**Qualitative approximation of solutions to** **difference equations**

**Janusz Migda**

^{B}

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

Received 22 March 2015, appeared 28 May 2015 Communicated by Stevo Stevi´c

**Abstract.**We present a new approach to the theory of asymptotic properties of solutions
to difference equations. Usually, two sequencesx,yare called asymptotically equivalent
if the sequence x−yis convergent to zero i.e., x−y∈ c0, wherec0denotes the space
of all convergent to zero sequences. We replace the spacec_{0}by various subspaces ofc_{0}.
Our approach is based on using the iterated remainder operator. Moreover, we use the
regional topology on the space of all real sequences and the ‘regional’ version of the
Schauder fixed point theorem.

**Keywords:** difference equation, difference pair, prescribed asymptotic behavior, re-
mainder operator, Raabe’s test, Gauss’s test, Bertrand’s test.

**2010 Mathematics Subject Classification:** 39A05, 39A10, 39A12, 39A99.

**1** **Introduction**

Let **N**,**R** denote the set of positive integers and the set of real numbers, respectively. In this
paper we assume that

m∈** _{N,}** f:

**R**→

_{R,}*σ*:

**N**→

**lim**

_{N,}*σ*(n) =

_{∞,}and consider difference equations of the form

∆^{m}x_{n}=a_{n}f(x_{σ}_{(}_{n}_{)}) +b_{n} (E)
where a_{n},b_{n} ∈_{R.}

Letp ∈**N. We say that a sequence**x: **N**→** _{R}**is a p-solution of equation (E) if equality (E)
is satisfied for any n ≥ p. We say thatxis a solution if it is a p-solution for certain p∈

**xis a p-solution for any p∈**

_{N. If}**N, then we say that**xis a full solution.

In this paper, we present a new approach to the theory of asymptotic properties of solu- tions. The main concept, in our theory, is an asymptotic difference pair. The idea of the paper

BEmail: migda@amu.edu.pl

is based on the following observation. Ifx is a solution of (E), f is bounded and the sequence
ais ‘sufficiently small’, then∆^{m}x is close tob, andx is close to the set

∆^{−}^{m}b= ^{n}y∈_{R}** ^{N}** :∆

^{m}y =bo . This means that

x∈_{∆}^{−}^{m}b+Z (1.1)

whereZ is a certain space of ‘small’ sequences. UsuallyZ =c0 is the space of all convergent
to zero sequences. In this paper we replacec_{0} by various subspaces of**R**** ^{N}**.

More precisely, assume that AandZ are linear subspaces of**R**** ^{N}**such that A⊂

_{∆}

^{m}Zand uα ∈ A for any bounded sequenceuand any

*α*∈ A. If a ∈ Aandx is a solution of (E) such that the sequenceu= f◦x◦

*σ*is bounded, then

∆^{m}x =au+b∈ A+b⊂_{∆}^{m}Z+b.

Hence∆^{m}x = _{∆}^{m}z+bfor certain z ∈ Zand we get ∆^{m}(x−z) =b. Therefore x−z ∈ _{∆}^{−}^{m}b
and we obtain (1.1).

We say that (A,Z) is an asymptotic difference pair of order m (the precise definition is given in Section 3). In the classic case, for example in [7,14,15], we have

A= (

a∈_{R}** ^{N}**:

### ∑

∞ n=1n^{m}^{−}^{1}|an|<_{∞}
)

, Z=_{c}_{0}_{.}

In this paper we present some other examples of asymptotic difference pairs. Our purpose is
to present some basic properties of such pairs. Next, we use asymptotic difference pairs in
the study of asymptotic properties of solutions. For a given asymptotic difference pair(A,Z),
assuminga ∈ A, we obtain sufficient conditions under which for any solution x of (E) there
exists y ∈ _{∆}^{−}^{m}bsuch that x−y ∈ Z. Moreover, assuming Z ⊂ c0 and using the fixed point
theorem, we obtain sufficient conditions under which for anyy∈_{∆}^{−}^{m}bthere exists a solution
xof (E) such thaty−x ∈Z. Even more, we can ‘compute moduloZ’ some parts of the set of
solutions of (E) (see Theorem4.9and Theorem4.11in Section 4).

The concept of an asymptotic difference pair is an effect of comparing the results from some previous papers. In those papers, implicitly, some concrete asymptotic difference pairs are used (for details see Section 7). In fact, this paper is a continuation of a cycle of papers [14–20].

In the study of asymptotic properties of solutions to difference equations the Schauder
fixed point theorem is often used. This theorem is applicable to convex and compact subsets
of Banach spaces. But the space **R**** ^{N}** of all real sequences with usual ‘sup’ norm is not a
normed space. We introduce a topology in

**R**

**, which we call the regional topology. Next, in Theorem2.6, we present the ‘regional version’ of the Schauder fixed point theorem. This theorem is applicable to any convex and compact subset Q of**

^{N}**R**

**which is ordinary in the sense that kx−yk <**

^{N}_{∞}

_{for all}x,y ∈ Q. In fact, the regional topology is the topology of uniform convergence. For more information about the regional topology see [20].

The main technical tool in our investigations is the iterated remainder operator. The fun- damental theory of this operator is given in [19]. This approach to the study of asymp- totic properties of solutions to difference equations were inspired by the following papers [2–4,6,7,9,11,24–28]. Moreover, the papers [5,8,10,12,13,21–23] were the inspiration to the study of ‘continuous’ version of the iterated remainder operator. The basic properties of this

‘continuous’ operator are presented in [20]. Probably, using the ‘continuous’ version of the iterated remainder operator, some of the results from this paper can be transferred to the theory of ordinary differential equations.

The paper is organized as follows. In Section 2, we introduce notation and terminology.

In Section 3, we define asymptotic difference pairs and establish some of their basic proper- ties. In Section 4, we obtain our main results. In Section 5, we present some examples of difference pairs. In our investigations the spaces A(t)(see (2.1)) play an important role. In Section 6, we obtain some characterizations of A(t). These results extend some classic tests for absolute convergence of series and extend results from [19]. In Section 7, we present some consequences of our main results. Next we give some remarks.

**2** **Notation and terminology**

Let**Z**denote the set of all integers. If p,k ∈** _{Z,}** p≤k, then

**N**p,

**N**

^{k}

_{p}denote the sets defined by

**N**p ={p,p+1, . . .},

**N**

^{k}

_{p}= {p,p+1, . . . ,k}.

The space of all sequencesx: **N**→** _{R}**we denote by SQ. Moreover, by BS we denote the Banach
space of all bounded sequencesx∈ SQ equipped with ‘sup’ norm. We use the symbols

Sol(E), Sol_{p}(E), Sol_{∞}(E)

to denote the set of all full solutions of (E), the set of all p-solutions of (E), and the set of all solutions of (E) respectively. Note that

Sol(E)⊂Sol_{p}(E)⊂Sol_{∞}(E)
for any p∈** _{N}**. For p∈

**we define**

_{N}Fin(p) ={x∈SQ : xn=0 forn≥ p}. Moreover, let

Fin(_{∞}) =Fin=

[∞
p=_{1}

Fin(p). Note that all Fin(p)are linear subspaces of SQ and

0=Fin(1)⊂Fin(2)⊂Fin(3)⊂ · · · ⊂Fin(_{∞}).

