Mezocontinuous operators and solutions of difference equations

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Mezocontinuous operators and solutions of difference equations

Janusz Migda

^B

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

Received 16 January 2016, appeared 15 March 2016 Communicated by Stevo Stevi´c

Abstract. We attempt to unify and extend the theory of asymptotic properties of solutions to difference equations of various types. Usually in difference equations some functions are used which generate transformations of sequences. We replace these functions by abstract operators and investigate some properties of such operators. We are interested in properties of operators which correspond to continuity or boundedness or local boundedness of functions. Next we investigate asymptotic properties of the set of all solutions to ‘abstract’ and ‘functional’ difference equations. Our approach is based on using the iterated remainder operator and the asymptotic difference pair. More- over, 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, asymptotic difference pair, prescribed asymptotic behavior, paracontinuous operator, mezocontinuous operator.

2010 Mathematics Subject Classification: 39A05, 39A10, 39A70.

1 Introduction

LetN,Rdenote the set of positive integers and the set of real numbers, respectively. Moreover, let SQ=_R^N denote the space of all real sequencesx:N→R. In the paper we assume that

m∈_N, F: SQ →SQ and consider difference equations of the form

∆^mxn= anF(x)(n) +bn, (E) where an,bn∈R. We say that (E) is an abstract difference equation of orderm.

Let p∈ N. We say that a sequencex ∈SQ is a p-solution of equation (E) if equality (E) is satisfied for any n≥ p. We say that x is a solution if it is a p-solution for certain p ∈_{N. If} x is a p-solution for any p∈N, then we say thatx is a full solution.

BEmail: migda@amu.edu.pl

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As a special case of (E) we get equations of type:

∆^mx_n= a_nf(n,x_σ₁₍_n₎, . . . ,x_σ_k₍_n₎) +b_n, f :N×_R^k →_R, _(E1) wherek∈_N,σ₁, . . . ,σ_k :N→_{N, or}

∆^mx_n =a_nf(n,x_n,∆xn,∆²x_n, . . . ,∆^kx_n) +b_n, f :N×_R^k⁺¹ →_R, (E2) wherekis an arbitrary natural number (the casek> mis not excluded).

In the series of papers [15–23] a new method in the study of asymptotic properties of solutions to difference equations is presented. This method, based on using the iterated remainder operator, and the regional topology on the space of all real sequences, allows us to control the degree of approximation. In the paper [22], summarizing some earlier results, the notion of a difference asymptotic pair was introduced and the theory of such pairs was used to study the asymptotic properties of solutions to autonomous difference equations of the form

∆^mxn= anf(x_σ(n)) +bn.

In this paper we extend the results from [22] to more general classes of equations. Our approach to the study of asymptotic properties of solutions were inspired by the papers [1–14]

and [24–33].

The paper is organized as follows. In Section 2, we introduce notation and terminology. In Section 3 we introduce the notion of mezocontinuous operator and we give some examples to show that the mezocontinuity of the operatorFin equation (E) corresponds to the continuity of the function f in equations (E1) and (E2). Main results are obtained in Section 4. In Section 5 we apply our results to ‘functional’ equations (E1) and (E2).

2 Notation and terminology

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

N(p) ={p,p+1, . . .}, N(p,k) ={p,p+1, . . . ,k}. We use the symbols

Sol(E), Solp(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. If x,y in SQ, then xy and |x| denote the sequences defined by xy(n) = x_ny_n and|x|(n) = |x_n|respectively. Let a,b∈ SQ, p∈ N. We will use the following notations

Fin(p) ={x∈SQ : xn=0 forn≥ p}, Fin=

∞

[

p=1

Fin(p),

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

∆⁻^mb= {y ∈SQ : ∆^my=b}, Pol(m−1) =_Ker∆^m. For a subsetY of a metric spaceXandc>0 let

B^∗(Y,c) = ^[

y∈Y

B^∗(y,c),

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where B^∗(y,c)denotes the closed ball of radiusccentered at y. Fory,ρ∈ SQ and p ∈ _Nwe define

B^∗(y,ρ,p) ={x∈SQ : |x−y| ≤ |ρ| andx_n=y_nforn< p}.

Assume that Z is a linear subspace of a linear space X. We say that a subset W of X is Z-invariant ifW+Z⊂W.

2.1 Regional topology

Let X be a real vector space. We say that a functionk · k : X→ [0,∞]is aregional normif the conditionkxk=0 is equivalent tox =0 and for anyx,y ∈Xandα∈ _Rwe have

kαxk=|α|kxk, kx+yk ≤ kxk+kyk.

Hence, the notion of a regional norm generalizes the notion of a usual norm. Note that a regional norm may take the value ∞. If a regional norm on X is given, then we say thatX is aregional normed space, if there exists a vectorx ∈ Xsuch thatkxk= ∞, then we say thatXis extraordinary.

