Approximative solutions of difference equations

Approximative solutions of difference equations

Janusz Migda

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

Received 29 September 2013, appeared 23 March 2014 Communicated by Stevo Stevi´c

Abstract.Asymptotic properties of solutions of difference equations of the form

∆^mxn =anf(n,x_σ(n)) +bn

are studied. Using the iterated remainder operator and fixed point theorems we obtain sufficient conditions under which for any solutionyof the equation∆^my=band for any reals≤0 there exists a solutionxof the above equation such that∆^kx =∆^ky+o(n^s−k) for any nonnegative integerk ≤ m. Using a discrete variant of the Bihari lemma and a certain new technique we give also sufficient conditions under which for a given real s ≤ m−1 all solutions xof the equation satisfy the condition x = y+o(n^s) wherey is a solution of the equation ∆^my = b. Moreover, we give sufficient conditions under which for a given natural k < m all solutions x of the equation satisfy the condition x=y+ufor a certain solutionyof the equation∆^my=band a certain sequenceusuch that∆^ku=o(1).

Keywords: difference equation, approximative solution, asymptotically polynomial so- lution, prescribed asymptotic behavior.

2010 Mathematics Subject Classification:39A10.

1 Introduction

LetN,Z,Rdenote the set of positive integers, the set of all integers and the set of real numbers, respectively. Letm∈_N. In this paper we consider the difference equation of the form

∆^mx_n =a_nf(n,x_σ(n)) +b_n (E) n ∈_N, a_n,b_n∈ _R, f: N×_R→_R, σ: N→_N, limσ(n) =_∞.

We assume there is a given functiong: [0,∞)→[0,∞)and a sequencewof real numbers such that

|f(n,t)| ≤g(|twn|) for (n,t)∈_N×_R. (G) By a solution of (E) we mean a sequence x: N → _R satisfying (E) for all large n. If (E) is satisfied for alln∈_Nwe say thatxis a full solution of (E).

BEmail: migda@amu.edu.pl

The purpose of this paper is to study the asymptotic behavior of solutions of equation (E).

In the study of solutions with prescribed asymptotic behavior some fixed point theorems are often used. Then there appear multiple sums of the form

∑

∑

. . .

∑

x_im. (R)

The reason is shown below. LetZdenote the space of all convergent to zero sequences. Then the operator

∆^m|_Z: Z→_∆^m(Z)

is bijective (it is a consequence of the equalityZ∩_Ker∆^m = 0). Moreover, if x ∈ _∆^m(Z)then the sum (R) is convergent and we may define a map

r^m: ∆^m(Z)→Z, r^m(x)(n) =

∑

∑

. . .

∑

x_im.

Thenr^m(the iterated remainder operator) is a linear operator and(−1)^mr^m is inverse to∆^m|Z.

Hence forx∈ _∆^m(Z)we have

∆^m((−1)^mr^m(x)) =x.

The last equality plays a crucial role in the application of fixed point theorems to the study of solutions of difference equations. Hence the operatorr^m is very important. In Section 3, we establish some basic properties of this operator. It is easy to see thatr^m is nondecreasing. This allows us to use the Knaster–Tarski fixed point theorem (see Section 6). The continuity ofr^mis more subtle. The operatorr^m is discontinuous (see Remark4.6) but restrictions r^m|Sto some important sets S are continuous (see Lemma 4.5). This allows us to use the Schauder fixed point theorem (see Section 5). Multiple sums of the form (R) are used in many papers, see for example [15,20,24,25,43]. If the series

∑

n^m−1|x_n|

is convergent, thenx∈ _∆^m(Z)and we may rewrite the sum (R) in the more comfortable form

∑

m+k−₁ m−1

x_n+k

of a single sum (see Lemma4.2). This is used to obtain fundamental properties of the operator r^m. The fact thatr^m(x) = o(1) is often used to obtain results of type ‘for a given sequencey there exists a solutionxsuch thatx−y=o(1)’, see, for example, [10, Theorem 1], [11, Theorem 1], [28, Theorem 1] or [39, Theorem 2.1]. Ifs∈(−_{∞, 0}]and

∑

n^m−1−s|x_n|<_∞, (1.1) thenx ∈ _∆^m(Z)andr^m(x) = o(n^s)(see Lemma4.2). Using this fact one can obtain results of type ‘for a given sequencey there exists a solution x such that x−y = _o(n^s)’, see theorems in Section 3 of [29] and theorems in Sections 5 and 6 of this paper. Obviously, ifs < t ≤ 0 then the conditionx−y = o(n^s)is more restrictive thanx−y = o(n^t). Hence we obtain an approximative solutionyand we may control the ‘degree’ of approximation.