If x,yin SQ, thenxydenotes the sequence defined by pointwise multiplication
xy(n) =x_{n}y_{n}.

Moreover,|x|denotes the sequence defined by|x|(n) =|xn|for every n.

**Remark 2.1.** A sequencex∈ SQ is ap-solution of (E) if and only if

∆^{m}x∈ a(f◦x◦*σ*) +b+_{Fin}(p)
and, consequently,x is a solution of (E) if and only if

∆^{m}x∈ a(f◦x◦*σ*) +b+Fin.

We use the symbols ‘big O’ and ‘small o’ in the usual sense but fora ∈SQ we also regard o(a)and O(a)as subspaces of SQ. More precisely, let

o(1) ={x ∈SQ : xis convergent to zero}, O(1) ={x∈SQ : xis bounded} and fora∈SQ let

o(a) =ao(1) +Fin= {ax: x∈o(1)}+Fin, O(a) =aO(1) +Fin= {ax: x∈O(1)}+Fin.

Moreover, let

o(n^{−}^{∞}) = ^{\}

s∈_{R}

o(n^{s}) =

∞

\

k=1

o(n^{−}^{k}), O(n^{∞}) = ^{[}

s∈_{R}

O(n^{s}) =

∞

[

k=1

O(n^{k}).
Note that ifa_{n} 6=0 for anyn, then

o(a) =ao(1), O(a) =aO(1). Forb∈SQ andX⊂SQ we define

∆^{−}^{m}b={y∈SQ : ∆^{m}y=b}, ∆^{−}^{m}X={y∈SQ : ∆^{m}y∈ X}.
Moreover, let

Pol(m−_{1}) =_{∆}^{−}^{m}_{0}=_{Ker∆}^{m} ={x ∈_{SQ :}_{∆}^{m}x= _{0}}_{.}
Then Pol(m−1)is the space of all polynomial sequences of degree less thanm.

For a subset Aof a metric spaceXand*ε*>0 we define an*ε-framed interior of* Aby
Int(A,*ε*) ={x∈ X: B(x,*ε*)⊂ A}

where B(x,*ε*)denotes a closed ball of radius*ε*about x.

We say that a subsetUofXis a uniform neighborhood of a subsetZofX, if there exists a
positive number*ε* such thatZ⊂Int(U,*ε*). For a positive constantM let

|f ≤ M|= {t∈** _{R}**:|f(t)| ≤ M}.
Let

A(1):= (

a∈SQ :

### ∑

∞ n=1|an|< _{∞}
)

. Fort∈ [1,∞)we define

A(t):= (

a∈SQ :

### ∑

∞ n=1n^{t}^{−}^{1}|a_{n}|<_{∞}
)

= (n^{1}^{−}^{t})A(1). (2.1)
Moreover, let

A(_{∞}) = ^{\}

t∈[_{1,∞})

A(t) =

\∞ k=1

A(k). Obviously any A(t)is a linear subspace of o(1)such that

O(1)A(t)⊂A(t). Note that if 1≤t ≤sthen

A(_{∞})⊂A(s)⊂A(t)⊂A(1).

**Remark 2.2.** If p∈ _{N,}*λ*∈ (_{0, 1})_{,}t∈ [_{1,}_{∞})_{,}s∈ (_{0,}_{∞})_{, and}*µ*>_{1, then}

0=Fin(1)⊂Fin(p)⊂Fin⊂o(*λ*^{n})⊂O(*λ*^{n})⊂o(n^{−}^{∞}) =A(_{∞}),

A(_{∞})⊂A(t)⊂A(1)⊂o(1)⊂O(1)⊂o(n^{s})⊂O(n^{s})⊂O(n^{∞})⊂o(*µ*^{n})⊂O(*µ*^{n}).

**2.1** **Unbounded functions**

We say that a function g: **R** → ** _{R}** is unbounded at a point p ∈ [−

_{∞,}

_{∞}] if there exists a sequence x∈SQ such that lim

_{n}→

_{∞}x

_{n}= pand the sequence g◦xis unbounded. Let

U(g) ={p∈[−_{∞},∞]: gis unbounded at p}.

A function g: **R** → ** _{R}** is called locally bounded if for anyt ∈

**there exists a neighborhood U of t such that the restriction g|U is bounded. Note that any continuous function and any monotonic function g:**

_{R}**R**→

**are locally bounded.**

_{R}**Remark 2.3.** Assumeg:**R**→** _{R. Then}**
(a) gis bounded if and only ifU(g) =

_{∅},

(b) gis locally bounded if and only ifU(g)⊂ {_{∞,}−_{∞}}.
**Example 2.4.** Assumeg: **R**→** _{R,}**T ={t

_{1},t2, . . . ,tn} ⊂

_{R. Then}(a) U(max(1,t)) =U(t+|t|) =U(e^{t}) ={_{∞}},
(b) U(min(1,t)) =U(t− |t|) =U(e^{−}^{t}) ={−_{∞}},

(c) if gis a nonconstant polynomial, thenU(g) ={−_{∞},∞},
(d) if g(t) =1/t fort6=0, thenU(g) ={0},

(e) if g(t) = ((t−t_{1})· · ·(t−t_{n}))^{−}^{1} fort ∈/T, thenU(g) =T.

**Remark 2.5.** Assumeg,h: **R**→_{R. Then}

U(g+h)⊂U(g)∪U(h), U(gh)⊂U(g)∪U(h).

This follows from the fact that ifgandhare bounded at a point p, then g+handghare also
bounded at p. Note also that ifU(g)∩U(h) =_{∅}, then

U(g+h) =U(g)∪U(h).

This is a consequence of the fact that if exactly one of the functions g,his bounded at a point p, theng+his unbounded at p.

**2.2** **Regional topology**

For a sequence x∈SQ we define a generalized norm kxk ∈[0,∞]by
kxk=sup{|xn|:n∈** _{N}**}.

We say that a subset Q of SQ is ordinary if kx−yk < _{∞} for any x,y ∈ Q. We regard any
ordinary subset Qof SQ as a metric space with metric defined by

d(x,y) =kx−yk. (2.2)

Let U ⊂SQ. We say that Uis regionally open if U∩Qis open in Qfor any ordinary subset Q of SQ. The family of all regionally open subsets is a topology on SQ which we call the regional topology. We regard any subset of SQ as a topological space with topology induced by the regional topology. The basic properties of regional topology are presented in [20]. We will use the following ‘regional’ version of the Schauder fixed point theorem.

**Theorem 2.6.** Assume Q is an ordinary compact and convex subset ofSQ. Then any continuous map
F: Q→Q has a fixed point.

Proof. Let a ∈ Q, W = Q−a and T: Q → W, T(x) = x−a. Since Q is ordinary, we have W ⊂BS. Moreover,Tis an isometry ofQontoW. Note also thatTpreserves convexity. Hence W is a compact and convex subset of the Banach space BS. Let

H: W →W, H=T◦F◦T^{−}^{1}.

Then H is continuous and, by the usual Schauder fixed point theorem, there exist a point
y∈W such thatHy=y. Letx= T^{−}^{1}y. Then

x= T^{−}^{1}y= T^{−}^{1}Hy=T^{−}^{1}TFT^{−}^{1}y =FT^{−}^{1}y=Fx.

