ2)−2)+17 =· · ·= S⁽_j(^2,2,2_2,2,2⁾_)+32(l(

δ2)−2)+47= I, S⁽_j(^2,2,2_2,2,2⁾_)+32(l(

δ2)−1)+16= M⁽_l(^2,2,2)

δ2) , S⁽_j(^2,2,2_2,2,2⁾_)+32(l(δ

2)−1)+17=· · · =S⁽_p^2,2,2_(2,2,2⁾₎₋₁ = I.

We denoteR²_k = R⁽_k^2,2,2) = R⁽_k^2,1,2)S⁽_k^2,2,2) andS²_k = S⁽_k^2,1,1)S⁽_k^2,2,1)S⁽_k^2,1,2)S⁽_k^2,2,2) fork ∈ Z. It is the end of the second step.

We continue the construction in the same way. Before the n-th step, we have {Rⁿ_k⁻¹} = {A_kS_k¹S²_k· · ·Sⁿ_k⁻¹}with period

p(n−1,n−1,n−1):= [j(n−1,n−1,n−1) +4_n−1− 4_n−2]p(n−1,n−2,n−1). We denote

d(x,y,z):=2^{(x−1)x6(2x−1)+y+(z−1)x}, x∈ _N, y,z ∈ {1, 2, . . . ,x}, (5.4) δ_j := ¹

j²+ε· ^ε sup

k∈_Z

kA_kk^{, j}∈_N, (5.5)

4_j :=

∏

d(i,i,i)l(δ_i), j∈_N. (5.6)

Let us consider the initial problem

x_k+1= Rⁿ_k⁻¹x_k, x₀ =u₁

and let{x_k^(n,1,1)}be its solution. Then there existsj(n, 1, 1)∈ _Ndivisible byd(n, 1, 1)such that j(n, 1, 1)> p(n−1,n−1,n−1). From Definition3.1, forx⁽_j(^n,1,1_n,1,1⁾₎₊₄

n, there exist matrices M₁^(n,1,1),M⁽₂^n,1,1), . . . ,M⁽_l(^n,1,1)

δn) ∈O_δn(I)_{, (5.7)}

where l(δ_n) can be taken in such a way that l(δ_n) ≥ l(δ_n−1). Next, we define the periodic sequence {S⁽_k^n,1,1)}with period

p(n, 1, 1):= [j(n, 1, 1) +4_n]p(n−1,n−1,n−1). Denote a(n,1,1) :=

x⁽_j(^n,1,1_n,1,1⁾₎

. Ifa(n,1,1)≤1/n, then we put S⁽₀^n,1,1) =· · ·=S⁽_p^n,1,1_(n,1,1⁾₎₋₁= I.

Ifa_(n,1,1) >1/nand

x⁽_j(^n,1,1_n,1,1⁾₎₊₄

n−x⁽_j(^n,1,1_n,1,1⁾₎

> ω(a_(n,1,1))/2, then S⁽₀^n,1,1) =· · ·=S⁽_p^n,1,1_(n,1,1⁾₎₋₁= I.

Ifa_(n,1,1) >1/nand

x⁽_j(^n,1,1_n,1,1⁾₎₊₄

n−x⁽_j(^n,1,1_n,1,1⁾₎

≤ ω(a_(n,1,1))/2, then S₀^(n,1,1)=· · · =S⁽_j(^n,1,1_n,1,1⁾₎₋₁ = I,

S⁽_j(^n,1,1_n,1,1⁾₎ =· · · =S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)/2−1}= I, S⁽_j(^n,1,1_n,1,1⁾_)+d(n,1,1)/2 = M⁽₁^n,1,1), S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)/2+1}=· · · =S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)+d(n,1,1)/2−1}= I,

S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)+d(n,1,1)/2} = M⁽₂^n,1,1), ...

S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)(l(δ}

n)−2)+d(n,1,1)_/2+₁ =· · · =S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)(l(δ}

n)−1)+d(n,1,1)_/2−1 = I, S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)(l(δ}

n)−1)+d(n,1,1)/2 = M⁽_l(^n,1,1_δ ⁾

n) , S⁽_j(^n,1,1_n,1,1⁾_{)+d(n,1,1)(l(}

δn)−1)+d(n,1,1)/2+1 =· · ·=S⁽_p^n,1,1_(n,1,1⁾₎₋₁= I.

We put R⁽_k^n,1,1) =Rⁿ_k⁻¹S⁽_k^n,1,1),k∈_Z.

Let us have the problem

x_k+1= R⁽_k^n,1,1)x_k, x0 =u₁

and its solution {x_k^(n,2,1)}. Let j(_{n, 2, 1}) be a positive integer divisible by d(_{n, 2, 1}) _{such that} j(n, 2, 1)> p(n, 1, 1). For vectorx⁽_j(^n,2,1_n,2,1⁾₎₊₄

n−4₁, there exist matrices (see Definition3.1) M₁^(n,2,1),M⁽₂^n,2,1), . . . ,M⁽_l(^n,2,1)

δn) ∈O_δn(I). (5.8)

We define the sequence{S⁽_k^n,2,1)}with period

p(n, 2, 1)_:= [j(n, 2, 1) +4_n− 4₁]p(n, 1, 1)