Qualitative approximation of solutions to discrete Volterra equations

Qualitative approximation of solutions to discrete Volterra equations

Janusz Migda

and Małgorzata Migda

1Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Pozna ´n, Poland

2Institute of Mathematics, Pozna ´n University of Technology, Piotrowo 3A, 60-965 Pozna ´n, Poland

Received 11 November 2017, appeared 9 February 2018 Communicated by Stevo Stevi´c

Abstract.We present a new approach to the theory of asymptotic properties of solutions to discrete Volterra equations of the form

∆^mxn=bn+

∑

K(n,k)f(k,x_σ(k)).

Our method is based on using the iterated remainder operator and asymptotic differ- ence pairs. This approach allows us to control the degree of approximation.

Keywords: Volterra discrete equation, difference pair, prescribed asymptotic behavior, asymptotically polynomial solution, bounded solution.

2010 Mathematics Subject Classification: 39A10, 39A22, 39A24.

1 Introduction

Let N, R denote the set of positive integers and real numbers respectively. Let m ∈ _{N. We} consider the nonlinear discrete Volterra equations of non-convolution type

∆^mx_n=b_n+

∑

K(n,k)f(k,x_σ(k)), (E) bn∈_R, f :N×_R→_R, K:N×_N→_R, σ:N→_N, σ(k)→_∞.

By asolutionof (E) we mean a sequencex:N→_Rsatisfying (E) for every largen. We say that x is a full solutionof (E) if (E) is satisfied for every n. Moreover, if p ∈ _Nand (E) is satisfied for every n ≥ p, then we say thatx is a p-solution. In this paper we regard equation (E) as a generalization of the equation

∆^mx_n =a_nf(n,x_σ(n)) +b_n. (1.1)

BCorresponding author. Email: malgorzata.migda@put.poznan.pl

Indeed, ifK(n,k) = 0 fork 6= n, then denotinga_n = K(n,n)we may rewrite (E) in the form (1.1). Hence the ordinary difference equation (1.1) is a special case of (E).

Volterra difference equations appeared as a discretization of Volterra integral and integro- differential equations. They also often arise during the mathematical modeling of some real life situations where the current state is determined by the whole previous history. Therefore, many papers have been devoted to these types of equations during the last few years. For example, the boundedness of solutions of such equations was studied in [6,12,17–22,25,39–41, 44]. Some results on the boundedness and growth of solutions of related difference equations were proved also in [45–47]. The periodicity was investigated, e.g., in [1,9–11,16,22,37,43].

Several fundamental results on the stability of linear Volterra difference equations, of both convolution and non–convolution type, can be found in [7,8,15]; see also [2,5,23,24,26,40,48].

Some related results on dynamic equations can be found in [3] and [4].

In recent years the first author presented a new theory of the study of asymptotic proper- ties of the solutions to difference equations. This theory is based mainly on the examination of the behavior of the iterated remainder operator and on the application of asymptotic differ- ence pairs. This approach allows us to control the degree of approximation. The theory was formed in three stages:

(S1) the approximation of solutions with accuracy o(1), (papers [27,28]),

(S2) the approximation with accuracy o(n^s),s∈(−_∞, 0], (papers [29,30,32,34,35]),

(S3) the approximation with accuracy determined by a certain asymptotic difference pair (papers [33,36]).

In papers [34,35] this new theory was applied to the study of neutral type equations. The application to the discrete Volterra equations was presented in [38] (stage (S1)) and in [37]

(stage (S2)). In this paper we continue those investigations by applying asymptotic difference pairs and we generalize the main results from [27–31,33,37,38]. Moreover, we generalize some earlier results, for example, from [13,14,25,42,49].

The paper is organized as follows. In Section 2, we introduce notation and terminology. In Section 3, in Theorems3.1and3.2, we obtain our main results. In Section 4, we present some consequences of our main results. These consequences concern asymptotically polynomial solutions. In the next section we use our results to investigate bounded solutions. In Section 6, we give some remarks. In particular, we present some tests that are helpful in verifying whether a given kernelKfulfills the assumptions of the main theorems. In the last section we present some applications.

2 Notation and terminology

In the paper we regardN×_Ras a metric subspace of the Euclidean planeR². The spaceR^N of all real sequences we denote also by SQ. Moreover

SQ^∗ ={x∈SQ : xn 6=0 for anyn}. For integers p,qsuch that 0≤ p≤q, we define

N(p) ={p,p+_1,p+_{2, . . .}}_{, N}(p,q) ={p,p+_{1, . . . ,}q}_.

We use the symbols

Sol(E), Solp(E)

to denote the set of all solutions of (E), and the set of all p-solutions of (E) respectively. If x,y in SQ, then

xy and |x|

denotes the sequences defined by xy(n) =xnyn and|x|(n) =|xn|respectively. Moreover kxk=sup

n∈_N

|x_n|.

If there exists a positive constantλsuch thatx_n≥λfor any n, then we write x_0.

Let a,b,w∈SQ,p∈ _N, t∈[1,∞),X⊂SQ. We will use the following notations Fin(p) ={x ∈SQ :x_n=0 forn≥ p}, Fin=

∞

[

p=1

Fin(p).

o(1) ={x ∈SQ : xis convergent to zero}, O(1) ={x ∈SQ : xis bounded}, o(a) ={ax: x∈o(1)}+Fin, O(a) ={ax: x ∈O(1)}+Fin,

O(w,σ) ={y∈SQ :y◦σ∈ O(w)}, A(t):=

(

a∈SQ :

∑

n^t−1|an|<_∞ )

, A(_∞) = ^\

t∈[_1,∞)

A(t),

∆^−mb={y∈SQ : ∆^my=b}, ∆^−mX= {y∈SQ : ∆^my∈X}, Pol(m−1) =_Ker∆^m =_∆^−m0.

Note that Pol(m−1)is the space of all polynomial sequences of degree less thanm. Moreover for any y∈_∆^−mbwe have

∆^−mb=y+Pol(m−1). Note also that

[

λ∈(0,1)

O(λⁿ)⊂A(_∞)⊂ ^\

s∈_R

o(n^s). For L:N×_N→_R, A⊂SQ, andt∈[1,∞]we define

L⁰ ∈SQ, L⁰(n) =

∑

|L(n,k)|, K(A) =ⁿL∈_R^N×N: L⁰ ∈ Ao

, K(t) =K(A(t)). For a subsetYof a metric space Xandε>0 we define anε-framedinterior ofY by

Int(Y,ε) ={x ∈X: B(x,ε)⊂Y}

where B(x,ε) denotes a closed ball of radius ε centered at x. We say that a subset U of X is a uniform neighborhood of a subset Z of X, if there exists a positive number ε such that Z⊂Int(U,ε). We say that a functionh: N×_R→_R is unbounded at a point p∈ [−_∞,∞]if there exist sequences x:N→_Randn:N→_Nsuch that

klim→_∞x_k = p, lim

k→_∞n_k =_{∞ and lim}

k→_∞h(n_k,x_k) =_∞.