**2.3** **Remainder operator**

In this subsection, we recall from [19] some basic properties of the iterated remainder operator.

Let S(m)denote the set of all sequences a∈SQ such that the series

### ∑

∞ i1=1### ∑

∞ i2=i1· · ·

### ∑

^{∞}

im=im−1

a_{i}_{m}.

is convergent. For anya∈S(m)we define the sequencer^{m}(a)by
r^{m}(a)(n) =

### ∑

∞ i_{1}=n

### ∑

∞ i_{2}=i

_{1}

· · ·

### ∑

^{∞}

im=i_{m}−1

a_{i}_{m}. (2.3)

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) (2.4)

is a linear operator which we call the iterated remainder operator of order m. The value
r^{m}(a)(n) we denote also by r^{m}_{n}(a) _{or simply} r^{m}_{n}a. If |a| ∈ S(m)_{, then} a ∈ S(m)_{and} r^{m}(a) _{is}
given by

r^{m}(a)(n) =

### ∑

∞ j=nm−1+j−n m−1

a_{j}.
Note that ifm=1, then

r(a)(n) =r^{1}(a)(n) =

### ∑

∞ j=na_{j}

is the n-th remainder of the series ∑^{∞}j=_{1}a_{j}. The following lemma is a consequence of [19,
Lemma 3.1].

**Lemma 2.7.** Assume x,u∈SQand p ∈** _{N. Then}**
(001) if|x| ∈S(m), then x∈S(m)and|r

^{m}x| ≤r

^{m}|x|, (002) |x| ∈S(m)if and only if∑

^{∞}n=1n

^{m}

^{−}

^{1}|x

_{n}|<

_{∞}, (003) if|x| ∈S(m), then r

^{m}

_{p}|x| ≤

_{∑}

^{∞}

_{n}

_{=}

_{p}n

^{m}

^{−}

^{1}|x

_{n}|, (

_{004}) |x| ∈

_{S}(m)if and only if x∈

_{A}(m),

(005) x∈A(m)if and only ifO(x)⊂S(m),
(006) _{∆}^{m}o(1) =S(m), r^{m}S(m) =o(1),
(007) if x∈ S(m), then∆^{m}r^{m}x= (−1)^{m}x,
(_{008}) if x∈ o(_{1}), then r^{m}∆^{m}x = (−1)^{m}x,

(009) if Z is a linear subspace ofo(1), then r^{m}∆^{m}Z=Z,
(010) if A is a linear subspace ofS(m), then∆^{m}r^{m}A= A,
(011) if x∈ A(m), u∈ O(1), then|r^{m}(ux)| ≤ |u|r^{m}|x|,
(012) if ux∈A(m),∆u≥0and u>0, then ur^{m}|x| ≤r^{m}|ux|,

(013) if x,y∈S(m)and xn ≤yn for n≥ p, then r^{m}_{n}x≤r^{m}_{n}y for n≥ p,
(014) r^{m}Fin(p) =Fin(p) =_{∆}^{m}Fin(p), r^{m}Fin=Fin=_{∆}^{m}Fin,

(015) if x≥0, then r^{m}x is nonnegative and nonincreasing.

**3** **Asymptotic difference pairs**

Let Zbe a linear subspace of SQ. We say that a subsetW of SQ isZ-invariant ifW+Z⊂W. We say, that a subset Xof SQ is:

asymptotic if Xis Fin-invariant, evanescent ifX⊂o(1),

modular if O(1)X⊂ X, c-stable ifXis o(1)-invariant.

We say that a pair (A,Z) of linear subspaces of SQ is a difference asymptotic pair of order
mor, simply, m-pair if Z is asymptotic, A is modular and A ⊂ _{∆}^{m}Z. We say that an m-pair
(A,Z)is evanescent ifZis evanescent.

**Remark 3.1.** For anya ∈SQ the spaces o(a)and O(a)are asymptotic and modular.

**Remark 3.2.** IfW is an asymptotic subset of SQ, x ∈ W and x^{0} is a sequence obtained from
x by changing finite number of terms, then x^{0} ∈ W. Moreover, a linear subspace Z of SQ is
asymptotic if and only if Fin⊂Z.

**Remark 3.3.** If(A,Z)is an evanescentm-pair, then, using Lemma2.7(006), we have
A⊂ _{∆}^{m}Z⊂_{∆}^{m}o(1) =S(m)⊂o(1).

Hence the space Ais evanescent.

**Remark 3.4.** Ifa∈SQ, then the sequence sgn◦ais bounded and|a|= (_{sgn}◦a)a. Hence, ifW
is a modular subset of SQ, then|a| ∈W for anya ∈W. In particular, if(A,Z)is an evanescent
m-pair and a∈ A, then

|a| ∈ A⊂_{∆}^{m}Z⊂_{∆}^{m}_{o}(_{1}) =_{S}(m)_{.}

Therefore A⊂_{A}(m)and, for anya∈ A, the sequencesr^{m}aandr^{m}|a|are defined.

**Lemma 3.5.** Assume(A,Z)is an m-pair, a,b∈SQ, and a−b∈ A. Then

∆^{−}^{m}a+Z=_{∆}^{−}^{m}b+Z.

Proof. We have a−b ∈ A ⊂ _{∆}^{m}Z. Hence there exists z_{0} ∈ Z such that a−b = _{∆}^{m}z_{0}. Let
x∈ _{∆}^{−}^{m}aandz∈Z. Then

∆^{m}(x−z0) =_{∆}^{m}x−_{∆}^{m}z0= a−(a−b) =b.

Thereforex+z= x−z_{0}+z_{0}+z∈ _{∆}^{−}^{m}b+Z. Thus

∆^{−}^{m}a+Z⊂_{∆}^{−}^{m}b+Z.

Sinceb−a=−(a−b)∈ A, we have

∆^{−}^{m}b+Z⊂_{∆}^{−}^{m}a+Z.

The proof is complete.

**Lemma 3.6.** Assume(A,Z)is an m-pair and b∈ A. Then

∆^{−}^{m}b+Z =_{Pol}(m−_{1}) +Z.

Proof. This lemma is an immediate consequence of the previous lemma.

**Lemma 3.7.** Assume(A,Z)is an m-pair, a ∈ A, b,x∈_{SQ}and

∆^{m}x∈O(a) +b.

Then x∈_{∆}^{−}^{m}b+Z.

Proof. The conditiona ∈ Aimplies O(a)⊂ A. Hence,

∆^{m}x−b∈O(a)⊂ A⊂ _{∆}^{m}Z.

Therefore, there existsz∈Zsuch that∆^{m}x−b=_{∆}^{m}_{z. Then}

∆^{m}(x−z) =_{∆}^{m}x−_{∆}^{m}z=b.

Thusx−z∈ _{∆}^{−}^{m}band we obtainx= x−z+z∈ _{∆}^{−}^{m}b+Z.

**Lemma 3.8**(Comparison test). Assume A is an asymptotic, modular linear subspace ofSQ, b∈ A,
a∈SQ, and|a_{n}| ≤ |b_{n}|for large n. Then a∈ A.

Proof. Assume|an| ≤ |bn|forn ≥ p. Let hn=

(0 ifbn=0
a_{n}/b_{n} if b_{n} 6=0.