In the study of asymptotic behavior of solutions of difference equations, asymptotically polynomial solutions play an important role. It is related to the fact that the solutions of the

‘simplest’ difference equation∆^mx = 0 are polynomial sequences. Analogously, if the differ- ence∆^mxis ‘sufficiently small’, thenxis asymptotically polynomial. This effect is used in many papers, see for example Theorems 2 and 3 in [34] or Theorem 5 in [28]. The ‘method of small difference’ has been developed in [29]. It is based on [29, Theorem 2.1] which states that if s∈(−_∞,m−1],∑^∞n=1n^m−1−s|a_n|<_∞and∆^mx_n =O(a_n), then

x ∈Pol(m−1) +o(n^s) ={ϕ+u: ϕ∈_Ker∆^m, un=o(n^s)}. (1.2) In Lemma3.11we extend this result and, in Section 7, we use the ‘method of small difference’ to establish sufficient conditions under which all solutions of (E) are asymptotically polynomial.

Asymptotically polynomial solutions appear in the theory of both differential and differ- ence equations. In particular, in the theory of second order equations, so called asymptotically linear solutions are considered. In the theory of differential equations, asymptotic linearity of solutionx, usually means one of the following two conditions

x(t) =at+b+o(1) or x(t) =at+o(t) as t →_∞. (1.3) In [32] the condition of the form x(t) = at+o(t^d) for certain d ∈ (0, 1) is also considered.

In some papers in addition to (1.3), some properties of derivative x⁰ are also considered. For example, in [32] Mustafa and Rogovchenko consider solutionsxsuch that

x(t) =at+o(t) and x⁰(t) =a+o(1) as t→_∞.

Ehrnström in [13] considers solutionsxsuch that

|x(t)−at−b|+|x⁰(t)−a| →0 as t→_∞. (1.4) A discrete analog of (1.4) may be written in the form

xn =an+b+o(1) and ∆xn= a+o(1). (1.5) We generalize (1.5) as follows. We say that a sequencexis regularly asymptotically polynomial if

x∈Pol(m) +_∆^−ko(1) ={ϕ+u:ϕ∈_Ker∆^m+1, ∆^ku=o(1)} (1.6) for some integersm ≥ −1 andk ∈ [0,m+1]. By Remark3.3, the condition (1.6) is equivalent to

∆^px∈ Pol(m−p) +o(n^k−p) for any p∈ {0, 1, . . . ,k}.

When k = mwe obtain a special case. By Lemma3.8the conditionx ∈ Pol(m) +_∆^−mo(1)is equivalent to the convergence of the sequence∆^mx_nand to the condition

nlim→_∞

p!∆^m−pz_n

n^p =λ (1.7)

for certain fixed real λ and any p ∈ {0, 1, . . . ,m}. Convergence of the sequence ∆^mx_n is comparatively easy to verify and condition (1.7) appears in many papers, see for example [9, 14, 25, 30, 42] or the proof of Theorem 3.1 in [41]. Our ‘small difference method’ covers both the case of usually asymptotically polynomial sequences (1.2) and the case of regularly

asymptotically polynomial sequences (1.6). Moreover, we extend the method to the case of

‘forced’ equations (see Lemma3.11).

This paper is a continuation of the papers [25,28] and [29]. The origin of our studies goes back to the results of Popenda and Werbowski [34], Hooker and Patula [19], Popenda [35], Drozdowicz and Popenda [11], Cheng and Patula [6], Popenda and Schmeidel [36] and Li and Cheng [22]. Our results show certain similarities to the results obtained in continuous case by Philos, Purnaras and Tsamatos in [33]. See also [2,8,17,18,21,40] and papers on asymptoti- cally linear solutions of second order differential equations: [12,23,31,37]. Our methods also bear some similarities to the methods used in the studies of asymptotic behavior of solutions to difference equations of neutral type, see, for example, [15,20,24,26,27,41,43]. Some closely re- lated results on difference and dynamic equations, including ones on approximative solutions, can be found in [4,5], and [38].

The paper is organized as follows. In Section 2 we introduce notation and terminology. In Sections 3 and 4 we present some preliminary results. Section 3 is devoted to asymptotically polynomial sequences. We establish some fundamental properties of the spaces of asymptot- ically polynomial sequences and regularly asymptotically polynomial sequences. At the end of Section 3 we obtain Lemma3.11which is the base of our ‘small difference method’. This method will be used in the proofs of Theorems7.4and7.5.

In Section 4 we establish some properties of the iterated remainder operator. These results will be used in the proofs of Theorems5.1, 5.2, 6.1 and6.2. Moreover, using the Schauder’s fixed point theorem we obtain a certain fixed point lemma (Lemma4.7) which will be used in the proofs of Theorems5.1and5.2. Using the Knaster–Tarski fixed point theorem we obtain another fixed point lemma (Lemma4.9) which will be used in the proofs of Theorems6.1and 6.2.

The main results appear in Sections 5, 6 and 7. In Section 5, assuming the function f is continuous, we establish conditions under which for anyy ∈ _∆^−mbsuch that(y◦σ)w= O(1) there exists a solution (or full solution)xof (E) such thatx= y+o(n^s). In Section 6 we obtain analogous results under the assumption that the function f is monotonic with respect to the second variable. In Section 7 we obtain the conditions under which all solutions of (E) are asymptotically polynomial or ‘translated’ asymptotically polynomial.