Leth:N×_R→_R,x ∈SQ. We will use the following notations U(h) ={p∈ [−_∞,∞]: his unbounded at p},

L(x) ={p∈ [−_∞,∞]: pis a limit point ofx}. Letg:[0,∞)→[0,∞)andw∈SQ^∗, we say that f is(g,w)-dominatedif

|f(n,t)| ≤g(|tw⁻_n¹|) for(n,t)∈_N×_R. (2.1) We say that a functiong :[0,∞)→[0,∞)is ofBihari typeif

Z _∞

1

dt

g(t) =_∞. (2.2)

2.1 Remainder operator Let

S(m) = (

a ∈SQ : the series

∑

∑

· · ·

∑

im=i_m−1

a_im is convergent )

. For anya∈S(m)we define the sequencer^m(a)by

r^m(a)(n) =

∑

∑

· · ·

∑

im=im−1

a_im.

Then S(m)is a linear subspace of o(1),r^m(a)∈o(1)for any a∈S(m)and r^m : S(m)→o(1)

is a linear operator which we call the remainder operator of order m. The value r^m(a)(n) we denote also byr^m_n(a)or simplyr^m_na. If a∈A(m), thena∈S(m)and

r^m(a)(n) =

∑

m−1+j−n m−1

a_j. (2.3)

for anyn ∈ N. The following lemma is a consequence of [31, Lemma 3.1, Lemma 4.2, and Lemma 4.8].

Lemma 2.1. Assume a∈A(m), u∈O(1), k∈ {0, 1, . . . ,m}, and p∈_{N. Then} (a) O(a)⊂A(m)⊂o(n^1−m), |r^m(ua)| ≤ kukr^m|a|, ∆r^m|a| ≤0,

(b) |r^m_pa| ≤r^m_p|a| ≤_∑^∞_n=pn^m−1|an|, r^ka ∈A(m−k), (c) _∆^mr^ma= (−1)^ma, r^mFin(p) =Fin(p) =_∆^mFin(p).

For more information about the remainder operator see [31].

2.2 Asymptotic difference pairs

We say that a pair (A,Z) of linear subspaces of SQ is anasymptotic difference pairof order m or, simply,m-pairif

Fin+Z⊂Z, O(1)A⊂ A, A⊂ _∆^mZ.

We say that anm-pair(A,Z)isevanescentifZ⊂o(1). IfA⊂SQ and(A,A)is anm-pair, then we say that Ais anm-space. We will use the following lemma.

Lemma 2.2. Assume(A,Z)is an m-pair, a,b,x∈SQ, and W⊂SQ. Then (a) if Z+W ⊂W and b−a∈ A, then W∩_∆^−mb+Z=W∩_∆^−ma+Z, (b) if a ∈ A and∆^mx∈O(a) +b, then x∈_∆^−mb+Z,

(c) if Z⊂o(1), then A ⊂A(m)and r^mA⊂ Z.

Proof. Lety∈W∩_∆^−ma. Then

∆^my−b=a−b∈ A⊂_∆^mZ.

Hence∆^my−b=_∆^mzfor somez ∈Z. Therefore∆^m(y−z) = band we obtainy−z∈ _∆^−mb.

Moreovery−z∈W+Z⊂W. Hencey−z∈W∩_∆^mb. Ifz₁∈ Z, then y+z₁ =y−z+z+z₁ ∈W∩_∆^−mb+Z and we obtain

W∩_∆^−ma+Z⊂W∩_∆^−mb+Z.

Since Ais a linear space, the reverse inclusion follows by interchanging the letters aandbin the previous part of the proof. Hence we get (a). For the proof of (b) see [33, Lemma 3.7]. (c) is a consequence of [33, Remark 3.4].

Example 2.3. Assumes ∈_R,(s+1)(s+2). . .(s+m)6=0, andt ∈(−_∞,m−1]. Then (_o(_n^s)_{, o}(_n^s+m))_, (_O(_n^s)_{, O}(_n^s+m))_, (_A(_m−t)_{, o}(_n^t))

are m-pairs.

Example 2.4. Ifλ∈ (1,∞), then o(λⁿ)and O(λⁿ)arem-spaces.

Example 2.5. Ifk∈_N(0,m−1), then(A(m−k),∆^−ko(1))is an asymptoticm-pair.

Example 2.6. Assumes ∈(−_∞,−m), t∈(−_∞, 0], andu∈[1,∞). Then

(o(n^s), o(n^s+m)), (O(n^s), O(n^s+m)), (A(m−t), o(n^t)), (A(m+u), A(u)) are evanescentm-pairs.

Example 2.7. Ifλ∈ (0, 1), then o(λⁿ), O(λⁿ), and A(_∞)are evanescentm-spaces.

For more information about difference pairs see [33].

2.3 Fixed point lemma

We will use the following fundamental lemma.

Lemma 2.8. Assume y∈ SQ,ρ∈_o(₁), and

S={x∈SQ : |x−y| ≤ |ρ|}.

Then the formula d(x,y) = sup_n∈N|x_n−y_n| defines a metric on S such that any continuous map H :S→_Shas a fixed point.

Proof. The assertion is a consequence of [32, Theorem 3.3 and Theorem 3.1].

3 The set of solutions

In this section, in Theorems3.1and3.2, we obtain our main results.

For a sequence x∈SQ we define sequencesF(x)andG(x)by F(x)(k) = f(k,x_σ(k))_, G(x)(n) =

∑

K(n,k)f(k,x_σ(k))_{. (3.1)} Let K ∈ K(m)and p ∈ N. We say that a sequence y ∈ SQ is (K,f,p)-regular if there exist a subsetUof RandM >0 such that

y(_N)⊂Int(U,Mr^m_pK⁰), |f(n,t)| ≤M for any (n,t)∈_N×U, (3.2) and the restriction f|_N×U is continuous. We say thaty is f -regular if there exist a uniform neighborhoodUofy(_N)such that the restriction f|_N×Uis continuous and bounded.

We say that a subset W of SQ is (f,σ)-ordinary if for any y ∈ W the sequence F(y) _is bounded. If anyy∈W is f-regular, then we say thatW is f-regular.