AssumeXis a regional normed space. We say that a subsetZof Xisordinaryifkx−yk<

∞ for any x,y ∈ Z. We regard any ordinary subset Z of X as a metric space with metric defined by

d(x,y) =kx−yk_.

We say that a subsetUof Xis regionally open ifU∩Zis open inZfor any ordinary subsetZ of X. The family of all regionally open subsets is a topology onXwhich we call the regional topology. We regard any subset of X as a topological space with topology induced by the regional topology. Let

Reg(0) ={x∈ X:kxk<_∞}.

Obviously Reg(0) is a linear subspace of X. Moreover, the regional norm induces a usual norm on Reg(0). We say that Xis a Banach regional space if Reg(0)is complete. Let x ∈ X.

We say that the set

Reg(x) =x+Reg(0)

is a regionof x. If y ∈ X and kx−yk < ∞, then Reg(x) = Reg(y). Any region is ordinary and open in X. Moreover, any region is connected and is metrically equivalent to the normed space Reg(0). From a topological point of view, the spaceX is a disjoint union of all regions.

Note that if x ∈X, then the region Reg(x)is the ordinary component ofx and the connected component of x. Hence any ordinary subset of X is a subset of a certain region, and any connected subset of Xis ordinary. Moreover, if H: SQ→SQ is continuous andx∈SQ, then

H(Reg(x))⊂Reg(H(x)).

We say that a subset Y of X is regional if Reg(y) ⊂ Y for any y ∈ Y. The basic properties of the regional topology are presented in [21]. We will use the following theorem (see [21, Theorem 3.1]).

Theorem 2.1 (Generalized Schauder theorem). Assume Q is a closed and convex subset of a regional Banach space X, a map H : Q → Q is continuous and the set HQ is ordinary and totally bounded. Then there exists a point x∈ Q such that Hx= x.

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We will use the standard regional norm on SQ defined by kxk=sup{|x_n|:n∈_N}. Moreover, we will use the following fixed point theorem.

Theorem 2.2. Assume y ∈ SQ,ρ ∈ o(₁)_{, p} ∈ _{N, and B} = _B^∗(_y,_ρ,_p). Then any continuous map H :B→Bhas a fixed point.

Proof. By [21, Theorem 3.3], B is ordinary, convex and compact. Hence the assertion is a consequence of Theorem2.1.

2.2 Remainder operator

Fort∈ [1,∞)andm∈_Nlet

A(t):=

a∈SQ :

∑

∞ n=₁

n^t⁻¹|a_n|<_∞

,

r^m : A(m)→o(1), r^m(a)(n) =

∑

∞ j=n

m−1+j−n m−1

a_j.

Thenr^m is a linear operator which we call the iterated remainder operator of order m. The valuer^m(a)(n)we denote also byr_n^m(a)_{or simply}r^m_na. The following lemma is a consequence of [20, Lemma 3.1], [20, Lemma 4.2], and [20, Lemma 4.8].

Lemma 2.3. Assume a∈A(m), u∈O(1), k∈ {0, 1, . . . ,m}, and p∈_{N. Then} (a) O(a)⊂A(m)⊂o(n¹⁻^m), |r^m(ua)| ≤ kukr^m|a|, ∆r^m|a| ≤0,

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

For more information about the remainder operator see [20].

2.3 Asymptotic difference pairs

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

Fin+Z⊂Z, O(1)A⊂ A, A⊂_∆^mZ.

We say that anm-pair(A,Z)isevanescentifZ⊂o(1). The following lemma is a consequence of [22, Lemma 3.5 and Lemma 3.7].

Lemma 2.4. Assume(A,Z)is an m-pair, a,b,x∈ SQ. Then (a) if a−b∈ A, then∆⁻^ma+Z =_∆⁻^mb+Z,

(b) if a∈ A and∆^mx∈O(a) +b, then x∈ _∆⁻^mb+Z, (c) if Z⊂o(1), then A⊂A(m)and r^mA⊂ Z.

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Example 2.5. Assumes ∈_R,t∈ (−_∞,m−1],λ∈(1,∞), and (s+1)(s+2)· · ·(s+m)6=0.

Then

(_o(n^s)_{, o}(n^s⁺^m))_, (_O(n^s)_{, O}(n^s⁺^m))_, (_A(m−t)_{, o}(n^t))_, (o(λⁿ), o(λⁿ)), (O(λⁿ), O(λⁿ))

are m-pairs.

Example 2.6. Assumes ∈(−_∞,−m), t∈(−_{∞, 0}],u∈ [1,∞), andλ∈ (0, 1). Then (_o(n^s)_{, o}(n^s⁺^m))_, (_O(n^s)_{, O}(n^s⁺^m))_, (_A(m−t)_{, o}(n^t))_,

(A(m+u), A(u)), (o(λⁿ), o(λⁿ)), (O(λⁿ), O(λⁿ)) are evanescentm-pairs.