2 Notation and terminology

Ifp,k∈ _Z,p ≤k, thenN(p),N(p,k)denote the sets defined by

N(p) ={p,p+_{1, . . .}}_{, N}(p,k) ={p,p+_{1, . . . ,}k}_.

The space of all sequencesx: N → _Rwe denote by SQ. For any x ∈ SQ we denote by ¯xthe sequence defined by

x¯n = _f(_n,xσ(n))_{. (2.1)}

The Banach space of all bounded sequencesx ∈SQ with the norm kxk=sup{|x_n|:n∈_N}

we denote by BS. Ifx,yin SQ, thenxydenotes the sequence defined by pointwise multiplica- tion

xy(n) =x_ny_n.

Moreover,|x|denotes the sequence defined by|x|(n) =|x_n|for everyn.

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) ={ax:x ∈o(1)}, O(a) =aO(1) ={ax: x∈O(1)}. Form∈_N(0)we define

Pol(m−1) =_Ker∆^m = {x∈SQ :∆^mx=0}.

Then Pol(m−1)is the space of all polynomial sequences of degree less thanm. Note that Pol(−1) =_Ker∆⁰=0

is the zero space. For a subsetXof SQ let

∆^mX={_∆^mx: x∈ X}, ∆^−mX={z∈SQ : ∆^mz ∈X}

denote respectively the image and the inverse image ofXunder the map∆^m: SQ→SQ. Ifb∈ SQ, then∆^−m{b}we also denote simply by∆^−mb. Now, we can define spaces of asymptotically polynomial sequences and regularly asymptotically polynomial sequences

Pol(m−1) +o(n^s), Pol(m−1) +_∆^−ko(1),

wheres∈(−_∞,m−1]_andk ∈_N(_0,m−1). Moreover, we will also use the sets of ‘translated’

asymptotically polynomial sequences

∆^−mb+o(n^s), ∆^−mb+_∆^−ko(1).

Remark 2.1. Note that ifyis an arbitrary element of∆^−mb, then∆^−mb=y+Pol(m−1)and so

∆^−mb+o(n^s) =y+Pol(m−1) +o(n^s),

∆^−mb+_∆^−ko(1) =y+Pol(m−1) +_∆^−ko(1). Hence∆^−mb+o(n^s)and∆^−mb+_∆^−ko(1)are affine subsets of the space SQ.

Now we define the spaces S(m)ofm-times summable sequences and the remainder opera- tor. Let

S(0) =o(1), S(1) =ⁿx∈SQ : the series

∑

xn is convergento . Forx ∈S(₁), we define the sequencer(x)by the formula

r(x)(n) =

∑

x_j. Thenr(x)∈S(₀)and we obtain the remainder operator

r: S(₁)→_S(₀)_.

Obviously the operatorr is linear. If m ∈ _N then, by induction, we define the linear space S(m+1)and the operator

r^m+1: S(m+1)→S(0) by

S(m+1) ={x ∈S(m): r^m(x)∈S(1)}, r^m+1(x) =r(r^m(x)). The valuer^m(x)(n)we denote also byr^m_n(x)or simplyr^m_nx. Note that

r^m_nx=

∑

∑

· · ·

∑

im=im−1

x_im for anyx∈S(m)and anyn∈_N.

A functionh: X→Yof a topological spaceXto metric spaceYis called locally bounded if for anyx∈ Xthere exists a neighborhoodUofxsuch thath|_U is bounded.

Remark 2.2. IfX ⊂ R, then every continuous, every monotonic and every bounded function h: X→_Ris locally bounded. Moreover, ifXis closed, thenhis locally bounded if and only if it is bounded on every bounded subset ofX.

Fork∈_N(1)we use the factorial notation

n^(k)=n(n−1). . .(n−k+1) with n⁽⁰⁾=1.

We say that the equation (E) is of continuous type if the function f is continuous (we regard N×_R as a metric subspace of the plane R²). If f is monotonic with respect to the second variable we say that (E) is of monotone type.

3 Asymptotically polynomial sequences

In this section we establish some basic properties of the spaces of asymptotically polynomial sequences. The main result of this section is Lemma3.11which will be used in the proofs of Theorems7.4and7.5.

Lemma 3.1. Assume m∈_N(0), k∈_N(0,m)and x ∈SQ. Then (a) _∆^mx∈_∆^ko(1)⇐⇒ x∈Pol(m−1) +_∆^k−mo(1);

(b) x∈ _∆^−mo(1)⇐⇒_∆^px∈o(n^m−p)for every p∈_N(0,m); (c) Pol(m−1)⊂_∆^−mo(1)⊂o(n^m).

Proof. (a) If∆^mx=_∆^ku,u=o(1), then choosing a sequencewsuch that∆^m−kw=uwe obtain

∆^mx=_∆^ku= _∆^k_∆^m−kw=_∆^mw.