Theorem 3.1. Assume(A,Z)is an m-pair, K∈K(A), and W ⊂SQ. It follows that (A1) if W is(f,σ)-ordinary, then W∩Sol(E)⊂ _∆^−mb+Z.

Moreover, assume that the pair(A,Z)is evanescent, y∈ _∆^−mb, and p∈N. It follows that (A2) if y is(K,f,p)-regular, then y ∈Sol_p(E) +Z,

(A3) if y is f -regular, then y∈Sol(E)+Z,

(A4) if W is f -regular and Z+W ⊂W, then W∩Sol(E)+Z=W∩_∆^−mb+Z, (A5) if Z+W ⊂W and f is continuous and bounded, then

W∩Sol₁(_E) +_Z=_W∩Sol(_E) +_Z=_W∩_∆^−mb+_Z.

Theorem 3.2. Assume (A,Z) is an m-pair, K ∈ K(A), w ∈ SQ^∗, g : [0,∞) → [0,∞), and f is (g,w)-dominated. It follows that

(B1) if g is locally bounded, then O(w,σ)∩Sol(E)⊂_∆^−mb+Z,

(B2) if g is nondecreasing,σ(n)≤ n for large n, K ∈ K(1), b∈ A(1), w⁻¹ ∈ O(n^1−m), and g is of Bihari type, then

Sol(E)⊂_∆^−mb+Z.

Moreover, assume that the pair (A,Z)is evanescent, y∈ _∆^−mb, W ⊂O(w,σ), L,M >0, p ∈_{N, f} is continuous, and Z+W ⊂W. It follows that

(B3) if g[0,L]⊂[0,M]and|y◦σ| ≤L|w| −Mr^m_pK⁰, then y∈Sol_p(E) +Z.

Moreover, assume that g is locally bounded and|w| 0. It follows that (B4a) W∩_∆^−mb+Z=W∩Sol(E) +Z,

(B4b) _O(w,σ)∩_∆^−mb+Z=_O(w,σ)∩_Sol(_E) +Z,

(B4c) if w◦σ∈O(w), thenO(w)∩_∆^−mb+Z=O(w)∩Sol(E) +Z.

Moreover, assume that g is bounded. It follows that

(B5a) W∩_∆^−mb+Z=W∩Sol(E) +Z=W∩Sol₁(E) +Z,

(B5b) O(w,σ)∩_∆^−mb+Z=O(w,σ)∩Sol(E) +Z=O(w,σ)∩Sol₁(E) +Z,

(B5c) if w◦σ∈O(w), thenO(w)∩_∆^−mb+Z=O(w)∩Sol(E) +Z=O(w)∩Sol₁(E) +Z.

The following, final, theorem is a curiosity. It concerns all the solutions of equation (E);

moreover there are no conditions placed on the function f. This theorem generalizes [33, Theorem 4.2].

Theorem 3.3. Assume(A,Z)is an m-pair, K∈K(A), and x ∈Sol(E). Then x ∈_∆^−mb+Z or L(x)∩U(f)6=_∅.

3.1 The proof of Theorem3.1

(A1) AssumeW is(f,σ)-ordinary and x ∈ W∩Sol(E). Let M = kF(x)k. By (3.1),|G(x)| ≤ MK⁰. Hence

∆^mx∈ G(x) +b+Fin⊂O(K⁰) +b+Fin=O(K⁰) +b.

Moreover K⁰ ∈ A. Therefore, using Lemma2.2, we obtainx ∈_∆^−mb+Z.

(A2) Choose a positive constant M and a subset U of R such that (3.2) is satisfied and f is continuous onN×U. Leta=K⁰. Defineρ∈SQ andS⊂SQ by

ρ_n =

(Mr^m_na forn≥ p,

0 forn< p, S={x∈SQ :|x−y| ≤ρ}. (3.3) Since the sequencer^m|a|is nonincreasing, we haveρ_n≤ ρ_pfor anyn. Assumex∈ S. Ifk∈ _N, then|x_σ(k)−y_σ(k)| ≤ρ_σ(k)≤ ρ_pand we obtain

x_σ(k)∈B(y_σ(k),ρ_p)⊂U.

Hence|f(k,x_σ(k))| ≤ M. Therefore, forn∈_N, we get

|G(x)(n)|=

∑

K(n,k)f(k,x_σ(k))

≤

∑

|K(n,k)||f(k,x_σ(k))| ≤ Ma_n.

Thus, for anyx ∈S, we haveGx∈ O(a)⊂ A⊂A(m). Let H:S→SQ, H(x)(n) =

(y_n forn< p

yn+ (−1)^mr^m_nGx forn≥ p. (3.4) Ifx∈Sandn≥ p, then

|H(x)(n)−yn|=|r^m_nGx| ≤r^m_n|Gx| ≤ Mr^m_na =ρn. HenceHS⊂S. Let ε>0. Chooseq∈_Nandβ>0 such that

M

∑

n^m−1a_n<ε and β

∑

n^m−1a_n<_{ε. (3.5)} Let

D={(n,t)∈_N×_R:n∈_N(p,q) and |t−y_σ(n)| ≤ρ_n}.

Then D is a compact subset of R². Hence f is uniformly continuous on D and there exists δ>0 such that if(n,s),(n,t)∈Dand|s−t|<δ, then

|f(n,s)− f(n,t)|< β.

Letx,z∈S,kx−zk<δ. Using Lemma2.1we obtain kHx−Hzk= kr^m(Gx−Gz)k=sup

n≥p

|r_n^m(Gx−Gz)| ≤sup

n≥p

r^m_n|Gx−Gz|

= r^m_p|Gx−Gz| ≤

∑

n^m−1|G(x)(n)−G(z)(n)|

≤

∑

n^m−1|G(x)(n)−G(z)(n)|+

∑

n^m−1|G(x)(n)−G(z)(n)|

≤ β

∑

n^m−1a_n+

∑

n^m−1|G(x)(n)|+

∑

n^m−1|G(z)(n)|

≤ ε+M

∑

n^m−1a_n+M

∑

n^m−1a_n≤ 3ε.

Hence the map H : S → S is continuous. By Lemma 2.8, there exists an x ∈ S such that Hx =x. Then, forn≥ p, we getxn =yn+ (−1)^mr^m_nGx. Hence

x−y−(−1)^mr^mGx∈Fin(p). (3.6) Therefore, by Lemma2.1,

∆^mx−b−Gx∈_∆^mFin(p) =Fin(p). Thusx ∈Sol_p(E). Moreover,Gx∈O(a)⊂ A. By (3.6), we have

y∈ x+_r^m_A+_Fin(_p)⊂ x+_Z⊂Solp(_E) +_Z.