Then h ∈ O(1). Moreover, if n ≥ p and b_{n} = 0, then a_{n} = 0. Hence a_{n} = h_{n}b_{n} for n ≥ p.

Thereforea−hb∈Fin(p). Letz= a−hb. Then

a= hb+z∈ O(1)A+Fin⊂ A+A= A.

**4** **Solutions**

In this section, in Theorems 4.9 and4.11, we obtain our main results. First we introduce the notion of f-ordinary and f-regular sets. We use these sets in Theorem4.11. At the end of the section we present some examples of f-regular sets.

We say that a subsetW of SQ is f-ordinary if for anyx∈W the sequence f◦xis bounded.

We say that a subsetW of SQ is f-regular if for any x∈W there exists an index psuch that f
is continuous and bounded on some uniform neighborhood of the set x(_{N}_{p}). For x∈SQ let

L(x) ={p∈[−_{∞,}_{∞}]_{:} pis a limit point ofx}_{.}
**Lemma 4.1.** If x∈SQ, then

f ◦x∈_{O}(_{1}) or L(x)∩_{U}(f)6=_{∅.}

Proof. Assume the sequence f ◦x is unbounded from above. Then there exists a subsequence
x_{n}_{k} such that

klim→_{∞} f(xn_{k}) =_{∞.}

Lety_{k} =x_{n}_{k} and let p∈ L(y). There exists a subsequence y_{k}_{i} such that

ilim→_{∞}y_{k}_{i} = p.

Then limi→_{∞} f(y_{k}_{i}) = _{∞} and we obtain p ∈ U(f)_{. Since} y is a subsequence of x, we have
L(y)⊂ L(x). Hence p ∈ U(f)∩L(x). Analogously, if the sequence f◦x is unbounded from
below, thenU(f)∩L(x)6= _{∅.}

Note that if the sequence f◦xis bounded, then f◦x◦*σ*is also bounded.

**Theorem 4.2.** Assume(A,Z)is an m-pair, a∈ A, and x ∈_{Sol}_{∞}(_{E}). Then
x ∈_{∆}^{−}^{m}b+Z or L(x)∩U(f)6=_{∅.}

Proof. Assume L(x)∩_{U}(f) = ∅. Then, by Lemma4.1, the sequence f ◦x is bounded. Hence
the sequence f◦x◦*σ*is bounded too. By Remark2.1,

∆^{m}x∈ a(f◦x◦*σ*) +b+Fin.

Hence

∆^{m}x∈ aO(1) +Fin+b=O(a) +b.

Using Lemma3.7we obtain x∈_{∆}^{−}^{m}b+Z. The proof is complete.

**Corollary 4.3.** Assume(A,Z)is an m-pair, a∈ A, B∪C=**R, C is closed inR, f is bounded on B,**
U(f)⊂**R, and x is a solution of** (E). Then

x∈_{∆}^{−}^{m}b+Z or L(x)∩C6=_{∅.}

Proof. Using the relation U(f) ⊂ ** _{R}** we see that U(f) ⊂ C. Hence the assertion is a conse-
quence of Theorem4.2.

**Example 4.4.** Assume(A,Z)_{is an}m-pair,a∈ A, and f(t) =t^{−}^{1}fort6=0. Ifxis a solution of
(E) such that 0 is not a limit point ofx, then, by Theorem4.2,x ∈_{∆}^{−}^{m}b+Z.

**Example 4.5.** Assume (A,Z) is an m-pair, a ∈ A, f is continuous and there exists a proper
limit limt→_{∞} f(t). Then, by Theorem 4.2, for any bounded below solution x of (E) we have
x∈ _{∆}^{−}^{m}b+Z.

**Theorem 4.6.** Assume(A,Z)is an m-pair, a ∈ A, and W⊂SQis f -ordinary. Then
W∩Sol_{∞}(E)⊂ _{∆}^{−}^{m}b+Z.

Proof. Letx ∈W∩Sol_{∞}(E). By Remark2.1,

∆^{m}x∈ a(f◦x◦*σ*) +b+Fin.

Sincex∈W, we have f◦x =O(1). Hence f◦x◦*σ*=O(1)and

∆^{m}x∈ aO(1) +Fin+b=O(a) +b.

Now, the assertion follows from Lemma3.7.

**Theorem 4.7.** Assume(A,Z)is an evanescent m-pair, a∈ A, M >0, p∈_{N,}

y∈ _{∆}^{−}^{m}b, R= Mr^{m}|a|, (y◦*σ*)(_{N}_{p})⊂Int(|f ≤ M|,Rp), (4.1)
and f is continuous on|f ≤ M|. Then y ∈Solp(E) +Z.

Proof. For x∈SQ let

x^{∗} = f ◦x◦*σ.*

Moreover, let*ρ*∈SQ,

*ρ*n=

(0 forn< p,

Rn forn≥ p, S= {x∈SQ :|x−y| ≤*ρ*}. (4.2)
By Lemma2.7(015), the sequence Ris nonincreasing. Hence*ρ*_{n}≤ R_{p}for any n.

Assumex∈ S. Then

|x_{σ}_{(}_{n}_{)}−y_{σ}_{(}_{n}_{)}| ≤*ρ*_{σ}_{(}_{n}_{)} ≤R_{p}

for any n. By (4.1), x_{σ}_{(}_{n}_{)} ∈ B(y_{σ}_{(}_{n}_{)},R_{p}) ⊂ |f ≤ M| for n ≥ p. Hence |x^{∗}_{n}| ≤ M for n ≥ p.

Therefore the sequence x^{∗} is bounded. Since A is a modular space, we have ax^{∗} ∈ A. By
Remark3.3, A⊂S(m). Hence the sequencer^{m}(ax^{∗})is defined for anyx∈S. Let

H: S→SQ, H(x)(n) =

(y_{n} forn< p,

y_{n}+ (−1)^{m}r_{n}^{m}(ax^{∗}) forn≥ p. (4.3)
Ifx∈Sandn≥ p, then, using Lemma2.7(001) and (013), we get

|H(x)(n)−y_{n}|= |r_{n}^{m}(ax^{∗})| ≤r^{m}_{n}|ax^{∗}| ≤r^{m}_{n}|Ma|= Mr_{n}^{m}|a|=R_{n}=*ρ*_{n}.

HenceHS⊂ S. We will show thatHis continuous. By Remark3.4, A⊂A(m). Therefore, by Lemma2.7 (004) and (002),

### ∑

∞ n=1n^{m}^{−}^{1}|an|<_{∞.}

Let*ε* >0. There existq> pand*α*>0 such that
2M

### ∑

∞ n=qn^{m}^{−}^{1}|a_{n}|<*ε* and *α*

### ∑

q n=pn^{m}^{−}^{1}|a_{n}|<*ε.* (4.4)
Let

W =

q

[

n=p

[y_{σ}_{(}_{n}_{)}−R_{p},y_{σ}_{(}_{n}_{)}+R_{p}].

