Qualitative approximation of solutions to difference equations

Qualitative approximation of solutions to difference equations

Janusz Migda

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 spacec₀by various subspaces ofc₀. 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→_N, limσ(n) =_∞, and consider difference equations of the form

∆^mx_n=a_nf(x_σ(n)) +b_n (E) where a_n,b_n ∈_R.

Letp ∈N. We say that a sequencex: N→_Ris 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∈ _{N. If} xis a p-solution for any p∈N, then we say thatxis 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∆^mx is close tob, andx is close to the set

∆^−mb= ⁿy∈_R^N :∆^my =bo . This means that

x∈_∆^−mb+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 replacec₀ by various subspaces ofR^N.

More precisely, assume that AandZ are linear subspaces ofR^Nsuch that A⊂ _∆^mZand uα ∈ A for any bounded sequenceuand any α∈ A. If a ∈ Aandx is a solution of (E) such that the sequenceu= f◦x◦σis bounded, then

∆^mx =au+b∈ A+b⊂_∆^mZ+b.

Hence∆^mx = _∆^mz+bfor certain z ∈ Zand we get ∆^m(x−z) =b. Therefore x−z ∈ _∆^−mb 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^m−1|an|<_∞ )

, Z=_c0.

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 ∈ _∆^−mbsuch that x−y ∈ Z. Moreover, assuming Z ⊂ c0 and using the fixed point theorem, we obtain sufficient conditions under which for anyy∈_∆^−mbthere 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^N, 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 R^N which is ordinary in the sense that kx−yk < _{∞ 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

LetZdenote the set of all integers. If p,k ∈_Z, p≤k, thenNp,N^k_p denote the sets defined by Np ={p,p+1, . . .}, N^k_p= {p,p+1, . . . ,k}.

The space of all sequencesx: N→_Rwe 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∈_Nwe define

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_ny_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

∆^mx∈ a(f◦x◦σ) +b+_Fin(p) and, consequently,x is a solution of (E) if and only if

∆^mx∈ 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

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

Pol(m−₁) =_∆^−m₀=_Ker∆^m ={x ∈_{SQ :∆}^mx= ₀}_. Then Pol(m−1)is the space of all polynomial sequences of degree less thanm.

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

where B(x,ε)denotes a closed ball of radiusεabout x.

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

|f ≤ M|= {t∈_R:|f(t)| ≤ M}. Let

A(1):= (

a∈SQ :

∑

|an|< _∞ )

. Fort∈ [1,∞)we define

A(t):= (

a∈SQ :

∑

n^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(λⁿ)⊂O(λⁿ)⊂o(n^−∞) =A(_∞),

A(_∞)⊂A(t)⊂A(1)⊂o(1)⊂O(1)⊂o(n^s)⊂O(n^s)⊂O(n^∞)⊂o(µⁿ)⊂O(µⁿ).

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 ∈ _R 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→_Rare locally bounded.

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 ={t₁,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−t₁)· · ·(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⁻¹y. Then

x= T⁻¹y= T⁻¹Hy=T⁻¹TFT⁻¹y =FT⁻¹y=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

∑

∑

· · ·

∑

im=im−1

a_im.

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

∑

∑

· · ·

∑

im=i_m−1

a_im. (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_na. If |a| ∈ S(m)_{, then} a ∈ S(m)_and r^m(a) _is given by

r^m(a)(n) =

∑

m−1+j−n m−1

a_j. Note that ifm=1, then

r(a)(n) =r¹(a)(n) =

∑

a_j

is the n-th remainder of the series ∑^∞j=₁a_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^mx| ≤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=pn^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) _∆^mo(1) =S(m), r^mS(m) =o(1), (007) if x∈ S(m), then∆^mr^mx= (−1)^mx, (₀₀₈) if x∈ o(₁), then r^m∆^mx = (−1)^mx,

(009) if Z is a linear subspace ofo(1), then r^m∆^mZ=Z, (010) if A is a linear subspace ofS(m), then∆^mr^mA= 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_nx≤r^m_ny for n≥ p, (014) r^mFin(p) =Fin(p) =_∆^mFin(p), r^mFin=Fin=_∆^mFin,

(015) if x≥0, then r^mx 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 ⊂ _∆^mZ. 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 x⁰ is a sequence obtained from x by changing finite number of terms, then x⁰ ∈ 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⊂ _∆^mZ⊂_∆^mo(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⊂_∆^mZ⊂_∆^m_o(₁) =_S(m)_.

Therefore A⊂_A(m)and, for anya∈ A, the sequencesr^maandr^m|a|are defined.

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

∆^−ma+Z=_∆^−mb+Z.

Proof. We have a−b ∈ A ⊂ _∆^mZ. Hence there exists z₀ ∈ Z such that a−b = _∆^mz₀. Let x∈ _∆^−maandz∈Z. Then

∆^m(x−z0) =_∆^mx−_∆^mz0= a−(a−b) =b.