(A3)Now, we assume that y is f-regular. Choose a uniform neighborhood U of y(_N)such that the restriction f|_N×U is continuous and bounded. There exists a positive constant c such thaty(_N)⊂Int(U,c). Let

M =_sup{|f(n,t)|_:n∈_N,t ∈U}_.

Sincer^mK⁰ ∈o(1), there exists an index psuch thatMr^m_pK⁰ ≤c. Then y(_N)⊂Int(U,c)⊂Int(U,Mr^m_pK⁰)⊂U.

This means thatyis(K,f,p)-regular. By (A2), we gety∈Sol_p(E) +Z⊂Sol(E) +Z.

(A4)Now, we assume thatW is f-regular andZ+W ⊂W. Let S =Sol(E), Y=_∆^−mb.

Obviously, W is (f,σ)-ordinary. If w ∈ W∩S, then, by (A1), w = y+z for somey ∈ Y and z∈Z. Hencey=−z+w∈ Z+W ⊂W. Thereforew=y+z∈W∩Y+Zand we obtain

W∩S+Z⊂W∩Y+Z.

If w ∈ W∩Y, then, by (A3), w = x+z for some x ∈ S and z ∈ Z. Hence x = −z+w ∈ Z+W ⊂W. Therefore w= x+z∈W∩S+Zand we obtain

W∩Y+Z⊂W∩S+Z.

(A5)Now we assume that f is continuous and bounded and Z+W ⊂W. By (A4) we have W∩Sol(E)+Z=W∩_∆^−mb+Z.

Since Sol₁(E)⊂Sol(E), we get

W∩Sol₁(E)+Z⊂W∩_∆^−mb+Z.

Let M =sup{|f(n,t)|:(n,t)∈_N×_R}and letU=R. Then for anyy∈ SQ we have y(_N)⊂_R=Int(U,Mr^m₁K⁰).

Since f is continuous onR, anyy ∈SQ is(K,f, 1)-regular. Hence, by (A2), we obtain W∩_∆^−mb+Z⊂W∩_Sol1(E)+Z.

3.2 The proof of Theorem3.2 We will use the following three lemmas.

Lemma 3.4([35, Lemma 4.1]). Assumeα,u∈SQare nonnegative, p∈_{N, g}:[0,∞)→[0,∞), 0≤c<_β, g(c)>_0, u_n ≤c+

n−1 j

∑

α_jg(u_j) for n≥ p,

∑

α_j ≤

Z _β

c

dt g(t)^, and g is nondecreasing. Then u_n≤βfor n≥ p.

Lemma 3.5([30, Lemma 7.3]). If x is a sequence of real numbers, m∈_Nand p∈_N(m)then there exists a positive constant L= L(x,p,m)such that

|x_n| ≤n^m−1 L+

n−1 i

∑

|_∆^mx_i|

!

for n≥ p.

Lemma 3.6. Let w∈SQ

(1) if|w| 0, thenO(w) +O(1)⊂O(w), andO(w,σ) +O(1)⊂O(w,σ), (2) if y∈O(w,σ), thenO(y)⊂O(w,σ),

(3) if w◦σ∈O(w), thenO(w)⊂O(w,σ).

Proof. Lety∈O(w)andu∈O(1). Choose positive δ,L,M∈ _Rsuch that

|wn| ≥δ, |un| ≤ L, and |yn| ≤ M|wn| for anyn. Then

|y_n+u_n| ≤ |y_n|+|u_n| ≤M|w_n|+L= M|w_n|+Lδ⁻¹δ ≤ M|w_n|+Lδ⁻¹|w_n|= (M+Lδ⁻¹)|w_n| for any n. Hence O(w) +O(1) ⊂ O(w). Similarly O(w,σ) +O(1) ⊂ O(w,σ). Assume y∈_O(w,σ)_andx ∈_O(y). There exist positive constants M,Psuch that

|y(σ(n))| ≤ M|w_n|, |x_n| ≤P|y_n|

for largen. Then |x(σ(n))| ≤ P|y(σ(n))| ≤ PM|w_n| for largen. Hencex ∈ O(w,σ) and we get (2). (3) is a consequence of (2).

Now we start the proof of Theorem3.2.

(B1)Assumeg is locally bounded. LetP be a positive constant. For anyt ∈ [0,P]there exist a neighborhoodU_t of t and a positive constant Q_t such that |g(s)| ≤ Q_t for any s ∈ U_t. By compactness of[0,P]we can chooset₁,t2, . . . ,tnsuch that[0,P]⊂Ut₁∪Ut2∪ · · · ∪Utn. Then

g(s)≤Q=max{Q_t1, . . . ,Q_tn} (3.7) for anys ∈[0,P]. Let y ∈O(w,σ). Theny◦σ ∈ O(w). Sincew ∈SQ^∗, there exists a positive constantPsuch that

|y_σ(n)| ≤P|wn| (3.8)

for anyn. Using (2.1), (3.8), and (3.7) we get

|F(x)(n)|= |f(n,y_σ(n))| ≤g |y_σ(n)|

|wn|

!

≤ Q.

Hence the set O(w,σ)is(f,σ)-ordinary and, by Theorem3.1(A1), we obtain O(w,σ)∩Sol(_E)⊂_∆^−mb+Z.

(B2)Assumex is a solution of (E). Since K∈K(1), we haveK⁰ ∈A(1). Hence

∑

∑

|K(i,j)|=

∑

∑

|K(i,j)|=

∑

K⁰(i)<_∞.

ChooseM >0 such that|w_n⁻¹| ≤ Mn^1−m. For j∈_Nlet u_j = x_σ(j)w⁻_j ¹

, α_j = M

∑

|K(i,j)|.

Using the condition: K(i,j) =0 fori<jwe obtain

∑

α_j = M

∑

∑

|K(i,j)| ≤M

∑

∑

|K(i,j)|= M

∑

∑

|K(i,j)|

= M

∑

∑

|K(i,j)|= M

∑

K⁰(i)<_∞.

By Lemma3.5, there exists a positive constant Lsuch that

|x_σ(n)| ≤σ(n)^m−1 L+

σ(n)−1 i

∑

|_∆^mx_i|

!