Hence

x−w∈_Ker∆^m =Pol(m−1) and w∈_∆^k−mo(1). Therefore

x = (x−w) +w∈_Pol(m−₁) +_∆^k−m_o(₁)_.

Assumex=u+w,u∈Pol(m−1)andw∈_∆^k−mo(1). Then

∆^mx=_∆^mu+_∆^mw=_∆^mw=_∆^k_∆^m−kw∈ _∆^ko(1). (b) Ifx∈_∆^−mo(1), then∆^mx=o(1)and

∆∆^m−1x_n

∆n = _∆^mxn=o(1). By the Stolz–Cesàro theorem∆^m−1xn=o(n). Hence

∆∆^m−2x_n

∆n² = ^n∆∆

m−2x_n n∆n² = ^∆

m−1x_n n

n

∆n² −→0.

Again by the Stolz–Cesàro theorem∆^m−2xn =o(n²). Analogously∆^m−3xn =o(n³)and so on.

Inverse implication is obvious.

(c) Obviously Pol(m−1)⊂_∆^−mo(1). Takingp =0 in (b) we obtain∆^−mo(1)⊂o(n^m). Remark 3.2. The inclusion

∆^−mo(₁)⊂_o(n^m) is proper for anym∈_N(1). For example, ifa_n= (−1)ⁿ, then

a∈o(n^m), ∆^ma_n=2^m(−1)^m+n ∈/o(1) and soa∈/_∆^−m_o(₁)_.

Remark 3.3. Ifm∈ _N(0)andk∈_N(0,m), then by Lemma3.1we have

x ∈Pol(m) +_∆^−k_o(₁)⇐⇒ _∆^px∈ Pol(m−p) +_o(n^k−p) _{for any} p∈_N(_0,k)_.

In the next two lemmas we describe elements of the spaces of asymptotically polynomial sequences and elements of the spaces of regularly asymptotically polynomial sequences.

Lemma 3.4. Assume m∈_N(0), k∈_N(0,m)and x∈SQ. Then x∈ _Pol(m) +_o(n^k)

if and only if there exist constants c_m, . . . ,c_kand a sequence w∈o(n^k)such that x_n =c_mn^m+c_m−1n^m−1+· · ·+c_kn^k+w_n. Moreover, the constants c_m, . . . ,c_kand the sequence w are unique.

Proof. If Pol(m,k)denotes the subspace of Pol(m)generated by sequences (n^m),(n^m−1), . . . ,(n^k),

then

Pol(m) +o(n^k) =Pol(m,k) +o(n^k) and Pol(m,k)∩o(n^k) =0.

Hence

Pol(m) +o(n^k) =Pol(m,k)⊕o(n^k) and we obtain the result.

Lemma 3.5. Assume m∈_N(0), k∈_N(0,m)and x ∈SQ. Then x ∈Pol(m) +_∆^−ko(1)

if and only if there exist constants cm, . . . ,c_k and a sequence w∈o(_n^k)_{such that} x_n=c_mn^m+c_m−1n^m−1+· · ·+c_kn^k+w_n and∆^pwn=o(n^k−p)for any p∈_N(0,k).

Proof. The result is a consequence of Lemma3.4and Lemma3.1(b).

Remark 3.6. We can compare the spaces of asymptotically polynomial sequences. Let P(m,k) =Pol(m) +o(n^k) and D(m,k) =Pol(m) +_∆^−ko(1).

Then, using Lemma3.1and the fact that ifs,t∈ R, then the condition o(n^s)⊂o(n^t)

is equivalent to the conditions ≤t, we obtain a diagram

P(m, 0) −−−−→ P(m, 1) −−−−→ P(m, 2) −−−−→ . . . −−−−→ P(m,m) −−−−→ P(m,m+1) x





x





x





x





x



 D(m, 0) −−−−→ D(m, 1) −−−−→ D(m, 2) −−−−→ . . . −−−−→ D(m,m) −−−−→ D(m,m+1) where arrows denote inclusions. Note thatD(m, 0) =Pol(m) +o(1) =P(m, 0),

P(m,k) =o(n^k) and D(m,k) =_∆^−ko(1) for k >m.

Now, we describe the elements of the space

D(m−1,k) =Pol(m−1) +_∆^−ko(1) in a different way than in Lemma3.5.

Lemma 3.7. Let m∈_N(0), k∈_N(0,m)and z∈ SQ. The following conditions are equivalent:

(1) z∈Pol(m−1) +_∆^−ko(1). (2) _∆^mz∈_∆^m−ko(1).

(3) _∆^kz∈Pol(m−k−1) +o(1).

(4) _∆^pz∈Pol(m−p−1) +_∆^p−ko(1)for certain p∈_N(0,k). (5) _∆^pz∈Pol(m−p−1) +_∆^p−ko(1)for every p∈_N(0,k).

Proof. Implications (1) ⇒ (4), (3) ⇒ (4),(5) ⇒ (1), (5) ⇒ (3)and(5) ⇒ (4)are obvious.

