, where χ is either of the symbols s, s

Vol. 44, No. 1, 2014, 9-20

SOLVABILITY OF CERTAIN SEQUENCE SPACES EQUATIONS WITH OPERATORS

Bruno de Malafosse¹

Abstract. In this paper we deal with special sequence space equa- tions (SSE) with operators,which are determined by an identity whose each term is a sum or a sum of products of sets of the form χ_a(T) and χ_f(x)(T) wheref mapU⁺ to itself andχis any of the symbolss, s⁰, or s^(c). Among other things under some conditions we solve (SSE) with operatorsχ_a(C(λ)Dτ)+χ_x(C(µ)Dτ) =χ_b, andχ_a(C(λ)C(µ))+

χ_x(C(λσ)C(µ)) =χ_bwhereχ∈{ s, s⁰}

, andχ_a(C(λ)Dτ)+s⁰x(C(µ)Dτ) = χ_bwhereχis either of the symbolss, ors^(c) andC(ν)Dτ is a factorable matrix.

AMS Mathematics Subject Classification(2010): 40C05, 46A45

Key words and phrases: matrix transformations, BK space, multiplier of sets of sequences, sequence space inclusion equations, sequence space equations with operators

1. Introduction

In [21] Wilansky introduced sets of the forma⁻¹∗χ, where a= (an)_n≥1is a sequence satisfying an ̸= 0 for all n, and χ is any set of sequences. Recall that x= (xn)_n≥1 belongs toa⁻¹∗χ if (anxn)_n≥1 belongs to χ. In this way, for any strictly positive sequence a, are defined the sets s⁰_a, s^(c)a and s_a by a⁻¹∗χ where χ is either of the sets c₀, c, and ℓ_∞ respectively. In [5, 8] the sum s_a+s_b and the product s_a∗s_b of the sets s_a and s_b were defined, and characterizations of matrix transformations mapping in the setss_a+s⁰_b(∆^q) and sa+s^(c)_b (∆^q) were given, where ∆ is the operator of the first difference. In [16]

de Malafosse and Malkowsky gave among other things properties of the matrix of weighted means considered as operator in the set s_a. Characterizations of matrix transformations mapping ins⁰_α

(

(∆−λI)^h )

+s^(c)_β (

(∆−µI)^l )

withλ, µ,h,l∈Ccan be found in [9]. There are many other results using the setss⁰_a, s^(c)a ands_a, let us cite for instance applications to the following topics,σ−core, [7], solvability of infinite tridiagonal systems, [6],measure of noncompactness, [18], Hardy theorem, [20] andstatistical convergence, [19].

In this paper our aim is to solve specialsequence spaces equations (SSE), which are determined by an identity whose each term is a sum or a sum of products of sets of the form χ_a(T)and χ_f(x)(T) wheref maps U⁺ to itself, andχis any of the symbolss,s⁰, ors^(c), the sequencexis the unknown andTis

1LMAH Universit´e du Havre, I.U.T Le Havre BP 4006 76610, Le Havre, France, e-mail:

[email protected]

a given triangle. The resolution of such (SSE) consists in determining the set of all sequencesxsatisfying the identity, see for instance [13, 11, 15, 12, 17, 14, 10].

This paper is organized as follows. In Section 2 we recall some results on the sum and the product of sets of the formχ_a, whereχis either of the symbols s, ors⁰. In Section 3 we solve the sequence spaces equationsχ_a(C(λ)D_τ) + χ_x(C(µ)D_τ) = χ_b and χ_a(

N_q) +χ_x(

N_pD_q/p)

=χ_b for χ ∈{ s, s⁰}

, where N_q is the operator of weighted means in some cases and we solve another type of (SSE) defined byχ_a(C(λ)D_τ) +s⁰_x(C(µ)D_τ) =s⁰_b whereχis either of the symbolss, ors^(c). In Section 4 we deal with (SSE) with operators represented by products of triangles of the form χ_a(C(λ)C(µ)) +χ_x(C(λσ)C(µ)) =χ_b whereC(ν)Dτ is a factorable matrix forχ∈{

s, s⁰} .