≤ n^m−1 L+

n−1 i

∑

|_∆^mx_i|

! . Letc= ML+M∑^∞i=1|b_i|. Then

u_n= x_σ(n)w_n⁻¹

≤ ML+M

n−1

∑

|_∆^mx_i|= ML+M

n−1 i

∑

b_i+

∑

K(i,j)f

j,x_σ(j)

≤ ML+M

∑

|b_i|+M

n−1 i

∑

∑

|K(i,j)|g(u_j) =c+M

n−1 i

∑

n−1 j

∑

|K(i,j)|g(u_j)

= c+M

n−1 j

∑

n−1 i

∑

|K(i,j)|g(u_j)≤ c+M

n−1 j

∑

∑

|K(i,j)|g(u_j)

= c+

n−1 j

∑

∑

M|K(i,j)|g(u_j) =c+

n−1 j

∑

α_jg(u_j).

Hence, by Lemma 3.4, the sequence u is bounded. Therefore, there exists a constant Q > 1 such thatg(u_i)≤ Qfor any iand we get

f(i,x_σ(i))

≤ g

x_σ(i)w⁻_i ¹

= g(u_i)≤Q for any i. Hence

∑

K(n,i)f(i,x_σ(i))

≤Q

∑

|K(n,i)|= QK⁰_n. For large nwe have

∆^mx_n =b_n+

∑

K(n,i)f(i,x_σ(i)).

Hence∆^mx ∈b+O(K⁰)andK⁰ ∈ A. By Lemma2.2, we havex ∈_∆^−mb+Z.

(B3)Leta =K⁰. Defineρ andSby (3.3). Letx∈S. Using the inequality

|y◦σ| ≤ L|w| −Mr^m_pa, we get

x_σ(n) wn

=

x_σ(n)−y_σ(n)+y_σ(n)

|wn|

≤ |x_σ(n)−y_σ(n)|+|y_σ(n)|

|wn| ≤ ^Mr

mpa+|y_σ(n)| wn

≤ L for any n. Using (2.1) and inclusiong[0,L]⊂ [0,M], we have

|F(x)(n)|=|f(n,x_σ(n))| ≤g |x_σ(n)| w_n

!

≤ M

for anyn. Therefore

|G(x)(n)|=

∑

K(n,k)F(x)(k)

≤

∑

M|K(n,k)| ≤Ma_n. Now, repeating the second part of the proof of Theorem3.1(A2), we obtain

y∈Solp(_E) +Z.

(B4a)Now, we assume that gis locally bounded,|w| 0,W ⊂O(w,σ), andZ+W ⊂W. Let y∈W∩_∆^−mb. Choose positive constants P,λsuch that|y◦σ| ≤ P|w|and|w|>λ. Let

L₁ =P+1 and α=inf{L₁|wn| − |y_σ(n)|:n∈_N}. Then

L₁|wn| − |y_σ(_n)|=P|wn| − |y_σ(_n)|+|wn| ≥P|wn| − |y_σ(_n)|+λ≥λ

for anyn. Hence α≥λ>0. Similarly as in (3.7) there exists a positive constant M₁such that g[0,L₁]⊂ [0,M₁]. Since lim

n→_∞r^m_n|a|=0, there exists an index psuch that M₁r^m_p|a| ≤_α.

ThenM₁r^m_p|a| ≤L₁w_n− |y_σ(n)|for anyn. Hence, by (B3),y∈Sol_p(E) +Zand we obtain W∩_∆^−mb⊂Sol(E) +Z.

By (B1), we haveW∩_Sol(_E)⊂_∆^−mb+Z. Using [33, Lemma 4.10] we obtain W∩_∆^−mb+Z=W∩Sol(E) +Z.

(B4b)SinceZ⊂o(1), by Lemma3.6 (1), we have O(w,σ) +Z⊂O(w,σ). Hence, by (B4a), we get

O(w,σ)∩_∆^−mb+Z=O(w,σ)∩Sol(E)+Z.

(B4c)By Lemma3.6(1) and (3) we have

O(w) +Z⊂O(w) and O(w)⊂O(w,σ). Hence (B4c) is a consequence of (B4a).

(B5a)Since Sol₁(E)⊂Sol(E) we have

W∩_Sol1(E)+Z⊂W∩_Sol(E)+Z. (3.9) Choose M,δ ∈ (0,∞) such that|g| ≤ M and|w| ≥δ. Lety ∈ W∩_∆^−mb. Sincey ∈ O(w,σ), there exists a positivePsuch that|y◦σ| ≤P|w|. Let

L= P+δ⁻¹Mr^m₁K⁰. Then

|y◦σ| ≤P|w|=L|w| −δ⁻¹|w|Mr^m₁K⁰ ≤ L|w| −Mr^m₁K⁰. Moreoverg[0,L]⊂ [0,M]. Hence, by (B3),y∈Sol₁(E)+Zand we obtain

W∩_∆^−mb⊂_Sol1(E)+Z

Letw∈W∩_∆^−mb. Choosex∈Sol₁(E) andz∈Zsuch thatw=x+z. Then x= w−z∈W+Z⊂W.

Hencew∈W∩Sol₁(E)+Zand we obtain

W∩_∆^−mb⊂W∩Sol₁(E)+Z. (3.10) By (B4a) we have

W∩_∆^−mb+Z=W∩Sol(_E) +Z (3.11) Using (3.9), (3.10), and (3.11) we obtain (B5a).

(B5b)Analogously to the proof of (B4b), we can see that (B5b) is a consequence of (B5a).

(B5c)The assertion is a consequence of (B5a) and Lemma3.6(1) and (2).

3.3 The proof of Theorem3.3 Assume

L(x)∩U(f) =_∅. (3.12)

We will show that the sequence F(x)is bounded. If lim sup

n→_∞

F(x)(n) =lim sup

n→_∞

f(n,x_σ(n)) =_∞, then there exists an increasing sequence (_nk)of natural numbers such that

klim→_∞ f(n_k,x_σ(nk)) =_∞.

Lety_k =x_σ(nk) and let p∈L(y). There exists a subsequence(y_ki)of (y_k)such that

ilim→_∞y_ki = p.

Then lim_i→_∞ f(n_ki,y_ki) = _{∞. Hence} p ∈ U(f). Since y_k = x_σ(nk) and σ(n) → _{∞, we have} L(y)⊂L(x). Therefore p∈L(x)which contradicts (3.12). Analogously lim infF(x)(n)>−_∞ and so F(x)is bounded. Sincex ∈_Sol(_E)_{we have}

∆^mx ∈aF(x) +b+Fin⊂O(a) +b+Fin=O(a) +b and, by Lemma2.2(b), we obtain x∈_∆^−mb+Z.

4 Asymptotically polynomial solutions

In this section we apply our main results to investigate asymptotically polynomial solutions of equation (E). We assume that g:[0,∞)→[0,∞)andw∈SQ^∗.