For more information about difference pairs see [22].

3 Mezocontinuous operators

AssumeW ⊂X⊂SQ and H:X→SQ. We definekHk ∈[0,∞]by kHk=sup{|H(x)(n)|:x∈ X,n∈_N}.

We say, that His bounded if kHk< _{∞. Let}P be a property of operators. We say thatH has the property P on W if the restriction H|W has the property P. Recall that we regard any subset of SQ as a topological space with topology induced by the regional topology. Hence, the continuity of His defined by a standard way. We say, that His:

paracontinuousif for any ε> _{0 and any}n ∈ _Nthere exists aδ =δ(n,ε)> 0 such that if x,z∈ X, andkx−zk<δ, then|H(x)(n)−H(z)(n)|<ε,

mezocontinuousif it is paracontinuous on any bounded subset ofX, regionally boundedif it is bounded onX∩Reg(_x)_{for any}_x∈ X.

We say that a mapG:X→Yfrom a subsetXof SQ to a metric spaceYisuniformly continuous if it is uniformly continuous on any ordinary subset ZofX.

Remark 3.1. For n∈_Nlet ev_n denote the evaluation (projection operator) defined by evn : SQ→_R, evn(x) =xn.

If X ⊂ SQ, then an operator H : X → SQ is paracontinuous if and only if for any n the function evn◦H : SQ→_Ris uniformly continuous.

Remark 3.2. If X ⊂ SQ, H₁,H₂, . . . ,H_k : X → SQ are paracontinuous, and a function ϕ:R^k →_Ris uniformly continuous, then the operator H:X→SQ defined by

H(x)(n) = ϕ(H₁(x)(n), . . . ,H_k(x)(n)) is paracontinuous.

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Example 3.3. Assumeϕ_n :R→_Ris a sequence of uniformly continuous functions. Then the operatorH: SQ→SQ defined byH(x)(n) =ϕ_n(xn)is paracontinuous.

Example 3.4. Assumek∈_N, f :N×_R^k →_Ris continuous andσ₁,σ₂, . . . ,σ_k :N→_N. Then the operator

H: SQ→SQ, H(x)(n) = f(n,x_σ₁₍_n₎, . . . ,x_σ_k₍_n₎) is mezocontinuous.

Justification. AssumeS is a bounded subset of SQ. Choose a positiveεand an indexn. Let S_n= {(x_σ₁₍_n₎, . . . ,x_σ_k₍_n₎):x∈ S}.

SinceS is bounded, the set Sn is a bounded subset ofR^k. Choose a compact interval I such thatS_n ⊂ I^k. The function

g:R^k →_R, g(t₁, . . . ,t_k) = f(n,t₁, . . . ,t_k)

is uniformly continuous on I^k. Choose δ > 0 such that if α,β ∈ I^k, and kα−βk < δ, then

|g(α)−g(β)|< ε. Now, assumex,z∈ Sandkx−zk< _{δ. Then}

x^∗ := (x_σ₁(n), . . . ,x_σ_k(n))∈ Sn, y^∗ := (y_σ₁(n), . . . ,y_σ_k(n))∈Sn

andkx^∗−y^∗k ≤ kx−yk<δ. Hence|g(x^∗)−g(y^∗)|<ε. This means that

|H(x)(n)−H(y)(n)|<ε.

ThereforeH is paracontinuous onS.

Example 3.5. Ifk∈_Nand f :N×_R^k⁺¹ →_Ris continuous, then the operator H: SQ→SQ, H(x)(n) = f(n,x_n,∆xn,∆²x_n, . . . ,∆^kx_n) is mezocontinuous.

Justification. AssumeS is a bounded subset of SQ. Choose a positiveεand an indexn. Let S_n ={(x_n,∆xn,∆²x_n, . . . ,∆^kx_n): x∈S}.

SinceSis bounded, the setS_nis a bounded subset ofR^k⁺¹. Choose a compact interval I such thatS_n ⊂ I^k⁺¹. The function

g:R^k⁺¹ →_R, g(_t₀_,_t₁_{, . . . ,}_t_k) = _f(_n,_t₀_,_t₁_{, . . . ,}_t_k) is uniformly continuous onI^k⁺¹. Note that ifx,y∈SQ, then

k_∆x−_∆yk ≤2kx−yk, . . . ,k_∆^kx−_∆^kyk ≤2^kkx−yk.