Assume(₂)_{and let}p∈_N(_0,k)_{. Then}

∆^m−p∆^pz= _∆^mz ∈_∆^m−ko(1) and (m−k)−(m−p) =p−k.

Hence, by Lemma3.1, we have

∆^pz∈_Pol(m−p−₁) +_∆^p−k_o(₁)_.

Therefore(2)⇒(5). Assume

∆^pz∈ Pol(m−p−1) +_∆^p−ko(1) =Pol(m−p−1) +_∆^{(m−k)−(m−p)}o(1). Then, by Lemma3.1, we have

∆^m−p∆^pz∈ _∆^m−ko(1).

Hence∆^mz ∈_∆^m−ko(1). Therefore(4)⇒(2)The proof is complete.

In the next lemma we describe the elements of the space D(m,m) =Pol(m) +_∆^−mo(1).

Lemma 3.8. Assume z∈SQand m∈_N(0). The following conditions are equivalent.

(a) _∆^m+1z ∈_∆o(1).

(b) The sequence∆^mz is convergent.

(c) z∈Pol(m) +_∆^−mo(1).

(d) There exists a constantλ∈_Rsuch that for any p∈_N(0,m)we have

nlim→_∞

p!∆^m−pz_n

n^p = _lim

n→_∞

p!∆^m−pz_n n^(p) =_λ.

Proof. Equivalences(a) ⇔ (b) ⇔ (c)easily follow from Lemma 3.7. Taking p = 0 in(d)we obtain(d)⇒(b). Note that

nlim→_∞

∆^m−pz_n

n^p = lim

n→_∞

∆^m−pz_n n^(p) lim

n→_∞

n^(p)

n^p = lim

n→_∞

∆^m−pz_n n^(p) . Assume lim∆^mzn =λ. Then

∆∆^m−1zn

∆n =_∆^mzn→λ.

By the Stolz–Cesàro theorem limn⁻¹∆^m−1z_n=_{λ. Hence}

∆∆^m−2z_n

∆n² = ^n∆∆

m−2z_n n∆n² = ^∆

m−1z_n n

n

∆n² −→ ^λ 2.

By the Stolz–Cesàro theorem limn⁻²∆^m−2z_n=λ/2=λ/2!. Similarly from the equality lim n²

∆n³ = ¹ 3

we obtain limn⁻³∆^m−3zn = (λ/2)(1/3) = λ/3! and so on. We obtain the implication(b) ⇒ (d). The proof is complete.

The next two lemmas are used in the proof of Lemma3.11.

Lemma 3.9. Assume s∈ (−1,∞), m∈ _N(1)and∆^mx_n =o(n^s). Then x_n=o(n^s+m). Proof. See Lemma 2.1 in [29].

Lemma 3.10. Assume u is a positive and nondecreasing sequence, m∈ _N(1)and

∑

n^m−1un|an|<_∞.

Then there exists a sequence w∈o(u⁻¹)such that∆^mw=a.

Proof. See Lemma 2.3 in [29].

The following lemma is a base of our ‘small difference method’ which we use in Section 7.

This lemma extends Theorem 2.1 of [29].

Lemma 3.11. Assume a,b,x ∈ SQ, m ∈ _N(1), s ∈ (−_∞,m−1], k ∈ _N(0,m−1), c = |a|+|b| and

∆^mx∈O(a) +b.

Then

(a) if ∑^∞n=1n^m−1−s|an|<_∞, then x∈_∆^−mb+o(n^s), (b) if ∑^∞n=1n^m−1−s|cn|<_∞, then x∈Pol(m−1) +o(n^s), (c) if ∑^∞n=1n^m−1−k|an|<_∞, then x∈_∆^−mb+_∆^−ko(1), (d) if ∑^∞n=1n^m−1−k|c_n|<_∞, then x∈Pol(m−1) +_∆^−ko(1).

Proof. (a) Assume∑^∞n=1n^m−1−s|an|<_{∞. Let}s≤0. The condition∆^mx−b∈O(a)implies

∑

n^m−1−s|_∆^mx_n−b|< _∞.

Letu_n = n^−s. By Lemma 3.10, there exists a sequencew = o(n^s)such that∆^mw = _∆^mx−b.

Then∆^m(x−w) =b. Hencex−w∈_∆^−mband

x= x−w+w∈_∆^−mb+o(n^s).

Lets∈ (_0,m−₁]_{. Choose}k ∈_N(_1,m−₁)_{such that}k−₁<s ≤k. Then

∑

n^m−k−1n^k−s|_∆^mx−b|< _∞

and by Lemma3.10there existsw=o(n^s−k)such that∆^m−kw=_∆^mx−b. Choosez ∈SQ such that∆^kz= w. Sinces−k>−1, so by Lemma3.9we obtainz=o(n^s). Moreover

∆^mz =_∆^m−k_∆^k_z= _∆^m−k_w=_∆^m_x−b and x=_x−z+_z∈_∆^−mb+_o(_n^s)_. (b) If∑^∞n=1n^m−1−s(|a_n|+|b_n|)<_∞, then∆^mx ∈0+O(|a|+|b|)and by (a) we obtain

x∈ _∆^−m0+o(n^s) =Pol(m−1) +o(n^s). (c) Assume∑^∞n=1n^m−1−k|a_n|<∞. The condition∆^mx−b∈O(a)implies

∑

n^m−k−1|_∆^mxn−b|<_∞.