2. Sum and product of sequence spaces of the form χ

, where χ is either of the symbols s, s

2.1. The sets χ_a, where χ is either of the symbols s,s⁰, or s^(c) for a∈U⁺

We write s, ℓ_∞, c andc0 for the sets of all complex, bounded, convergent and convergent to naught sequences, respectively. For a given infinite matrix Λ = (λ_nm)_n,m≥1we define the operators Λ_n, for any integern≥1, by Λ_n(ξ) =

∑_∞

m=1λ_nmξ_m, whereξ= (ξ_m)_m≥1, and the series are assumed convergent for alln. So we are led to the study of the operator Λ defined by Λξ= (Λ_n(ξ))_n≥1 mapping a sequence space into another sequence space.

A Banach spaceE of complex sequences with the norm∥∥_E is aBK space if each projection Pn:E→Cdefined byPnξ=ξ_n is continuous. A BK space E is said to have AK if every sequence ξ ∈ E has a unique representation ξ=∑_∞

n=1ξ_ne⁽ⁿ⁾ wheree⁽ⁿ⁾ is the sequence with 1 in then-th position, and 0 otherwise.

Letabe a nonzero sequence. Using Wilansky’s notations we write 1/a∗E for the set of all ξ= (ξ_n)_n≥1 such that (anξ_n)_n≥1∈E. LetU⁺ be the set of all real sequencesξwithξ_n>0 for alln. Ifξ∈swe defineDξ as the diagonal matrix defined by [Dξ ]_nn =ξ_n for all n, we haveDa∗E = (1/a)⁻¹∗E and it can be easily shown Λ ∈(Da∗E, Db∗F) if and only if D1/bΛDa ∈(E, F) where E, F ⊂s. Recall that fora∈U⁺ we have sa =Da∗ℓ_∞, s⁰_a =Da∗c0

ands^(c)a =D_a∗c . Each of the previous sets is a BK spacenormed by ∥ξ∥s_a, where ∥ξ∥sa = sup_n(|ξ_n|/a_n) < ∞. So we can define s_a as the set of all sequences ξ such that (ξ_n/an)_n ∈ ℓ_∞, s⁰_a as the set of all sequences ξ such that ξ_n/an → 0 (n→ ∞) and s^(c)a as the set of all sequences ξ such that ξ_n/a_n → l (n→ ∞) for some l ∈ C, (cf. [3, 4]). If a = (rⁿ)_n≥1, we write χ_a =χ_r whereχ is any of the symbolss,s⁰, ors^(c) to simplify. Whenr= 1, we obtains₁=ℓ_∞, s⁰₁=c₀ ands^(c)₁ =c. If we lete= (1,1, ...), then we have s_e =s₁ =ℓ_∞, s⁰_e =s⁰₁=c₀ and s^(c)e =s^(c)₁ =c. When Λ maps E into F we write Λ∈(E, F), see [2]. So we have Λξ∈F for allξ∈E, (Λξ∈F means that for eachn≥1 the series defined by Λn(ξ) =∑_∞

m=1λnmξ_m is convergent and (Λn(ξ))_n≥1∈F). The setSaof all infinite matrices Λ = (λnm)n,m≥1such that

∥Λ∥S_a= sup_n≥1(

a⁻_n¹∑_∞

m=1|λnm|am

)<∞is a Banach algebra with identity normed by ∥Λ∥S_a.Recall that if Λ∈(sa, sa), then∥Λξ∥s_a ≤ ∥Λ∥S_a∥ξ∥s_a for allξ∈s_a. It is well-known thatS_a =(

s⁰_a, s_a)

= (

s^(c)a , s_a )

= (s_a, s_a).

2.2. Sum of sets of the form χ_a where χ is either of the symbols s⁰, or s.

In this subsection we recall some properties of thesum E+F of sets of the form s⁰_a, orsa.

LetE, F ⊂s be two linear vector spaces. We write E+F for the set of all sequencesξ=ζ+ζ^′ whereζ∈Eandζ^′∈F. In the next result we use the notation [max (a, b)]_n = max (an, bn). We prove the following results.