Letk∈_N(0,m). We say that a sequence ϕisasymptotically polynomial of type(m,k)if ϕ∈Pol(m) +o(n^k).

Moreover, if

ϕ∈Pol(m) +_∆^−ko(1),

then we say that ϕ is regularly asymptotically polynomial of type (m,k). Note that, by [30, Lemma 3.1 (b)], we have

∆^−ko(1) ={x ∈o(n^k): ∆^px∈o(n^k−p) for any p∈_N(0,k)}.

Corollary 4.1. Assume(A,Z)is an m-pair, K ∈K(A), b∈ A, and x is an(f,σ)-ordinary solution of (E). Then

x∈Pol(m−1) +Z.

Proof. By Theorem 3.1 (A1), we have x ∈ _∆^−mb+Z. Since b−0 ∈ A, taking W = SQ in Lemma2.2 (a), we obtain∆^−mb+Z=_∆^−m0+Z=Pol(m−1) +Z.

Note that if k ∈ _N(0,m−1) and Z ⊂ o(n^k), then by Corollary 4.1, any (f,σ)-ordinary solution of (E) is asymptotically polynomial of type(m−1,k).

Corollary 4.2. Assume s∈(−_∞,m−1], K∈K(m−s), b∈A(m−s), and x is an(f,σ)-ordinary solution of (E). Then

x∈Pol(m−1) +o(n^s).

Proof. By Example 2.3, (A(m−s), o(n^s)) is an asymptotic m-pair. Hence the assertion is a consequence of Corollary4.1.

Corollary 4.3. Assume k ∈ _N(0,m−1), K ∈ K(m−k), and b ∈ A(m−k). Than any (f,σ)- ordinary solution x of (E)is regularly asymptotically polynomial of type(m−1,k).

Proof. By Example2.5, (A(m−k), ∆^−ko(1))is an asymptoticm-pair. Hence, by Corollary4.1 we obtain

x ∈Pol(m−1) +_∆^−ko(1).

Corollary 4.4. Assume s ∈ (−_∞,m−1], K ∈ K(m−s), b ∈ A(m−s). Then for any (f,σ)- ordinary solution x of (E)there exist a sequenceϕ∈ Pol(m−1)and z∈o(n^s)such that x= ϕ+z and∆^pz_n=o(n^s−p)for any p∈_N(1,m).

Proof. By [33, Example 5.3], (A(m−s),r^mA(m−s)) is an m-pair. Hence, by Corollary 4.1, there exist a sequence ϕ ∈ Pol(m−1) and z ∈ r^mA(m−s) such that x = ϕ+z. By [30, Lemma 4.2], we have∆^pzn=o(n^s−p)for any p∈ _N(0,m).

Corollary 4.5. Assume K∈K(1), b∈ A(1), and x is an(f,σ)-ordinary solution of (E). Then there exists a constantλ∈_Rsuch that

nlim→_∞

∆^m−p−1x_n n^p = ^λ

p! (4.1)

for any p∈_N(0,m−1).

Proof. Takingk= m−1 in Corollary4.3we obtain

x ∈Pol(m−1) +_∆^−m+1o(1). (4.2) The existence ofλfollows from [30, Lemma 3.8].

Note that if condition (4.1) is satisfied, then by (4.2),x is regularly asymptotically polyno- mial of type(m−1,m−1)_.

Corollary 4.6. Assume (A,Z) is an m-pair, K ∈ K(A), b ∈ A, g is locally bounded, and f is (g,w)-dominated. Then

O(w,σ)∩Sol(E)⊂Pol(m−1) +Z.

Proof. Note thatb−0∈ A. LetW =SQ. By Lemma2.2 (a), we have

∆^−mb+Z=W∩_∆^−mb+Z=W∩_∆^−m0+Z=Pol(m−1) +Z.

Hence the assertion is a consequence of Theorem3.2 (B1).

Corollary 4.7. Assume s∈ (−_∞,m−1], K∈K(m−s), b∈A(m−s), g is locally bounded, and f is(g,w)-dominated. Then

O(w,σ)∩Sol(E)⊂Pol(m−1) +o(n^s).

Proof. Since(A(m−s), o(n^s))is an asymptoticm-pair, the assertion is a consequence of Corol- lary4.6.

Corollary 4.8. Assume s ∈ (−_∞,m−1], K ∈ K(m−s), b ∈ A(m−s), k ∈ [s,m−1]∩_N(0), w_n=n^k,σ(n) =O(n), g is locally bounded, and f is(g,w)-dominated. Then

O(n^k)∩Sol(E)⊂Pol(k) +o(n^s).

Proof. Lety∈O(n^k)∩Sol(E). Choose positive constantsQandLsuch that σ(n)≤ Qn and |y_n| ≤Ln^k

for large n. Then |y_σ(n)| ≤ Lσ(n)^k ≤ LQ^kn^k. Hence y◦σ ∈ O(n^k) = O(w_n). Therefore y ∈ O(w,σ) and, by Corollary4.7, we havey ∈ Pol(m−1) +o(n^s). Choose ϕ ∈ Pol(m−1) andz∈o(n^s)such thaty= ϕ+z. Thenϕ=y−z∈O(n^k)and we obtainϕ∈Pol(k).

Corollary 4.9. Assume k∈_N(0,m−1), K∈K(m−k), b∈ A(m−k), g is locally bounded, and f is(g,w)-dominated. Then

O(w,σ)∩Sol(E)⊂Pol(m−1) +_∆^−ko(1).

Proof. Since(A(m−k), ∆^−mo(1))is an asymptoticm-pair and b∈ A, we have

∆^−kb+_∆^−ko(1) =Pol(m−1) +_∆^−ko(1). Hence the assertion is a consequence of Corollary4.6.

Corollary 4.10. Assume (A,Z) is an m-pair, K ∈ K(A), b ∈ A, A ⊂ A(1), g is nondecreasing, σ(n)≤ n for large n, n^m−1 =O(w_n), f is(g,w)-dominated, and g is of Bihari type. Then

Sol(E)⊂Pol(m−1) +Z.

Proof. Since∆^−mb+Z = Pol(m−1) +Z, the assertion is a consequence of Theorem3.2(B2).

Corollary 4.11. If (A,Z) is an evanescent m-pair, K ∈ K(A), b ∈ A, and ϕ ∈ Pol(m−1) is f -regular, thenϕ∈Sol(E)+Z.

Proof. Note thatb∈ A⊂A(m). Letz = (−1)^mr^mb, and lety= ϕ+z. Then

∆^my= _∆^mϕ+_∆^mz=₀+b=b.