By Lemma3.10, there exists a sequencew=o(1)such that∆^m−kw=_∆^mx−b. Choosez ∈SQ such that∆^kz=w. Thenz ∈_∆^−ko(1)and

∆^mz=_∆^m−k_∆^kz =_∆^m−kw=_∆^mx−b. So x= x−z+z∈ _∆^−mb+_∆^−ko(1). (d) If∑^∞n=1n^m−1−k(|a_n|+|b_n|)<_{∞, then∆}^mx∈₀+_O(|a|+|b|)and by (c) we obtain

x∈ _∆^−m0+_∆^−ko(1) =Pol(m−1) +_∆^−ko(1). The proof is complete.

4 The iterated remainder operator and fixed point lemmas

In the first three lemmas we present some basic properties of the iterated remainder operator.

Next we obtain some fixed point lemmas (Lemma4.7and Lemma4.9). These lemmas will be used in Sections 5 and 6.

Lemma 4.1. Assume x,y∈SQ, m ∈_N(1)and p∈_N(0). Then (a) if|x| ∈S(m), then x ∈S(m)and|r^mx| ≤r^m|x|,

(b) |x| ∈S(m)if and only if∑^∞n=1n^m−1|x_n|<_∞,

(c) if|x| ∈S(m), then r^m_k|x| ≤_∑^∞_n=kn^m−1|x_n|for any k∈_N(1), (d) if x∈S(m), then∆^mr^mx= (−1)^mx,

(e) if x=_o(₁), then∆^mx∈ _S(m)and r^m∆^mx = (−₁)^mx, (f) _∆^m(S(0)) =S(m), r^m(S(m)) =S(0),

(g) _∆^p(S(m)) =S(m+p), r^p(S(m+p)) =S(m),

(h) if x,y∈_S(m)and x_n ≤y_nfor n≥ p, then r^m_nx≤r^m_ny for n≥ p,

(i) if x∈S(m)and y_n=x_nfor n≥ p, then y∈S(m)and r^m_ny=r^m_nx for n≥ p.

Proof. The assertion (a) is proved in Lemma 1 of [28] and (b) is proved in Lemma 3 of [28]. The assertion (c) follows from Lemma 2 of [28] and from the proof of Lemma 3 in [28], while (d) is proved in Lemma 5 of [28]. The assertion (e) follows from Lemma 6 of [28], while (f) is an easy consequence of (e) and (d). The assertion (g) is a consequence of (f) and (e). The assertion (h) is obvious form=_{1. For}m> 1 it can be easily proved by induction. The assertion (i) is an easy consequence of (h).

Lemma 4.2. Assume x∈ SQ, m∈_N(1), s∈(−_{∞, 0}]and

∑

n^m−1−s|x_n|<_∞. (4.1)

Then x ∈S(m), r^mx =o(n^s)and r^m_nx=

∑

∑

. . .

∑

x_im =

∑

m+k−1 m−₁

x_n+k. (4.2)

Moreover, if k∈_N(_0,m), then∆^kr^mx =_o(n^s−k).

Proof. Letu_n =n^−s. Then by Lemma3.10there exists a sequencew=o(n^s)such that∆^mw=x.

By Lemma4.1(f) we obtain

x=_∆^mw∈_∆^mo(n^s)⊂_∆^mo(1) =S(m).

By Lemma4.1(e) we obtainr^mx=r^m∆^mw= (−1)^mw∈o(n^s). The assertion (4.2) follows from Lemma 2 of [28]. Letk ∈_N(_0,m)_{. Then}m−₁−s=m−k−₁−(s−k)_and

∑

n^{(m−k)−1−(s−k)}|x_n|< _∞.

Hence, by the first part of the proof we obtainr^m−kx=o(n^s−k). On the other hand

∆^kr^mx=_∆^kr^kr^m−kx= (−1)^kr^m−kx.

Hence∆^kr^mx =o(n^s−k).

Remark 4.3. In general the conditionz∈o(n^s)does not imply the condition∆^mz∈o(n^s−m). Example 4.4. Lett,s ∈_R,t <s,m∈_N,m>s−tandz_n = (−1)ⁿn^t. Then

|_∆^mz_n|=

∑

(−1)^m+k m

k

z_n+k

=

∑

(−1)^m+k m

k

(−1)^n+k(n+k)^t

=

(−1)^m+n

∑

m k

(n+k)^t

≥n^t and we obtain

|_∆^mzn|

n^s−m = |_∆^mzn|

n^t+s−t−m =n^m−s+t|_∆^mzn|

n^t ≥n^m−s+t →_∞.