Then W is compact and, by (4.1), W ⊂ |f ≤ M|. Hence f is uniformly continuous on W.
Choose a positive*δ* such that for s,t ∈ W the condition|s−t|< *δ* implies |f(s)− f(t)| < *α.*

Assume x,z ∈S,kx−zk<*δ* and letu= a(x^{∗}−z^{∗}). Then, using Lemma2.7(001) and (015),
we have

kHx−Hzk=sup

n≥1

|H(x)(n)−H(z)(n)|=sup

n≥p

|r_{n}^{m}u| ≤sup

n≥p

r^{m}_{n}|u|=r^{m}_{p}|u|.
Hence, by Lemma2.7 (003),

kHx−Hzk ≤

### ∑

^{∞}

n=p

n^{m}^{−}^{1}|u_{n}| ≤

### ∑

q n=pn^{m}^{−}^{1}|u_{n}|+

### ∑

∞ n=qn^{m}^{−}^{1}|u_{n}|.

Note that |u_{n}| ≤ *α*|a_{n}|forn∈ _{N}^{q}_{p}. Moreover, |x^{∗}n| ≤ M and|z^{∗}n| ≤ M forn≥ q. Hence, by
(4.4), we get

kHx−Hzk ≤*α*

### ∑

q n=pn^{m}^{−}^{1}|an|+2M

### ∑

∞ n=qn^{m}^{−}^{1}|an|<*ε*+*ε.*

Therefore His continuous. Obviously the setSis ordinary and convex. We will show thatSis
compact. Note that, by (4.1) and (2.4), we haveR= Mr^{m}|a|=o(1). Hence, by (4.2),*ρ*=o(1).
Let

T={x ∈BS :|x| ≤*ρ*}.

ThenTis a closed subset of BS. Choose an*ε*>0. Then there exists an indexqsuch that*ρ*_{n}<*ε*
forn≥q. For n=_{1, . . . ,}qletG_{n}denote a finite*ε-net for the interval*[−*ρ*_{n},*ρ*_{n}]_{and let}

G= {x∈ T:xn∈ Gn forn ≤qandxn=0 forn>q}.

ThenGis a finite*ε-net for*T. HenceTis a complete and totally bounded metric space and so,
Tis compact. LetF: T→Sbe given byF(x)(n) =x_{n}+y_{n}. ThenFis an isometry ofTontoS.

Hence Sis compact. By Theorem 2.6, there exists a sequence x ∈ Ssuch that Hx = x. Then, by (4.3), forn≥ p, we have

x_{n}=y_{n}+ (−1)^{m}r_{n}^{m}(ax^{∗}). (4.5)
Hence ∆^{m}x_{n} = _{∆}^{m}y_{n}+_{∆}^{m}r_{n}^{m}((−1)^{m}ax^{∗}) for n ≥ p. Using the fact that y ∈ _{∆}^{−}^{m}b and
Lemma2.7(007), we obtain

∆^{m}xn =bn+anx^{∗}_{n}=bn+anf(x_{σ}_{(}_{n}_{)})
forn≥ p. Thus

x∈_{Sol}_{p}(_{E})_{.}

By (4.5),y−x+ (−1)^{m}r^{m}(ax^{∗})∈ Fin(p). Since ax^{∗} ∈ A, we have

y−x ∈r^{m}A+_{Fin.} _{(4.6)}

Using the definition of an evanescentm-pair and Lemma2.7(009), we have
r^{m}A⊂r^{m}∆^{m}Z=Z.

Now, by (4.6),y−x ∈Z+Fin. By Remark3.2,Z+Fin=Z. Hence y∈x+Z.

**Corollary 4.8.** Assume(A,Z)is an evanescent m-pair, a∈ A, y∈_{∆}^{−}^{m}b and{y}is f -regular. Then
y ∈Sol_{∞}(_{E}) +_{Z.}

Proof. There exist a positive M and*δ* >0 such that

(y◦*σ*)(** _{N}**)⊂Int(|f ≤ M|,

*δ*). LetR= Mr

^{m}|a|. ThenR=o(1)andR

_{p}<

*δ*for certain p. Hence

Int(|f ≤ M|,*δ*)⊂Int(|f ≤ M|,Rp)
and, by Theorem4.7,y∈Solp(_{E}) +Z.

The next theorem is our first main result. We assume that f is continuous and bounded.

This assumption is very strong but our result is also strong.

**Theorem 4.9.** Assume (A,Z) is an evanescent m-pair, a ∈ A, p ∈ ** _{N}**, and f is continuous and
bounded. Then

Sol(E) +Z=Sol_{p}(E) +Z=Sol_{∞}(E) +Z=_{∆}^{−}^{m}b+Z.

Proof. Choose Msuch that|f| ≤ M. Then|f ≤ M|=** _{R. Hence}**
Int(|f ≤ M|,

*δ*) =

**for any positive**

_{R}*δ. By Theorem*4.7we have

∆^{−}^{m}b⊂Sol_{p}(E) +Z
for any p. For a given p∈** _{N}**we obtain

∆^{−}^{m}b+Z⊂Sol(E) +Z⊂Solp(E) +Z⊂Sol_{∞}(E) +Z.

On the other hand, by Theorem4.6, takingW =SQ we obtain
Sol_{∞}(E) +Z⊂_{∆}^{−}^{m}b+Z.

The proof is complete.

**Lemma 4.10.** Assume Z is a linear subspace of a linear space X, D,S,W ⊂ X, W is Z-invariant,
W∩S⊂ D+Z and W∩D⊂S+Z. Then

W∩S+Z=W∩D+Z.

Proof. Assume w ∈ W∩S. SinceW∩S ⊂ D+Z, we have w ∈ W ∩(D+Z). Hence, there existd∈ Dandz∈ Zsuch thatw= d+z. SinceW isZ-invariant, we obtain

d= w−z∈W+Z⊂W.

Hencew= d+z∈ (W∩D) +Z. ThereforeW∩S⊂W∩D+Zand we obtain W∩S+Z⊂W∩D+Z+Z=W∩D+Z.

Analogously, we obtainW∩D+Z⊂W∩S+Z.

Now we are ready to prove our second main result.

**Theorem 4.11.** Assume(A,Z)is an evanescent m-pair, a∈ A, and W ⊂SQ. Then
(a) if W is f -ordinary, then W∩_{Sol}_{∞}(_{E})⊂_{∆}^{−}^{m}b+Z,

(b) if W is f -regular, then W∩_{∆}^{−}^{m}b⊂Sol_{∞}(E) +Z,
(c) if W is f -regular and Z-invariant, then

W∩Sol_{∞}(E) +Z=W∩_{∆}^{−}^{m}b+Z.

Proof. Assertion (a) is a special case of Theorem 4.6. (b) is a consequence of Corollary 4.8.

Using (a), (b), Lemma 4.10and the fact that any f-regular setW ⊂ SQ is also f-ordinary we obtain (c).

**Remark 4.12.** Any subset of an f-regular set is f-regular. If Z is a linear subspace of o(1),
then any c-stable subsetW of SQ is alsoZ-invariant.

**Remark 4.13.** AssumeW ⊂ SQ is f-regular and Z is a linear subspace of o(1). Then the set
W+Zis f-regular andZ-invariant.

**Example 4.14.** Assume f is continuous and bounded on a certain uniform neighborhood of a
setY⊂**R. Then the set**

W = {y∈SQ :y(** _{N}**)⊂Y}

is f-regular. Ifx∈SQ andz ∈o(1), then L(x+z) =L(x). Hence the sets
W_{1}= {y∈SQ : L(y)⊂Y}, W_{2}={y∈ SQ : limyn∈Y}
are f-regular and c-stable.