Sinceϕis f-regular, there exists a subsetUofRand a positive numberεsuch that ϕ(_N)⊂Int(U,ε)

and f|_N×U is continuous and bounded. Letµ ∈ (0,ε/2). Since z_n = o(1), there exists an indexpsuch that|zn| ≤µfor anyn≥ p. Then

(ϕ+z)(_N(p))⊂Int(U,µ). Let

y^∗(n) = (

ϕ(n) forn< p

(ϕ+z)(n) forn≥ p, b^∗(n) =

(∆^mϕ(n) forn< p b(n) forn≥ p.

Theny^∗ is f-regular and∆^my^∗ = b^∗. Hence, by Theorem3.1 (A3), there exists a solutionx of the equation

∆^mxn =b^∗(n) +

∑

K(n,k)f(k,x_σ(k))

such thaty^∗ ∈ x+Z. Sinceb^∗(n) =b_nforn≥ p, we get x∈Sol(E). By the definition ofy^∗ we haveϕ+z−y^∗ ∈Fin(p). Hence

ϕ∈y^∗−z+Fin(p)⊂y^∗+Z⊂ x+Z+Z=x+Z.

5 Bounded solutions

In this section we apply our main results to investigate the bounded solutions of equation (E).

We say that a function f : N×_R → _R islocally equibounded if for every t ∈ _R there exists a neighborhood U of t such that f is bounded on N×U. Obviously every bounded function f :N×_R→_R is locally equibounded.

Example 5.1. Let f₁(n,t) = t and f₂(n,t) = n. Then f₁ is continuous, unbounded and locally equibounded, f₂ is continuous but not locally equibounded.

Example 5.2. Assume g₁, . . . ,g_k :R→_R are continuous, α₁, . . . ,α_k ∈ O(1) and let f(n,t) =

∑

α_i(n)g_i(t). Then f is continuous and locally equibounded.

Lemma 5.3. If f is locally equibounded, thenO(1)is(f,σ)-ordinary.

Proof. Let x ∈O(1). Choosea,b ∈_R such thatx(_N)⊂ [a,b]. For anyt ∈ [a,b]there exist an open subsetU_tof Rand a positive constant M_t such that

|f(n,s)| ≤ Mt

for anys∈U_t and anyn∈N. There exists a finite subset{t₁, . . .t_n}such that [a,b]⊂U_t1∪ · · · ∪U_tn.

IfM =_max(M_t1, . . .M_tn)_{, then}|f(k,x_σ(k))| ≤ Mfor any k.

In the next corollary we identify the setRwith the space Pol(0)of constant sequences.

Corollary 5.4. Assume (A,Z) is an m-pair, K ∈ _K(A), w ∈ _O(₁), b = _∆^mw, and f is locally equibounded. Then

O(₁)∩_Sol(E) ⊂w+_R+Z.

Proof. Note that∆^−mb=w+Pol(m−1). Since the sequence wis bounded, we have

O(1)∩_∆^−mb=O(1)∩(w+Pol(m−1)) =w+Pol(0) =w+_R. (5.1) By Lemma 5.3 O(1) is(f,σ)-ordinary. Hence the assertion is a consequence of Theorem3.1 (A1).

Corollary 5.5. Assume(A,Z)is an evanescent m-pair, K ∈ K(A), w∈ O(1), b = _∆^mw, and f is continuous and locally equibounded. Then

O(₁)∩_Sol(E)+Z= w+_R+Z. (5.2)

Proof. If f is continuous and locally equibounded, then O(1)is f-regular. Hence, using (5.1), and Theorem 3.1(A4) we obtain (5.2).

Corollary 5.6. Assume(A,Z)is an evanescent m-pair, K ∈ K(A), w∈ O(1), b = _∆^mw, and f is continuous and bounded. Then

O(1)∩Sol₁(E) +Z=O(1)∩Sol(E) +Z =w+_R+Z. (5.3) Proof. Since the set O(₁) _is f-regular, the assertion is a consequence of Corollary 5.5 and Theorem3.1(A5).

Letk∈_NandZ⊂SQ. We define

Per(k) ={x∈SQ : xisk-periodic}, Val(k) ={x∈SQ : card(x(_N))≤ k}. Per(k,Z) =_Per(k) +Z, Val(k,Z) =_Val(k) +Z,

Corollary 5.7. Assume(A,Z) is an evanescent m-pair, K ∈ K(A), k ∈ N, and f is locally equi- bounded. Then

(1) if∆^−mb∩Per(k,Z)6= _{∅, then} O(1)∩Sol(E)⊂Per(k,Z), (2) if∆^−mb∩Val(k,Z)6=_{∅, then} O(1)∩Sol(E)⊂Val(k,Z). Proof. Ifw∈ _∆^−mb∩Per(k,Z), then by Corollary5.4

O(1)∩Sol(E) ⊂w+_R+Z⊂Per(k) +Z=Per(k,Z), and we obtain (1). Analogously we obtain (2).

Corollary 5.8. Assume f is continuous and locally equibounded, (A,Z) is an evanescent m-pair, K∈_K(A), and w∈ _∆^−mb. Then

(1) if w∈Per(k,Z), thenO(1)∩Sol(E)+Z=Per(k,Z)∩Sol(E)+Z=w+_R+Z, (2) if w∈Val(k,Z), thenO(1)∩Sol(E)+Z=Val(k,Z)∩Sol(E)+Z=w+_R+Z.

Proof. Since f is continuous and locally equibounded, the set O(1) is f-regular. Moreover, since the pair(A,Z)is evanescent, we have Z+O(1) ⊂ O(1). Using Theorem 3.1 (A4) and (5.1) we have

O(1)∩Sol(E)+Z=O(1)∩_∆^−mb+Z= w+_R+Z.

By Corollary5.7, O(1)∩Sol(E)⊂Per(k,Z). Hence

O(1)∩Sol(E)⊂Per(k,Z)∩Sol(E).

Since Per(k,Z)⊂O(1), we get O(1)∩Sol(E) =Per(k,Z)∩Sol(E) and we obtain (1). Similarly we obtain (2).

Corollary 5.9. Assume f is continuous and bounded,(A,Z)is an evanescent m-pair, K∈K(A), and w∈_∆^−mb. Then

(1) if w∈Per(k,Z), then

O(₁)∩Sol(E)+_Z=_O(₁)∩Sol1(E)+_Z=_Per(_k,Z)∩Sol(E)+_Z

=Per(k,Z)∩Sol₁(E)+Z= w+_R+Z, (2) if w∈Val(k,Z), then

O(1)∩Sol(E)+Z=O(1)∩Sol₁(E)+Z=Val(k,Z)∩Sol(E)+Z

=_Val(k,Z)∩_Sol1(E)+Z=w+_R+Z.