Thereforex+z= x−z₀+z₀+z∈ _∆^−mb+Z. Thus

∆^−ma+Z⊂_∆^−mb+Z.

Sinceb−a=−(a−b)∈ A, we have

∆^−mb+Z⊂_∆^−ma+Z.

The proof is complete.

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

∆^−mb+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∈_SQand

∆^mx∈O(a) +b.

Then x∈_∆^−mb+Z.

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

∆^mx−b∈O(a)⊂ A⊂ _∆^mZ.

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

∆^m(x−z) =_∆^mx−_∆^mz=b.

Thusx−z∈ _∆^−mband we obtainx= x−z+z∈ _∆^−mb+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_nb_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(_Np). 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_nk such that

klim→_∞ f(xn_k) =_∞.

Lety_k =x_nk and let p∈ L(y). There exists a subsequence y_ki such that

ilim→_∞y_ki = p.

Then limi→_∞ f(y_ki) = _∞ 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◦σis also bounded.

Theorem 4.2. Assume(A,Z)is an m-pair, a∈ A, and x ∈_Sol∞(_E). Then x ∈_∆^−mb+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◦σis bounded too. By Remark2.1,

∆^mx∈ a(f◦x◦σ) +b+Fin.

Hence

∆^mx∈ aO(1) +Fin+b=O(a) +b.

Using Lemma3.7we obtain x∈_∆^−mb+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∈_∆^−mb+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⁻¹fort6=0. Ifxis a solution of (E) such that 0 is not a limit point ofx, then, by Theorem4.2,x ∈_∆^−mb+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∈ _∆^−mb+Z.

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

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

∆^mx∈ a(f◦x◦σ) +b+Fin.

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

∆^mx∈ 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∈ _∆^−mb, R= Mr^m|a|, (y◦σ)(_Np)⊂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, letρ∈SQ,

ρn=

(0 forn< p,

Rn forn≥ p, S= {x∈SQ :|x−y| ≤ρ}. (4.2) By Lemma2.7(015), the sequence Ris nonincreasing. Henceρ_n≤ R_pfor 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)^mr_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^m−1|an|<_∞.

Letε >0. There existq> pandα>0 such that 2M

∑

n^m−1|a_n|<ε and α

∑

n^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 positiveδ 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^mu| ≤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| ≤

∑

n^m−1|u_n|+

∑

n^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

∑

n^m−1|an|+2M

∑

n^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 anε>0. Then there exists an indexqsuch thatρ_n<ε forn≥q. For n=_{1, . . . ,}qletG_ndenote a finiteε-net for the interval[−ρ_n,ρ_n]_{and let}

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

ThenGis a finiteε-net forT. 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)^mr_n^m(ax^∗). (4.5) Hence ∆^mx_n = _∆^my_n+_∆^mr_n^m((−1)^max^∗) for n ≥ p. Using the fact that y ∈ _∆^−mb and Lemma2.7(007), we obtain

∆^mxn =bn+anx^∗_n=bn+anf(x_σ(n)) forn≥ p. Thus

x∈_Solp(_E)_.

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

y−x ∈r^mA+_{Fin. (4.6)}

Using the definition of an evanescentm-pair and Lemma2.7(009), we have r^mA⊂r^m∆^mZ=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∈_∆^−mb and{y}is f -regular. Then y ∈Sol_∞(_E) +_Z.

Proof. There exist a positive M andδ >0 such that

(y◦σ)(_N)⊂Int(|f ≤ M|,δ). LetR= Mr^m|a|. ThenR=o(1)andR_p< δfor 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=_∆^−mb+Z.

Proof. Choose Msuch that|f| ≤ M. Then|f ≤ M|=_{R. Hence} Int(|f ≤ M|,δ) =_R for any positiveδ. By Theorem4.7we have

∆^−mb⊂Sol_p(E) +Z for any p. For a given p∈_Nwe obtain

∆^−mb+Z⊂Sol(E) +Z⊂Solp(E) +Z⊂Sol_∞(E) +Z.

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

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

W∩Sol_∞(E) +Z=W∩_∆^−mb+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 W₁= {y∈SQ : L(y)⊂Y}, W₂={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)_andp∈N. We say that a sequencex∈_{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 W₁ =ⁿx ∈SQ : lim sup

n→_∞

x_n< _∞^o _and W₂ =ⁿx ∈_{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 W₁= ⁿx ∈SQ : lim inf

n→_∞ xn>−_∞^o and W₂=ⁿx∈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→_Ris continuous,T={t₁,t2, . . . ,tn} ⊂_Rand f(t) = ^g(t)

(t−t₁)(t−t₂)_{. . .}(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^mO(a))is an evanescentm-pair.