**Example 4.15.** If f is bounded, then SQ is f-ordinary and c-stable. Moreover, if f is continu-
ous, then SQ is f-regular.

**Example 4.16.** If fis locally bounded, then the set O(_{1})of all bounded sequences is f-ordinary
and c-stable. Moreover, if f is continuous, then O(1)is f-regular.

**Example 4.17.** If f is locally bounded, then the setCof all convergent sequences is f-ordinary
and c-stable. Moreover, if f is continuous, then Cis f-regular.

**Example 4.18.** LetZbe a linear subspace of o(_{1})_{and}p∈**N. We say that a sequence**x∈_{SQ is}
(p,Z)-asymptotically periodic if there exists a p-periodic sequenceysuch thatx−y∈ Z. If f
is locally bounded, then the setW of all(p,Z)-asymptotically periodic sequences is f-ordinary
andZ-invariant. Moreover, if f is continuous, thenW is f-regular.

**Example 4.19.** If f(t) =e^{t}, then the sets
W_{1} =^{n}x ∈SQ : lim sup

n→_{∞}

x_{n}< _{∞}^{o} _{and} W_{2} =^{n}x ∈_{SQ : lim}

n→_{∞}x_{n}=−_{∞}^{o}
are f-regular and c-stable.

**Example 4.20.** If f is continuous and lim sup

t→_{∞}

|f(t)|<∞, then the sets
W_{1}= ^{n}x ∈SQ : lim inf

n→_{∞} xn>−_{∞}^{o} and W_{2}=^{n}x∈SQ : lim

n→_{∞}xn =_{∞}^{o}
are f-regular and c-stable.

**Example 4.21.** If f(t) = t^{−}^{1} fort 6=0, then the setW = {x ∈ SQ : 0 /∈ L(x)}is f-regular and
c-stable.

**Example 4.22.** Assumeg:**R**→** _{R}**is continuous,T={t

_{1},t2, . . . ,tn} ⊂

**and f(t) =**

_{R}^{g}(t)

(t−t_{1})(t−t_{2})_{. . .}(t−t_{n})

fort ∈/T. Then the setW ={x ∈SQ :T∩L(x) =_{∅}}is f-regular and c-stable.

**5** **Examples of difference pairs**

We say that a subset Aof SQ is anm-space, if (A,A)is anm-pair. In this section we present some examples of differencem-pairs andm-spaces. Next we establish some lemmas to justify our examples. Part of those lemmas are a mathematical folklore. We present the proofs of them for the convenience of the reader.

**Remark 5.1.** Assume that(A,Z)is anm-pair. IfZ^{∗}is a linear subspace of SQ such thatZ⊂Z^{∗},
then(A,Z^{∗}) is anm-pair. Analogously, if A∗ is a modular subspace of A, then (A∗,Z)is an
m-pair.

**Example 5.2.** Ifa∈A(m), then(O(a),r^{m}O(a))is an evanescentm-pair.

**Example 5.3.** IfXis an asymptotic and modular subspace of A(m)_{, then}(X,r^{m}X)is an evanes-
centm-pair.

**Example 5.4.** Lets∈ (−_{∞,}−m). The following pairs are evanescentm-pairs
(o(n^{s}), o(n^{s}^{+}^{m})), (O(n^{s}), O(n^{s}^{+}^{m})).

**Example 5.5.** Ifs∈_{R}_{and}(s+_{1})(s+_{2})· · ·(s+m)6=_{0, then}
(o(n^{s}), o(n^{s}^{+}^{m})), (O(n^{s}), O(n^{s}^{+}^{m}))
arem-pairs.

**Example 5.6.** Let*λ*∈(0, 1). The following spaces are evanescentm-spaces
Fin, o(*λ*^{n}), O(*λ*^{n}), o(n^{−}^{∞}).

**Example 5.7.** Let*λ*∈(_{1,}_{∞}). The following spaces arem-spaces
o(*λ*^{n}), O(*λ*^{n}), O(n^{∞}).

**Example 5.8.** Ifs∈(−_{∞, 0}], then (A(m−s), o(n^{s}))is an evanescentm-pair.

**Example 5.9.** Assumes ∈(−_{∞,}m−1], andq∈_{N}^{m}_{0}^{−}^{1}. Then

(A(m−s), o(n^{s})), (A(m−q), ∆^{−}^{q}o(1))
are m-pairs.

**Example 5.10.** Ift∈[_{1,}_{∞})_{, then}(_{A}(_{m}+_{t})_{, A}(_{t}))is an evanescentm-pair.

Note that Example 5.2 is a special case of Example 5.3. Note also that Lemma 2.7 (010) justifies Example5.3. To justify Examples5.4and5.5 we need the following four lemmas.

**Lemma 5.11** (Cesàro–Stolz lemma). Assume x,y ∈ SQ, y is strictly monotonic and one of the
following conditions is satisfied

(a) x=o(1)and y=o(1), (b) y is unbounded.

Then

lim inf ∆x

∆y ≤_{lim inf} ^{x}

y ≤_{lim sup}^{x}

y ≤_{lim sup}^{∆}^{x}

∆y.

Proof. If (a) is satisfied, then the assertion is proved in [1]. Assume (b) and y is unbounded
from above. Thenyis increasing and limy_{n}=_{∞}. Let

L=_{lim inf} ^{∆}^{x}^{n}

∆yn. If L=−∞, then the inequality

lim inf∆x

∆y ≤lim infx

y (5.1)

is obvious. Assume L > −∞. Choose a constant M such that M < L. Then there exists an indexpsuch that∆xn/∆yn≥ Mforn≥ p. We can assume thatyn>0 and∆yn>0 forn≥ p.

Ifn≥ p, then

xn−xp =_{∆x}_{p}+_{∆x}_{p}+_{1}+· · ·+_{∆x}_{n}_{−}_{1}

≥ M(_{∆}y_{p}+_{∆}y_{p}+1+· · ·+_{∆}y_{n}_{−}_{1}) =M(y_{n}−y_{p})_{.}
Hence xn≥ Myn+xp−Myp and

xn

y_{n} ≥ M+ ^{x}^{p}−Myp

y_{n}
forn≥ p. Since lim(1/y_{n}) =0, we have

lim infxn

yn

≥ M.

Therefore, we obtain (5.1). Similarly, one can prove the inequality lim supx

y ≤lim sup∆x

∆y.

Replacingy by−ywe obtain the result ifyis unbounded from below.

**Lemma 5.12.** Assume x∈SQ, s∈** _{R, and s}**> −1or x=o(1). Then

∆x =o(n^{s}) =⇒ x=o(n^{s}^{+}^{1}), ∆x=O(n^{s}) =⇒ x=O(n^{s}^{+}^{1}).

Proof. If s = −1, then, by assumption, x = o(1) = o(n^{s}^{+}^{1}). Hence the assertion is true for
s=−1. Assumes6= −1. Note that

∆xn

∆n^{s}^{+}^{1} = ^{∆x}^{n}
n^{s}

n^{s}

∆n^{s}^{+}^{1}.