Proposition 1. Let a,b∈U⁺ and assumeχ is either of the symbolss⁰, ors.

Then we have

(i)χ_a ⊂χ_b if and only if there is K >0such that a_n≤Kb_n for alln.

(ii)χ_a=χ_b if and only ifs_a =s_b, that is, there are K₁,K₂>0 such that K₁≤ b_n

an ≤K₂ for alln.

(iii)χ_a+χ_b=χ_a+b=χ_max(a,b).

(iv)χ_a+χ_b=χ_a if and only ifb/a∈ℓ_∞.

Proof. The caseχ=swas shown in [5, Proposition 1, p. 244], and [8, Theorem 4, p. 293]. The case χ=s⁰ can be shown similarly, since we have sa =sb if and only ifs⁰_a=s⁰_b.

Notice thatχ_a ⊂χ_b is equivalent toa∈s_b.

2.3. Solvability of the equation χ_a+χ_x=χ_b whereχ is either of the symbols s⁰, or s.

In the following we determine the set of all sequences x= (x_n)_n≥1 ∈ U⁺ such that yn = bnO(1) (n→ ∞) if and only if there are u, v ∈ s such that y=u+vandun=anO(1) andvn=xnO(1) (n→ ∞) for ally∈s. Similarly we determine the sequencesx∈U⁺ such thatyn=bno(1) if and only if there areu,v∈ssuch thaty=u+vandun=ano(1) andvn =xno(1) (n→ ∞). Theorem 2. Let a,b∈U⁺, and consider the equation

(1) χ_a+χ_x=χ_b

whereχis either of the symbolss⁰, orsandx= (xn)_n≥1∈U⁺ is the unknown.

Then

(i) if a/b∈c₀, then equation (1) holds if and only if there are K₁,K₂>0 depending onx, such thatK1bn≤xn≤K2bn for alln, that is sx=sb.

(ii) If a/b, b/a∈ℓ_∞, then equation (1) holds if and only if there isK >0 depending onxsuch that 0< x_n≤Kb_n for all n; that is, x∈s_b.

(iii) Ifa/b /∈ℓ_∞, then equation (1) has no solution inU⁺.

Proof. The case of equation (1) whereχ =s was shown in [1]. For equation (1) with χ=s⁰ it is enough to note thatsa+sx=sb can be written in the form s_a+x=s_b which is turn in s⁰_a+x=s⁰_b and s⁰_a+s⁰_x=s⁰_b. This concludes the proof.

In the following corollary we writecl(u),u >0, for the set of all sequences ξ such that Kuⁿ ≤ ξ_n ≤ K^′uⁿ for alln and for some K, K^′ > 0. This set is an equivalence class for the relationξRξ^′ if sξ =s_ξ′ with ξ^′ = (uⁿ)_n. The following (SSE) is completely solved.

Corollary 3. Letr,u >0. The setΛχ of all x∈U⁺ that satisfy the equation

(2) χ_r+χ_x=χ_u whereχ∈{

s⁰, s}

is defined by

Λ_χ =





cl(u) forr < u, s_u∩

U⁺ forr=u,

∅ forr > u.

2.4. Product of sequence spaces of the form χ_a for χ∈{ s⁰, s}

. In this subsection we will deal with some properties of theproduct E∗F of particular subsets E and F ofs. For any sequences ξ∈E andη ∈F we put ξξ^′ =(

ξ_nξ^′_n)

n≥1. Most of the following results were shown in [5].

For any sets of sequencesEandF, we writeE∗F for the set of all sequences ξξ^′such thatξ∈Eandξ^′ ∈F.We immediately have the following results where Sχ,χ∈{

s⁰, s}

, is constituted of all the sets of the formχ_a with a∈U⁺. Proposition 4. The set Sχ, where χ ∈ {

s⁰, s}

with multiplication ∗ is a commutative group withχ₁ as the unit element.

Proof. First it can easily be seen that χ_a ∗χ_b = χ_ab. We deduce the map ψ :U⁺ 7→ Sχ defined by ψ(a) =χ_a is a surjective homomorphism and since U⁺with the multiplication of sequences is a group it is the same forSχ. Then the unit element ofSχ isψ(e) =χ₁.