Example 5.3. IfXis an asymptotic and modular subspace of A(m)_{, then}(X,r^mX)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∈_Rand(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. Letλ∈(0, 1). The following spaces are evanescentm-spaces Fin, o(λⁿ), O(λⁿ), o(n^−∞).

Example 5.7. Letλ∈(_1,∞). The following spaces arem-spaces o(λⁿ), O(λⁿ), 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^m₀⁻¹. Then

(A(m−s), o(n^s)), (A(m−q), ∆^−qo(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} ^∆xn

∆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 =_∆xp+_∆xp+₁+· · ·+_∆xn−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+ ^xp−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 = ^∆xn 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∈_Rand 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 =

∑

z_n, x₁=0, x_n =z₁+· · ·+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∈_Rand(s+1)(s+2)· · ·(s+m)6=0. Then o(n^s) ⊂ _∆^mo(n^s+m), O(n^s) ⊂ _∆^mO(n^s+m). Proof. The assertion is an easy consequence of the previous lemma.

Lemma5.14justify Examples5.4 and5.5.

Lemma 5.15. Ifλ∈(0, 1)∪(1,∞), then

o(λⁿ)⊂_∆^m_o(λⁿ)_{, O}(λⁿ)⊂_∆^m_O(λⁿ)_.

Proof. Letx,w∈SQ and∆w= x. Since∆λⁿ=λⁿ⁺¹−λⁿ =λⁿ(λ−1), we have

∆wn

∆λⁿ = ^xn

∆λⁿ =Lx_n

λⁿ (5.2)

where, L = 1/(λ−1). Assumeλ ∈ (0, 1)and x ∈ o(λⁿ). 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(λⁿ)we have ∆wn/∆λⁿ → 0. Moreover, w_n → 0 and λⁿ → 0. By Lemma 5.11, we obtainw∈o(λⁿ). Hence

x=_∆w∈_∆o(λⁿ). Therefore o(λⁿ)⊂ _∆o(λⁿ)and, by induction,

o(λⁿ)⊂_∆^mo(λⁿ).

If x ∈ _O(λⁿ), then the sequence x_n/λⁿ is bounded and, by (5.2), the sequence ∆w_n/∆λⁿ is also bounded. Hence, by Lemma5.11,w∈O(λⁿ)and we obtain O(λⁿ)⊂_∆O(λⁿ). Moreover, by induction

O(λⁿ)⊂_∆^m_O(λⁿ)_.

Ifλ>_{1, then} λⁿ→_∞and using Lemma5.11(b) we obtain the result.

Lemma 5.16. o(n^−∞)⊂_∆^mo(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(_∞).

Thereforer²A(_∞) =rrA(_∞)⊂rA(_∞)⊂A(_∞)and so on. Aftermsteps we obtain r^mA(_∞)⊂A(_∞).

By Lemma2.7(007), we have∆^mr^mA(_∞) =A(_∞). Hence A(_∞) =_∆^mr^mA(_∞)⊂ _∆^mA(_∞). 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)⊂ _∆^mO(n^∞)for anys> m. Hence

O(n^∞) = ^[

s>m

O(n^s)⊂_∆^mO(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 = _∆^mx. By [16, Theorem 2.1] we have x∈Pol(m−1) +o(n^s). Hence

a=_∆^mx ∈_∆^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^m₀⁻¹, thenA(m−q)⊂ _∆^m_∆^−qo(1).

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

a=_∆^mx ∈_∆^m(Pol(m−1) +_∆^−qo(1)) =_∆^m_∆^−qo(1). Using Lemmas5.17and5.18we obtain Examples5.8 and5.9.

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

Proof. Choosek ∈_Nsuch thatk ≤t<k+1. Lets =t−k. Then

A(t) =n^1−tA(1) =n^1−(k+s)A(1) =n^−sn^1−kA(1) =n^−sA(k). Hence, fora ∈A(t)we haven^sa ∈A(k). By Lemma2.7(012),

a ∈_A(k) _and n^sr|a| ≤r|n^sa|_.

Since|n^sa| ∈A(k)andr(A(k))⊂A(k−1), we have r|n^sa| ∈A(k−1).

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

ra ∈n^−sA(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^mA(m+t)⊂A(t). Hence, using Lemma2.7 (007),

A(m+t) =_∆^mr^mA(m+t)⊂_∆^mA(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^−tfor 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−1a_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)^swe obtain

(₁+_x)^s=₁+_sx+_o(_x) _forx→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_nu_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^nt for largen. Letb_n = (n−₁)^−t_{. Since}

e<

1+ ¹ n−1

n

, we have

e^nt <

1+ ¹ n−1

t

= n

n−1 t

= ^bn b_n+1

. Hence

|a_n|

|a_n+1| ≤e^nt < ^bn b_n+1

and |a_n| bn

< |a_n+1| b_n+1

for largen. Hence, there exists aλ>0 such that|an|/bn>λfor 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).