By the proof of [16, Lemma 2.1], the sequence (n^{s}/∆n^{s}^{+}^{1})is convergent. Hence the assertion
follows from Lemma5.11.

**Lemma 5.13.** Assume s∈** _{R}**and s+16=0. Then

o(n^{s})⊂ _{∆o}(n^{s}^{+}^{1}), O(n^{s})⊂ _{∆O}(n^{s}^{+}^{1}).

Proof. Assumez = o(n^{s}). Choosex ∈ SQ such thatz = _{∆}x. If s > −1, then, by Lemma5.12,
x=o(n^{s}^{+}^{1}). Lets<−1. Then the series∑znis convergent. Let

*σ* =

### ∑

∞ n=1z_{n}, x_{1}=0, x_{n} =z_{1}+· · ·+z_{n}−1−*σ* forn>1.

Thenx = _{o}(_{1})_{,} _{∆}x = z and by Lemma5.12, we have x = _{o}(n^{s}^{+}^{1}). Hence we obtain o(n^{s})⊂

∆o(n^{s}^{+}^{1}). Analogously O(n^{s})⊂_{∆O}(n^{s}^{+}^{1}).

**Lemma 5.14.** Assume s∈** _{R}**and(s+1)(s+2)· · ·(s+m)6=0. Then
o(n

^{s}) ⊂

_{∆}

^{m}o(n

^{s}

^{+}

^{m}), O(n

^{s}) ⊂

_{∆}

^{m}O(n

^{s}

^{+}

^{m}). Proof. The assertion is an easy consequence of the previous lemma.

Lemma5.14justify Examples5.4 and5.5.

**Lemma 5.15.** If*λ*∈(0, 1)∪(1,∞), then

o(*λ*^{n})⊂_{∆}^{m}_{o}(*λ*^{n})_{,} _{O}(*λ*^{n})⊂_{∆}^{m}_{O}(*λ*^{n})_{.}

Proof. Letx,w∈SQ and∆w= x. Since∆λ^{n}=*λ*^{n}^{+}^{1}−*λ*^{n} =*λ*^{n}(*λ*−1), we have

∆wn

∆*λ*^{n} = ^{x}^{n}

∆*λ*^{n} =Lx_{n}

*λ*^{n} (5.2)

where, L = 1/(*λ*−1). Assume*λ* ∈ (0, 1)and x ∈ o(*λ*^{n}). Then the series ∑^{∞}n=1x_{n} is conver-
gent. Hence x ∈ S(1) = _{∆o}(1)and there existsw ∈ o(1)such that x = ∆w. Using (5.2) and
the fact that x ∈ o(*λ*^{n})we have ∆wn/∆λ^{n} → 0. Moreover, w_{n} → 0 and *λ*^{n} → 0. By Lemma
5.11, we obtainw∈o(*λ*^{n}). Hence

x=_{∆w}∈_{∆o}(*λ*^{n}).
Therefore o(*λ*^{n})⊂ _{∆o}(*λ*^{n})and, by induction,

o(*λ*^{n})⊂_{∆}^{m}o(*λ*^{n}).

If x ∈ _{O}(*λ*^{n}), then the sequence x_{n}/λ^{n} is bounded and, by (5.2), the sequence ∆w_{n}/∆*λ*^{n} is
also bounded. Hence, by Lemma5.11,w∈O(*λ*^{n})and we obtain O(*λ*^{n})⊂_{∆O}(*λ*^{n}). Moreover,
by induction

O(*λ*^{n})⊂_{∆}^{m}_{O}(*λ*^{n})_{.}

If*λ*>_{1, then} *λ*^{n}→_{∞}and using Lemma5.11(b) we obtain the result.

**Lemma 5.16.** o(n^{−}^{∞})⊂_{∆}^{m}o(n^{−}^{∞}).

Proof. Using [19, Lemma 4.8] we have rA(k+_{1}) ⊂ _{A}(k) _{for any} k ∈ ** _{N. Hence}** rA(

_{∞}) ⊂ rA(k+1)⊂A(k)and we get

rA(_{∞})⊂ ^{\}

k∈_{N}

A(k) =A(_{∞}).

Thereforer^{2}A(_{∞}) =rrA(_{∞})⊂rA(_{∞})⊂A(_{∞})and so on. Aftermsteps we obtain
r^{m}A(_{∞})⊂A(_{∞}).

By Lemma2.7(007), we have∆^{m}r^{m}A(_{∞}) =A(_{∞}). Hence
A(_{∞}) =_{∆}^{m}r^{m}A(_{∞})⊂ _{∆}^{m}A(_{∞}).
Now the result follows from the equality o(n^{−}^{∞}) =A(_{∞}).

Using Lemma2.7 (014), Lemma5.15and Lemma5.16we justify Example5.6. By Lemma
5.14, we have O(n^{s})⊂ _{∆}^{m}O(n^{∞})for anys> m. Hence

O(n^{∞}) = ^{[}

s>m

O(n^{s})⊂_{∆}^{m}O(n^{∞})_{.}
Therefore, using Lemma5.15, we obtain Example5.7.

**Lemma 5.17.** If s∈(−_{∞,}m−1], thenA(m−s)⊂_{∆}^{m}(o(n^{s})).

Proof. Let a ∈ A(m−s). Choose x ∈ SQ such that a = _{∆}^{m}x. By [16, Theorem 2.1] we have
x∈Pol(m−1) +o(n^{s}). Hence

a=_{∆}^{m}x ∈_{∆}^{m}(Pol(m−1) +o(n^{s}))

=_{∆}^{m}_{Pol}(_{m}−1) +_{∆}^{m}_{o}(_{n}^{s}) =_{∆}^{m}_{o}(_{n}^{s})_{.}

**Lemma 5.18.** If q∈_{N}^{m}_{0}^{−}^{1}, thenA(m−q)⊂ _{∆}^{m}_{∆}^{−}^{q}o(1).

Proof. Let a∈ A(m−q). Choose x ∈ SQ such thata = _{∆}^{m}x. By [17, Lemma 3.1 (d)] we have
x∈Pol(m−1) +_{∆}^{−}^{q}o(1). Hence

a=_{∆}^{m}x ∈_{∆}^{m}(Pol(m−1) +_{∆}^{−}^{q}o(1)) =_{∆}^{m}_{∆}^{−}^{q}o(1).
Using Lemmas5.17and5.18we obtain Examples5.8 and5.9.

**Lemma 5.19.** If t∈[m+1,∞), then r^{m}A(t)⊂A(t−m).

Proof. Choosek ∈** _{N}**such thatk ≤t<k+1. Lets =t−k. Then

A(t) =n^{1}^{−}^{t}A(1) =n^{1}^{−(}^{k}^{+}^{s}^{)}A(1) =n^{−}^{s}n^{1}^{−}^{k}A(1) =n^{−}^{s}A(k).
Hence, fora ∈A(t)we haven^{s}a ∈A(k). By Lemma2.7(012),

a ∈_{A}(k) _{and} n^{s}r|a| ≤r|n^{s}a|_{.}

Since|n^{s}a| ∈A(k)andr(A(k))⊂A(k−1), we have
r|n^{s}a| ∈A(k−1).