Remark 5. Note that the inverse ofχ_a isχ_1/a withχ∈{ s⁰, s}

.

As a direct consequence of Proposition 4 we deduce the following corollary.

Corollary 6. Leta,b,c∈U⁺ and letχ∈{ s⁰, s}

. Then (i) χ_a∗χ_b=χ_ab.

(ii) χ_a∗χ_b=χ_a∗χ_c if and only ifχ_b =χ_c.

(iii) The sequence x= (xn)_n≥1 ∈ U⁺ satisfies the equation χ_a∗χ_x =χ_b if and only if K1bn/an ≤xn ≤K2bn/an for all n and for some K1, K2 >0 depending onx.

Throughout this paper the unknown of each sequence spaces equation is a sequencex∈U⁺.

3. The (SSE) with operators represented by factorable matrices

In this section we deal with the resolution of (SSE) of the formχ_a(C(λ)Dτ) +χ_x(C(µ)Dτ) =χ_b andχ_a(

Nq

)+χ_x(

NpD_q/p)

=χ_b forχ∈{ s, s⁰}

where Nq is the operator of weighted means in some cases. Then we solve the (SSE) χ_a(C(λ)Dτ) +s⁰_x(C(µ)Dτ) =s⁰_b, whereχis either of the symbols s, ors^(c). 3.1. The operators C(η),∆ (η) and the sets bΓ,Γ and Cc₁

The infinite matrix T = (t_nm)_n,m≥1 is said to be a triangle if t_nm = 0 for m > n and t_nn ̸= 0 for all n. Now let U be the set of all sequences (u_n)_n≥1 ∈s, withu_n ̸= 0 for alln. The infinite matrix C(η) = (c_nm)_n,m≥1, forη= (η_n)_n≥1∈U, is defined by

c_nm=



 1

η_n ifm≤n,

0 otherwise.

It can be shown that the matrix ∆ (η) = (dnm)_n,m≥1with d_nm=





η_n ifm=n,

−η_n−1 ifm=n−1 andn≥2, 0 otherwise,

is the inverse of C(η), that is C(η) (∆ (η)ξ) = ∆ (η) (C(η)ξ) for all ξ ∈ s.

If η =ewe get the well known operator of the first difference represented by

∆ (e) = ∆. We then have ∆ξ_n=ξ_n−ξ_n−1 for alln≥1, with the convention ξ₀= 0. It is usually written Σ =C(e). Note that ∆ = Σ⁻¹ and ∆, Σ∈ SR

for anyR >1.

Consider the sets Cc1=

{

ξ∈U⁺: [C(ξ)ξ]_n= 1 ξ_n

∑n m=1

ξ_m=O(1) }

, bΓ =

{

ξ∈U⁺ : lim

n→∞

(ξ_n−1 ξ_n

)

<1 }

, Γ = {

ξ∈U⁺: lim sup

n→∞

(ξ_n−1 ξ_n

)

<1 }

and G1={

ξ∈U⁺: there existC >0 andγ >1 such thatξ_n≥Cγⁿ for alln} . By [4, Proposition 2.1, p. 1786] and [16, Proposition 2.2 p. 88], we obtain the next lemma.

Lemma 7. Γb⊂Γ⊂Cc1⊂G1. We also need the following results.

Lemma 8. [8, Proposition 9, p. 300] Leta,b∈U⁺. Then (i) the following statements are equivalent

(α) χ_a(∆) =χ_b whereχ is any of the symbolss, ors⁰, (β) a∈Cc₁ ands_a =s_b.