Proof. By Theorem3.1(A5) we have

O(1)∩Sol(E)+Z=O(1)∩Sol₁(E)+Z and

Per(k,Z)∩Sol(E)+Z=Per(k,Z)∩Sol₁(E)+Z.

Hence (1) is a consequence of Corollary5.8(1). Analogously we obtain (2).

6 Remarks

In this section, we present some examples of f-regular sets. These sets are used in Theorem 3.1. Next, we discuss the condition w◦σ ∈ O(w) which is important in Theorem 3.2. Fi- nally, we present some tests that are helpful in verifying whether a given kernelK fulfills the assumptions of Theorems3.1and3.2.

Remark 6.1. If K ∈ K(m), then, by (2.3), r^mK⁰ ∈ o(1). Hence for any f-regular sequence y there exists an indexpsuch thatyis(_K,f,p)_-regular.

We say that a subsetW of SQ is o(1)-invariantif o(₁) +_W ⊂W.

Note that ifW is o(1)-invariant and(A,Z)is an evanescentm-pair, thenZ+W ⊂W.

Example 6.2. If f is continuous and bounded, then SQ is f-regular and o(₁)-invariant. If f is continuous and locally equibounded, then O(1)is f-regular and o(1)-invariant.

Example 6.3. If f is continuous and locally equibounded, then the set of all convergent se- quences x∈SQ is f-regular and o(1)-invariant. More generally, the set

{x∈SQ : L(x)is finite} is f-regular and o(1)-invariant.

Example 6.4. AssumeUis a uniform neighborhood of a setY ⊂ _R and f is continuous and bounded onN×U. Then the sets

W_L={y∈ SQ : L(y)⊂Y}, W_∞ ={y∈SQ : limy∈Y} are f-regular and o(1)-invariant.

By Lemma3.6(3) the conditionw◦σ ∈_O(w)_{implies O}(w)⊂_O(w,σ). Moreover, subsets of O(w,σ)play an important role in Theorem 3.2. Below, we discuss the condition w◦σ ∈ O(w).

Example 6.5. Ifs∈(0,∞),wn=n^s, andσ(n) =O(n), thenw0 andw◦σ∈O(w).

Justification. Obviously, w 0. If M is a positive constant such that σ(n) ≤ Mn for any n.

thenw(σ(n)) = (σ(n))^s≤(Mn)^s= M^sw_n. Hencew◦σ∈O(w).

Example 6.6. If O(w_n+1) =O(w_n), and the sequenceσ(n)−nis bounded, thenw◦σ∈O(w). Justification. Choose k ∈ _N such that |σ(n)−n| ≤ k for any n. Since w_n+1 = O(w_n), there exists a constant M>1 such that|w_n+1| ≤ M|wn|for largen. Then

|w_n+2| ≤ M|w_n+1| ≤M²|wn|, . . . , |w_n+k| ≤M^k|wn|. Hence, for any p∈_N(0,k), we have

|w_n+p| ≤ M^k|w_n|

for largen. Analogously, since w_n=O(w_n+1), there exists a constantQ>1 such that for any p ∈_N(0,k), we have

|w_n−p| ≤Q^k|w_n|

for largen. Now, if L=max(M^k,Q^k), then|w(σ(n))| ≤L|w_n|for largen.

Remark 6.7. If s ∈ _R and wn = n^s, then O(w_n+1) = O(wn). Similarly, if λ ∈ (0,∞) and w_n=λⁿ, then O(w_n+1) =O(w_n). On the other hand, ifw_n=nⁿ, then(w_n+1)∈/O(w_n).

In our main theorems we assume that (A,Z) is an m-pair and K ∈ K(A). The basic example of an m-pair is (A(t), o(n^m−t)). Hence in our theory, the answer to the following question is very important: whether for a given kernel K : N×_N → _R the relation K ∈ K(A(t)) =K(t)is fulfilled? Below we present some lemmas concerning this problem. These lemmas are analogous to the classical tests for absolute convergence of series.

Forn∈_Nlet

K^∗(n) =nmax{|K(n, 1)|,|K(n, 2)|, . . . ,|K(n,n)|}, K∗(n) =nmin{|K(n, 1)|,|K(n, 2)|, . . . ,|K(n,n)|}. Note that

K∗ ≤K⁰ ≤K^∗. (6.1)

Moreover if|K|is nondecreasing with respect to second variable, then K∗(n) =n|K(n, 1)|, K^∗(n) =n|K(n,n)|

for anyn, if|K|is nonincreasing with respect to second variable, then K∗(n) =n|K(n,n)|, K^∗(n) =n|K(n, 1)|

for anyn.

Lemma 6.8(Comparison test 1). Assume a,b,c∈ SQ, and A is a linear subspace ofSQ, such that O(1)A⊂ A, andFin+A⊂ A. Then

(₁) if|b_n| ≤ |c_n|for large n and c∈ A, then b∈ A, (2) if|b_n| ≥ |a_n|for large n and a∈/ A, then b∈/ A.

Proof. For the proof of (1) see [33, Lemma 3.8]. (2) is a consequence of (1).

Lemma 6.9(Comparison test 2). Assume A is a linear subspace ofSQ, such that O(1)A⊂ A, and Fin+A⊂ A.

Moreover, let L∈K(A), c∈A, and

K⁰ ≤L⁰ or |K| ≤ |L| or K^∗ ≤ |c|. Then K∈K(A).

Proof. The assertion is an easy consequence of Lemma6.8.

Lemma 6.10(Logarithmic test). Assume t∈[1,∞), u∗(n) =−^lnK∗(n)

lnn , u^∗(n) =−^lnK

∗(n) lnn . Then

(1) if lim infu^∗(n)>t, then K∈K(t), (₂) if limu^∗(n) =_∞, then K∈_K(_∞), (3) if u∗(n)≤ t for large n, then K∈/K(t), (4) if lim supu∗(n)< t, then K∈/K(t).

Proof. The assertion is a consequence of (6.1), Lemma6.8and [33, Lemma 6.2].

Lemma 6.11(Raabe’s type test). Assume t∈ [1,∞), u∗(n) =n

K∗(n) K∗(_n+₁)−1

, u^∗(n) =n

K^∗(n) K^∗(_n+₁)−1

. Then

(₁) if lim infu^∗(n)>t, then K∈_K(t),