By the comparison test we obtainn^{s}r|a| ∈A(k−1). Using the inequality|ra| ≤r|a|we have
n^{s}|ra| ≤n^{s}r|a|. By comparison test,n^{s}|ra| ∈A(k−1). Hence

ra ∈n^{−}^{s}A(k−1) =A(t−1).
Therefore

r(A(t))⊂A(t−1) and, by induction, we obtain the result.

Now lett∈ [1,∞). By Lemma5.19we have r^{m}A(m+t)⊂A(t). Hence, using Lemma2.7
(007),

A(m+t) =_{∆}^{m}r^{m}A(m+t)⊂_{∆}^{m}A(t)
and we obtain Example5.10.

**6** **Absolute summable sequences**

In our investigations the spaces A(t)play an important role. In this section we obtain some characterizations of A(t). Our results extend some classical tests for absolute convergence of series and extend results from [19].

**Lemma 6.1.** Assume t∈[1,∞)and s∈** _{R. Then}**(n

^{s})∈A(t)⇔s <−t.

Proof. We have

(n^{s})∈A(t)⇔(n^{s})∈(n^{1}^{−}^{t})A(1)⇔(n^{t}^{+}^{s}^{−}^{1})∈A(1)⇔t+s−1< −1⇔s< −t.

**Lemma 6.2**(Generalized logarithmic test). Assume a∈SQ, t ∈[1,∞)and
u_{n} =−^{ln}|a_{n}|

lnn . Then

(1) if lim infu_{n}> t, then a∈A(t),
(2) if u_{n}≤t for large n, then a∈/A(t),
(_{3}) if lim supu_{n}<t, then a∈/_{A}(t),
(_{4}) if limu_{n}=_{∞}, then a∈_{A}(_{∞}).

Proof. If lim infu_{n} > t, then there exists a number s > t such thatu_{n} > s for large n. Then

|an| ≤ n^{−}^{s} for large n. Hence (1) follows from the comparison test and from the fact that
(n^{−}^{s})∈A(t). Ifu_{n}≤t for largen, then|a_{n}| ≥n^{−}^{t}for largen. Hence (2) follows from the fact
that(n^{−}^{t})∈_{/}_{A}(t). The assertion (3) follows immediately from (2) and (4) is a consequence of
(1).

**Lemma 6.3**(Generalized Raabe’s test). Assume a∈SQ, t∈[1,∞),
u_{n} =n

|a_{n}|

|a_{n}+1|−_{1}

. Then

(_{1}) if lim infu_{n}>t, then a∈_{A}(t),
(2) if u_{n}≤t for large n, then a∈/A(t),
(_{3}) if lim supu_{n}<t, then a∈/A(t),
(4) if limu_{n} =_{∞, then a}∈A(_{∞}).
Proof. Let

b_{n} =n^{t}^{−}^{1}a_{n}, w_{n}=n

|b_{n}|

|b_{n}+1|−_{1}

. Then

w_{n}= n

n^{t}^{−}^{1}|a_{n}|

(n+_{1})^{t}^{−}^{1}|a_{n}+1|−1

=n

n
n+_{1}

t−1 |a_{n}|

|a_{n}+1|−1

!

= n n

n+_{1}

t−1 |a_{n}|

|a_{n}+1|−

n+1 n

t−1!

=n n

n+_{1}

t−1 |a_{n}|

|a_{n}+1|−

1+ ^{1}
n

t−1!
.
If s∈**R, then using the Taylor expansion of the function**(1+x)^{s}we obtain

(_{1}+_{x})^{s}=_{1}+_{sx}+_{o}(_{x}) _{for}_{x}→0.

Hence

1+ ^{1}

n t−1

=1+ (t−1)^{1}

n +o(n^{−}^{1}).
Therefore

w_{n} =
n

n+1 t−1

n

|a_{n}|

|an+_{1}|−1

− n

n+1 t−1

(t−1−no(n^{−}^{1}))

=c_{n}u_{n}−c_{n}(t−1−o(1)), c_{n} =
n

n+_{1}
t−1

→1.

Thus

lim infw_{n}=lim infu_{n}−(t−1) =lim infu_{n}−t+1.

Hence, if lim infu_{n} >t, then lim infw_{n} >1 and by the usual Raabe’s test we obtain b∈ A(1)
i.e.,a ∈A(t). The assertion (1) is proved. Now, we assume thatun≤t for largen. Then

n

|a_{n}|

|a_{n}+1|−1

≤t i.e., |a_{n}|

|a_{n}+1| ≤ ^{t}

n +1 for largen.

Hence

w_{n} =n
n

n+_{1}

t−1 |a_{n}|

|a_{n}+1|−

1+ ^{1}
n

t−1!

≤n n

n+_{1}
t−1

t n +1−

1+ ^{1}

n t−1!

.

It is easy to see that ift ≥1 andx∈(0, 1), then(1+x)^{t} ≥1+tx. Hence

1+ ^{1}
n

t

≥1+ ^{t}

n, and t n +1−

1+ ^{1}

n 1+ ^{1}
n

t−1

≤0.

Therefore

t n+1−

1+ ^{1}

n t−1

≤ ^{1}
n

1+ ^{1}

n t−1

= ^{1}
n

n+_{1}
n

t−1

Hencew_{n} ≤1 for largenand, by the usual Raabe’s test, we obtainb∈/A(1)i.e.,a∈/A(t). The
assertion (2) is proved. (3) is an immediate consequence of (2). (4) follows from (1).

**Lemma 6.4**(Generalized Schlömilch’s test). Assume a∈SQ, t∈[1,∞),
u_{n}= nln |a_{n}|

|a_{n}+1|^{.}
Then

(1) if lim infu_{n}>t, then a∈A(t),
(2) if un≤t for large n, then a∈/A(t),
(_{3}) if lim supu_{n}<t, then a∈/_{A}(t),
(4) if limu_{n} =_{∞, then a}∈A(_{∞}).

Proof. If lim infu_{n}=b>tandc∈(t,b), then lim infu_{n}>cfor largen. Hence

|a_{n}|

|a_{n}+1| ≥expc
n

.
Sincee^{x} ≥1+xforx >0, we have

|an|

|a_{n}+1| >1+ ^{c}

n and n

|an|

|a_{n}+1|−1

>c> t for largen. Now, by Raabe’s test we obtain (1).

Assumeu_{n}≤ tfor largen. Then
ln |a_{n}|

|a_{n}+1| ≤ ^{t}

n and |a_{n}|

|a_{n}+1| ≤e^{n}^{t}
for largen. Letb_{n} = (n−_{1})^{−}^{t}_{. Since}

e<

1+ ^{1}
n−1

n

, we have

e^{n}^{t} <

1+ ^{1}
n−1

t

= n

n−1 t

= ^{b}^{n}
b_{n}+1

. Hence

|a_{n}|

|a_{n}+1| ≤e^{n}^{t} < ^{b}^{n}
b_{n}+1

and |a_{n}|
bn

< |a_{n}+1|
b_{n}+1

for largen. Hence, there exists a*λ*>0 such that|an|/bn>*λ*for largen. Therefore

|a_{n}|> *λb*_{n}>*λn*^{−}^{t}

for largen. Using the fact that(n^{−}^{t})∈/A(t)we havea∈/A(t)and we obtain (2). The assertion
(3) is an immediate consequence of (2). (4) follows from (1).