(ii)a∈bΓif and only if s^(c)a (∆) =s^(c)a .

rom the preceding results we deduce the following:

3.2. Application to the equation χ_a(C(λ)Dτ) + χ_x(C(µ)Dτ) = χ_b wherex is the unknown

Leta,b,λ,µ,τ∈U⁺ and consider the equation

(3) χ_a(C(λ)Dτ) +χ_x(C(µ)Dτ) =χ_b, whereχ=s⁰, or s

andx∈U⁺is the unknown. The operator represented byC(λ)D_τ =D_1/λΣD_τ is called a factorable matrix. For χ = s⁰ solving the (SSE) (3) consists of determining all sequencesx∈U⁺such that the conditiony_n/b_n→0 (n→ ∞) holds if and only if there areu,v∈ssuch thaty=u+v and

τ1u1+...+τnun

λ_na_n →0 and τ1v1+...+τnvn

µ_nx_n →0 (n→ ∞) for ally∈s.

We then have the following result.

Theorem 9. Let a,b,λ,µ,τ ∈U⁺. Then

(i) If bτ /∈Cc₁, then equation (3) wherexis the unknown has no solutions.

(ii) If bτ ∈Cc1 we then have

(a) if aλ/bτ ∈ c0, then equation (3) is equivalent to sx = sbτ /µ, that is K1bnτn/µ_n≤xn ≤K2bnτn/µ_n for allnand for someK1,K2>0.

(b) if aλ/bτ, bτ /aλ∈ℓ_∞, then the solutions of (3) are all sequences that satisfy x∈s_{bτ /µ}, that is,x_n≤Kb_nτ_n/µ_n for allnand for some K >0.

(c) If aλ/bτ /∈ℓ_∞, then (3) has no solution.

Proof. We have [C(λ)Dτ]⁻¹=D1/τ∆ (λ) and [C(µ)Dτ]⁻¹=D1/τ∆ (µ) then χ_a(C(λ)D_τ) = [C(λ)D_τ]⁻_a¹χ_a=D_1/τ∆ (λ)χ_a

andχ_x(C(µ)D_τ) =D_1/τ∆ (µ)χ_xand equation (3) is equivalent to D_1/τ∆ (λ)χ_a+D_1/τ∆ (µ)χ_x=χ_b,

that is D_1/τ∆(

χ_aλ+χ_µx)

=χ_b. Since ∆ (λ) = ∆Dλ and ∆ (µ) = ∆Dµ we deduce

(4) χ_aλ+χ_µx=χ_b(

D_1/τ∆)

=χ_bτ(∆).

Then (4) is equivalent to χ_aλ+µx =χ_bτ(∆) itself equivalent to χ_aλ+µx =χ_bτ and bτ ∈Cc1 by Lemma 8. So if bτ /∈ Cc1 equation (3) has no solution and if bτ ∈Cc1it is enough to apply Theorem 1 to the equationχ_aλ+χ_µx=χ_bτ.

We can state the following corollaries.

Corollary 10. Consider the equation

(5) χ₁(C(λ)Dτ) +χ_x(C(µ)Dτ) =χ₁ withχ=s⁰, ors.

(i) If τ /∈Cc1, then (5) has no solutions.

(ii) If τ∈Cc₁, then

(a) if λ∈s⁰_τ, then (5) is equivalent tos_x=s_{τ /µ};

(b) ifλ∈s_τ,τ∈s_λ, then the solutions of (SSE) (5) are all sequences that satisfy x∈s_{τ /µ};

(c) ifλ /∈sτ, then (5) has no solution.

Proof. It is enough to takea=b=ein Theorem 9.

In the following remark where C(λ) = C((n)_n) is the Ces`aro operator denoted by C1, the (SSE) is completely solved.

Remark 11. Consider the (SSE)

(6) χ₁(C₁D_τ) +χ_x(C₁D_τ) =χ₁ withχ=s⁰, ors.

If τ /∈Cc₁ then (6) has no solution. If τ ∈ Cc₁ the solutions of (6) are all the sequences that satisfy s_x=s_(τn/n)

n . This means that there are K₁, K₂>0 such thatK₁τ_n/n≤x_n≤K₂τ_n/nfor alln. Indeed, we haveλ_n=nand since τ ∈Cc₁ implies that there isγ >1 such thatτ_n ≥Kγⁿ for allnand for some K >0, we deduce thatn/τ_n→0 (n→ ∞). So it is enough to apply Corollary 10 (ii).

To state the next result, consider the equation

(7) χ_a(C(λ)) +χ_x(C(µ)) =χ_b withχ=s⁰, or s.