Hence

z ∈o(n^s) and ∆^mz∈/o(n^s−m). Lemma 4.5. Assume m∈_N(1),ρ∈SQ,|ρ| ∈S(m)and

S={x ∈SQ : |x| ≤ |ρ|}. Then S⊂S(m)and the map r^m|S is continuous.

Proof. By Lemma4.1(b) we have∑^∞n=1n^m−1|ρ_n|<_∞andS ⊂S(m). Letε >0. Choose p∈ _N andδ>0 such that

∑

n^m−1|ρ_n|< ε and δ

∑

n^m−1<ε.

Letx,y∈S,kx−yk<δ. Then

kr^mx−r^myk= kr^m(x−y)k=sup

n

|r^m_n(x−y)| ≤sup

n

r^m_n|x−y|

=r^m₁|x−y| ≤

∑

n=1

n^m−1|x_n−y_n| ≤

∑

n^m−1|x_n−y_n|+

∑

n^m−1|x_n−y_n|

≤ δ

∑

n^m−1+

∑

n^m−1(|ρ_n|+|ρ_n|)<3ε.

Remark 4.6. The operatorr^m: S(m)→S(0)is discontinuous for anym∈ _N(1). For example if u_k ∈SQ, u_k(n) =

(1 forn≤k 0 forn>k,

then u_k ∈ _S(m)_, ku_kk = _{1 and} kr^m(u_k)k ≥ kr(u_k)k = k. Hence r^m is a linear unbounded operator. Therefore it is discontinuous.

Lemma 4.7(Schauder’s fixed point lemma). Assume y,ρ∈SQ,ρ≥0, andlimρ_n=0. In the set S ={x∈SQ : |x−y| ≤ρ}

we define the metric by the formula d(x,z) =sup_n∈N|xn−zn|. Then every continuous map H:S→S

has a fixed point.

Proof. Assume H: S → Sis continuous and let T = {x ∈ BS : |x| ≤ ρ}. ObviouslyT is a convex and closed subset ofBS. Choose anε>0. Then there exists p∈ _N(1)such thatρ_n ≤ε for anyn≥ p. Forn=_{1, . . . ,}pletG_ndenote a finiteε-net for the interval[−ρ_n,ρ_n]_{and let}

G={x∈ T: xn∈ Gn for n≤ p and xn= 0 for n> p}.

ThenGis a finiteε-net forT. HenceTis a complete and totally bounded metric space and so, Tis compact. HenceTis a convex and compact subset of the Banach spaceBS. LetF: T → S be a map given by F(x)(n) = x_n+y_n. Then F is an isometry of T onto S. Let B: T → T, B = F⁻¹◦H◦F. ThenBis continuous. By Schauder’s fixed point theorem there existsu ∈ T such thatB(u) =u. Letx= F(u). Then

x=F(u) =F(B(u)) =F(F⁻¹HF(u)) = HF(u) =H(x). The proof is complete.

The following lemma is a version of the Knaster–Tarski fixed point theorem. This theorem may be found in [1] or in [16] but we use a simpler version. For the convenience of the reader we cite the proof from [3].

Lemma 4.8(Knaster–Tarski). If X is a complete partially ordered set and a map T: X →X is nonde- creasing then there exists x0∈ X such that T(_x0) =_x0.

Proof. LetZ= {z∈X : z≤T(z)},x₀=supZ. Ifz∈Zthen

z≤ x₀ =⇒T(z)≤T(x₀) =⇒z ≤T(z)≤ T(x₀).

Hence z ≤ T(x₀)_{for every}z ∈ Z. Thereforex₀ = _supZ ≤ T(x₀)_{. Thus} x₀ ∈ Z. Moreover x₀ ≤ T(x₀)implies T(x₀) ≤ T(T(x₀)). HenceT(x₀) ∈ Z. Therefore T(x₀) ≤ supZ = x₀. It follows thatT(x₀) =x₀.

Lemma 4.9(Knaster–Tarski fixed point lemma). Let y,ρ∈ SQand let S denote the set{x ∈SQ :

|x−y| ≤ ρ} with natural order defined by: x ≤ z if xn ≤ zn for any n ∈ _N(1). Then every nondecreasing map T: S→S has a fixed point.

Proof. By Lemma4.8 it follows that it is sufficient to show that the set S is complete; i.e. for every B⊂Sthere exists a supB∈ S. LetB⊂S. Forn∈ _NletB_n ={x_n :x ∈ B}. ThenB_nis a subset of the complete partially ordered set

Yn= [yn−ρn,yn+ρn]⊂_R.

Letsn =supBninYn. We obtain a sequences∈S. Obviouslys =supB.

5 Approximative solutions of continuous type equations

In this section, in Theorem5.1, assuming the function f is continuous, we establish conditions under which for anyy ∈ _∆^−mbsuch that(_y◦σ)_w = _O(₁)there exists a solutionxof (E) such thatx=y+o(n^s). In Theorem5.2we obtain an analogous result with a full solutionx.