Corollary 12. Let a,b,λ,µ∈U⁺. Then

(i) If b /∈Cc₁, then equation (7) has no solution.

(ii) Ifb∈Cc₁, then3 cases are possible,

(a) ifaλ/b∈c₀ then the solutionsx∈U⁺ of equation (7) are all sequences that satisfys_x=s_b/µ;

(b) if there are k₁, k₂ > 0 such that k₁ ≤ a_nλ_n/b_n ≤ k₂ for all n, then equation (7) is equivalent to x∈s_b/µ;

(c) ifaλ/b /∈ℓ_∞, then equation (7) has no solution.

Proof. This result follows from Theorem 9 withτ=e.

Whena=ewe obtain the next corollary where the (SSE) is totally solved.

Corollary 13. The equation χ₁(C1) +χ_x(C1) = χ_b with χ =s⁰, or s, has no solution if b /∈Cc1 and ifb∈Cc1 the solutions are determined byK1bn/n≤ xn≤K2bn/n for allnand for someK1,K2>0.

Proof. This result follows from Corollary 12 witha=e,λn=µ_n =nfor alln.

Indeed, the condition b∈Cc1 implies that there is γ >1 such thatbn ≥Kγⁿ for alln. Then we haveanλn/bn≤Knγ⁻ⁿ=o(1) (n→ ∞).

Now state the next result where we put λ₀ = (n)_n≥1. Here the (SSE) is also totally solved.

Corollary 14. Let r₁,r₂>0 and consider the equation

(8) χ_r1(C1Dλ₀) +χ_x(C1Dλ₀) =χ_r2 withχ=s⁰, ors.

(i) If r₂≤1, then equation (8) has no solution.

(ii) Ifr₂>1, then

(a) ifr₁< r₂, then equation (8) is equivalent to s_x=s_r2; (b) ifr1=r2, then equation (8) is equivalent to x∈sr₂; (c) ifr1> r2, then equation (8) has no solution.

Proof. i) If r2 ≤ 1, then we have (r₂ⁿ)_n≥1 ∈/ Cc1, since by Lemma 7 we have Cc1 ⊂ G1 and by Corollary 12 equation (8) has no solutions. ii) Case when r₂>1. (a) First we have

nlim→∞

(n−1 n

r₂ⁿ⁻¹ rⁿ₂

)

= 1 r2

<1

and sincebΓ⊂Cc₁we deduce (nrⁿ₂)_n≥1∈Cc₁. So by Theorem 9 we have a_nλ_n

bnτn

= nrⁿ₁ nrⁿ₂ =

(r₁ r2

)n

=o(1) (n→ ∞) andsx=sr₂.The cases (b) and (c) can be shown similarly.

3.3. The (SSE) with operators of the weighted means

In this subsection we use the operator of weighted meansN_q defined by the triangle [

Nq

]

nm =qm/Qn for m≤n, where Qn =∑n

m=1qm, for all n, with q∈U⁺.

Consider now the equation

(9) χ_a(

Nq

)+χ_x(

NpDq/p

)=χ_b whereχ=s⁰, or s

forp,q∈U⁺.The question in the case whenχ=sis: what are the sequences x∈U⁺ such that y_n =O(b_n) (n→ ∞) if and only if there areu, v∈s such thaty=u+v and

q1u1+...+qnun

Q_n =O(a_n) and q1v1+...+qnvn

P_n =O(x_n) (n→ ∞) for ally?

Since we haveNq =D_1/QΣDq =C(Q)Dq, it is enough to takeQ=λ,µ=P andτ=qin Theorem 9. We then have

Corollary 15. Let a,b,p,q∈U⁺. Then (i) If bq /∈Cc1, then (9) has no solution;

(ii) if bq∈Cc1, then

(a) aQ/bq∈c0 implies that (SSE) (9) is equivalent tosx=s_bq/P.

(b) If there are k1, k2 >0 such that k1 ≤anQn/bnqn ≤k2 for all n, the solutions of (9) are all the sequences that satisfy x ∈ sbq/P (that is, xn ≤ Kbnqn/Pn for alln).

(c) If aQ/bq /∈ℓ_∞, then (9) has no solution.

This result leads to the following application.

Example 16. Let R >0 and let S be the set of all sequences x∈U⁺ that satisfy the statement: y_n/Rⁿ →0 (n→ ∞) if and only if there are u, v such thaty=u+v and

1 2ⁿ−1

∑n m=1

2^mum→0 and 1 nx_n

∑n m=1

2^mvm→0 (n→ ∞) for ally∈s.

It can be shown that the setS is empty ifR <1; ifR= 1, it is equal tos(1/n)_n

and ifR >1 it is determined byK₁(2R)ⁿ/n≤x_n ≤K₂(2R)ⁿ/nfor alln.

To end this section consider a new type of (SSE) using the setss^(c)a .

3.4. On the (SSE) χ_a(C(λ)Dτ) +s⁰_x(C(µ)Dτ) =s⁰_b whereχ is either s, or s^(c)

Consider now another type of (SSE) with factorable matrices using the set s^(c)a and that are totally solved. Here we determine the set of all the sequences x∈U⁺ such that the conditionyn/bn→0 (n→ ∞) holds if and only if there areu,v∈ssuch thaty=u+vand

τ₁u₁+...+τ_nu_n

λnan →l and τ₁v₁+...+τ_nv_n

µ_nxn →0 (n→ ∞)

for ally∈sand for some scalarl.We state the next lemma, which is a direct consequence of [17, Theorem 4.4, p. 7].

Lemma 17. Leta,b∈U⁺ and consider the (SSE)

(10) χ_a+s⁰_x=s⁰_b, whereχ is eithers, ors^(c).

(i) ifa/b∈c₀, then the solutions of (10) are all the sequences that satisfy sx=sb.

(ii) ifa/b /∈c0, then (10) has no solution.

From Lemma 17 and Theorem 9 we deduce the resolution of the (SSE) (11) χ_a(C(λ)D_τ) +s⁰_x(C(µ)D_τ) =s⁰_b whereχis either s, ors^(c). Theorem 18. Leta,b,λ,µ,τ ∈U⁺. Then

(i) ifbτ /∈Cc1, then (SSE) (11) has no solution.

(ii) Ifbτ ∈Cc₁, then two cases are possible,

(a) ifaλ/bτ ∈c₀, then the solutions of (11) are all the sequences that satisfy s_x=s_{bτ /µ};

(b) ifaλ/bτ /∈c0, then (11) has no solution.

Proof. Let χ be any of the symbols s, or s^(c). Show that if x satisfies (11), then χ_aλ+s⁰_µx =s⁰_bτ and bτ ∈Cc1. Reasoning as in the proof of Theorem 9, we have that (11) is equivalent to

(12) χ_aλ+s⁰_µx=s⁰_b(

D1/τ∆)

=s⁰_bτ(∆), and since we haves⁰_aλ⊂χ_aλ⊂s_aλ ands⁰_µx⊂s_µx, we deduce

s⁰_aλ+µx=s⁰_aλ+s⁰_µx⊂χ_aλ+s⁰_µx⊂s_aλ+s_µx=s_aλ+µx. Then

s⁰_aλ+µx⊂s⁰_bτ(∆)⊂saλ+µx. The first inclusion is equivalent toI∈(

s⁰_aλ+µx, s⁰_bτ(∆) )

and toD_1/bτ∆D_aλ+µx∈ (c0, c0). Since (c0, c0)⊂(c0, s1) =S1, we deduce

anλn+µ_nxn

b_nτ_n ≤K for alln.

The second inclusion yields ∆⁻¹= Σ∈(

s⁰_bτ, saλ+µx

), that isD1/(aλ+µx)ΣDbτ ∈ (c0, ℓ_∞) =S1 and

b₁τ₁+...+b_nτ_n anλn+µ_nxn

≤K^′ for alln.

We deduce

b1τ1+...+bnτn

b_nτ_n = b1τ1+...+bnτn

a_nλ_n+µ_nx_n

anλn+µ_nxn

b_nτ_n ≤KK^′ for alln.