In this section we will use the following condition (A) f is continuous,gis locally bounded,

s∈(−_{∞, 0}]_,

∑

n^m−1−s|a_n|< _{∞ and} w=_O(₁)_. Theorem 5.1. Assume(A), y∈ SQ,∆^my= b and

(y◦σ)w=O(1). (5.1)

Then there exists a solution x of (E)such that x=y+o(n^s)and moreover,

∆^kx= _∆^ky+_o(n^s−k) for any k∈_N(0,m).

Proof. Recall that forx∈SQ we denote by ¯xthe sequence defined by

¯

x_n = f(n,x_σ(n))_. Let

T ={x∈ SQ :|x−y| ≤1}.

By(A)and (5.1), there exists a constantKsuch that ifx ∈Tandn∈_{N, then}

|w_nx_σ(n)|=|w_nx_σ(n)−w_ny_σ(n)+w_ny_σ(n)|

≤ |w_n||x_σ(n)−y_σ(n)|+|w_ny_σ(n)| ≤K.

By(A)there existsM >0 such thatg([0,K])⊂[0,M]. Therefore, using (G) we have

g(|wnx_σ(n)|)≤ M and |x¯n| ≤g(|x_σ(n)wn|)≤ M (5.2) forx∈ Tandn∈N. There existsp≥1 such thatMr^m_n|a| ≤1 forn≥ p. Let

µ(n) =0 for n< p, µ(n) =1 for n≥ p, ρ=µMr^m|a|. LetS⊂SQ andA:S →SQ be defined by

S={x∈_{SQ :} |x−y| ≤ρ}_, A(x) =y+ (−₁)^mµr^m(ax¯)_. ThenS⊂T. Ifx∈S, then, using Lemma4.1(a) and (h), we get

|Ax−y|=|µr^m(ax¯)| ≤µr^m|ax¯| ≤ρ.

HenceA(S)⊂S. Chooseε>0. There existq≥ pandα>0 such that 2M

∑

n^m−1|an|<ε and αq^m−1

∑

|an|<ε. (5.3)

Let

L =max{|y_σ(n)−y_n|:n∈_N(1,q)}, W ={(n,t)∈_R² :n∈_N(1,q),|t−y_n| ≤ L+1}. By compactness ofW, the function f is uniformly continuous onW. Hence, there existsδ >0 such that if (n,s),(n,t) ∈ W and |s−t| < δ, then |f(n,s)− f(n,t)| < α. Assume x,z ∈ S,

|x−z|<δ, andu= x¯−z. Then¯ |Ax−Az|=|µr^m(au)|. Using Lemma4.1(a) and (c) we get d(Ax,Az) =sup

n∈_N

|Axn−Azn|=sup

n∈_N

|r_n^m(au)| ≤sup

n∈_N

r^m_n|au| ≤

∑

n^m−1|anun|. Hence

d(Ax,Az)≤

∑

n^m−1|anun|+

∑

n^m−1|anun|. (5.4) By (5.2),|u| ≤2M. Ifn∈_N(1,q), then

|x_σ(n)−y_n| ≤ |x_σ(n)−y_σ(n)|+|y_σ(n)−y_n| ≤ρ(n) +L≤ L+1.

Hence(n,x_σ(n))∈W. Analogously(n,z_σ(n))∈W and|u_n| ≤αforn≤q. By (5.3) and (5.4) we get

d(Ax,Az)≤αq^m−1

∑

|a_n|+2M

∑

n^m−1|a_n|<ε+_ε.

Thus the map A is continuous and, by Lemma 4.7, there exists a sequence x ∈ S such that Ax= x. Then

x_n=y_n+ (−1)^mr_n^m(a_nf(n,x_σ(n))) for n ≥ p

and, by Lemma4.1(d),∆^mx_n = a_nf(n,x_σ(n)) +b_nforn ≥ p. Now, the assertion follows from Lemma4.2.

Theorem 5.2. Assume(A), the function g is bounded, y∈SQ,∆^my=b and w(y◦σ) =O(1).

Then there exists a full solution x of (E)such that

∆^kx=_∆^ky+o(n^s−k) for any k ∈_N(0,m).

Proof. By (2.1) and (G), there exists a constant M such that|x¯| ≤ M for every x ∈ SQ. Let ρ=r^m(|a|M+|b|),

S={x ∈SQ :|x−y| ≤ρ}, A(x) =y+ (−1)^mr^m(ax¯+b).

Similarly as in the proof of Theorem5.1it can be shown that there exists a sequencex ∈Ssuch that Ax=x. Then

x=y+ (−1)^mr^m(ax¯). Hence

∆^mx_n=a_nf(n,x_σ(n)) +b_n for anyn.

This means that x is a full solution of (E). Moreover, by Lemma4.2, we have ∆^kx = _∆^ky+ o(n^s−k)_{for any}k ∈_N(_0,m)_.