We concludebτ ∈Cc₁and by (12) and Lemma 8 we haveχ_aλ+s⁰_µx=s⁰_bτ. Conversely if χ_aλ+s⁰_µx =s⁰_bτ and bτ ∈Cc1, then (12) and (11) hold. We conclude the proof using Lemma 17.

Remark 19. Note that the (SSE) in (11) has solutions if and only if bτ ∈Cc₁ andaλ/bτ ∈c₀.

4. On the equation χ

(C (λ) C (µ)) + χ

(C (λσ) C (µ)) = χ

In this section fora,b,λ,µ,σ∈U⁺we consider an equation that generalizes (SSE) (3) and defined forb∈Cc1 by

(13) χ_a(C(λ)C(µ)) +χ_x(C(λσ)C(µ)) =χ_b,

where χis any of the symbolss, or s⁰. For χ=s⁰ the resolution of equation (13) consists in determining the set of allx∈U⁺ such that for everyy∈sthe conditiony_n/b_n→0 (n→ ∞) holds if and only if there areu,v∈ssuch that y=u+vand

(14) 1 λnan

∑n m=1

( 1 µ_m

∑m k=1

uk

)

→0 and 1 λnσnxn

∑n m=1

( 1 µ_m

∑m k=1

vk

)

→0 (n→ ∞). To solve equation (13) we state the following proposition.

Proposition 20. Assume that b∈Cc1. Then

(i) if b/µ /∈Cc1, then equation (13) has no solution.

(ii) Let b/µ∈Cc1. Then

(a) if aλµ/b∈c0, then equation (13) holds if and only ifsx=s_b/λσµ; (b) ifaλµ/b,b/aλµ∈ℓ_∞, then equation (13) holds if and only ifx∈s_b/λσµ; (c) ifaλµ/b /∈ℓ_∞, then equation (13) has no solution.

Proof. Equation (13) is equivalent to ∆ (µ) (∆ (λ)χ_a+ ∆ (λσ)χ_x) = χ_b, that is

(15) ∆ (λ)χ_a+ ∆ (λσ)χ_x=χ_b(∆ (µ)) =D_1/µχ_b(∆)

and since b ∈ Cc1, we have D1/µχ_b(∆) = D1/µχ_b = χ_b/µ. So equation (15) is equivalent to χ_aλ+χ_λσx = χ_b/µ(∆). Then by Lemma 8 equation (15) is equivalent to b/µ ∈Cc₁ and χ_aλ+χ_λσx =χ_b/µ. We conclude by Theorem 1 and Corollary 6 that if aλµ/b∈c0 equation,χ_aλ+χ_λσx =χ_b/µ is equivalent tosx=s_b/λσµ. The cases (b) and (c) follow immediately from Theorem 1.

Example 21. The set of all x∈U⁺ such that yn/2ⁿ =O(1) (n→ ∞) holds if and only if there are u,v∈ssuch that y=u+v and

(16) 1 n

∑n m=1

( 1 m

∑m k=1

uk

)

=O(1) and 1 xn

∑n m=1

( 1 m

∑m k=1

vk

)

= 1

nO(1) (n→ ∞) for ally is given by

(17) K₁2ⁿ≤x_n≤K₂2ⁿ for alln.

Indeed, the previous statement is equivalent to the equation

(18) ℓ_∞(

C₁²)

+sx(C((1/n)_n)C1) =s2.

We haveb= (2ⁿ)_n≥1∈Cc1,b/µ= (2ⁿ/n)_n≥1∈Cc1andanλnµ_n/bn=n²2⁻ⁿ → 0 (n→ ∞). So we obtain (17). Furthermore for eachxsatisfying (17), we have

(ℓ_∞( C₁²)

+sx(C((1/n)_n)C1), sα

)= (s2, sα) for α∈U⁺.

SoA∈( ℓ_∞(

C₁²)

+sx(C((1/n)_n)C1), sα

)if and only if

sup_n(

α⁻_n¹∑_∞

m=1|anm|2^m)

<∞.

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Received by the editors August 18, 2011