Chooseδ>0 such that ifα,β∈ I^k⁺¹, andkα−βk<2^kδ, then|g(α)−g(β)|<ε. Now, assume x,z∈Sandkx−zk<δ. Then

x^∗ := (x_n,∆xn,∆²x_n, . . . ,∆^kx_n)∈S_n, y^∗ := (y_n,∆yn,∆²y_n, . . . ,∆^ky_n)∈S_n andkx^∗−y^∗k<2^kδ. Hence|g(x^∗)−g(y^∗)|< ε. This means that

|H(x)(n)−H(y)(n)|<ε.

ThereforeH is paracontinuous onS.

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Example 3.6. AssumeBis a bounded subset ofR, f :R→R, the restriction f|Bis continuous but not uniformly continuous, p∈_N,W ={x∈SQ : x(_N)⊂B}, and

H:W →SQ, H(x)(n):=

(f(x_p) for n= p xn for n6= p.

Then His continuous but not mezocontinuous.

|H(z)(p)−H(x)(p)|=|f(t)− f(s)| ≥ε.

Hence His not paracontinuous. SinceW is bounded,His not mezocontinuous.

Example 3.7. Let f :R→_Rand H: SQ→SQ is given byH(x)(n) = f(x_n). Then:

(a) _if f is uniformly continuous then His uniformly continuous, (b) if f is continuous then His mezocontinuous,

(c) if f is not uniformly continuous then His discontinuous.

Justification. The assertion (a) is obvious, and (b) is a consequence of Example 3.4. Assume f is not uniformly continuous. Then there exists a positiveεsuch that for anyn∈_Nthere exist x_n,z_n∈ _Rsatisfying |x_n−z_n|< 1/nand|f(x_n)− f(z_n)| ≥ε. Letδ > 0. Choosek ∈ _Nsuch that δ<1/kand definey∈SQ by

yn=

(x_n for n≤k, zn for n>k.

Thenkx−yk<δ andkHx−Hyk ≥ε. HenceHis discontinuous at x.

4 Solutions of abstract equations

LetW ⊂_SQ,a∈_A(m)_{, and} p∈N. We say thatW is

(F,a,p)-regularif for any y ∈ W there exists a positive constant M such thatF is paracontinuous on B=B^∗(y,Mr^m|a|,p)andkF|Bk ≤M,

F-regular if for any y ∈ W there exist a positive constant c and an index q such that F|B^∗(y,c,q)is paracontinuous and bounded,

F-optimalifW is o(1)-invariant and F|W is mezocontinuous and regionally bounded, F-ordinaryifF(W)⊂O(1).

Theorem 4.1. Assume(A,Z)is an evanescent m-pair, a∈A, p∈_{N, M}>0, y∈_∆⁻^mb, ρ= Mr^m|a|, B=B^∗(y,ρ,p), kF|Bk ≤M, and F is paracontinuous or continuous on B. Then y ∈_Sol_p(_E) +Z.

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Proof. Ifx ∈B, then the sequenceFx is bounded. HenceaFx∈O(a)⊂A(m). Define H:S→SQ, H(x)(n) =

(y_n for n< p, y_n+ (−1)^mr^m_n(aFx) for n≥ p.

Ifn≥ p, then using Lemma2.3we have

|H(x)(n)−yn|=|r^m_n(aFx)| ≤r^m_n|aFx| ≤Mr^m_n|a|=ρn.

HenceHB⊂B. Letε>0. Assume thatFis paracontinuous onB. There existq≥ pandα>0 such that

M

∑

∞ n=q

n^k⁻¹|a_n|<ε and α

∑

q n=p

n^k⁻¹|a_n|<ε.

For anyn∈ {p, . . . ,q}there existsδ_n >0 such that ifx,z∈ Bandkx−zk<δ_n, then

|F(x)(n)−F(z)(n)|<α.

Letδ=min(δ_p,δ_p+1, . . . ,δ_q). Ifx,z∈ Bandkx−zk< δ, then using Lemma2.3, we obtain kHx−Hzk= sup

n≥1

|H(x)(n)−H(z)(n)|=sup

n≥p

|r^m_n(aFx)−r^m_n(aFz)|

= sup

n≥p

|r^m_n(aFx−aFz)| ≤sup

n≥p

r^m_n|aFx−aFz|

=r^m_p|aFx−aFz| ≤

∑

^∞

n=p

n^m⁻¹|a_nF(x)(n)−a_nF(z)(n)|

≤

∑

q n=p

n^m⁻¹|a_nF(x)(n)−a_nF(z)(n)|+

∑

∞ n=q

n^m⁻¹|a_nF(x)(n)−a_nF(z)(n)|

≤α

∑

q n=p

n^m⁻¹|a_n|+

∑

∞ n=q

n^m⁻¹|a_nF(x)(n)|+

∑

∞ n=q

n^m⁻¹|a_nF(z)(n)|<3ε.

Hence H is continuous. Now assume that F is continuous on B and x ∈ B. There exists a δ(x,ε) > 0 such that the condition kz−xk < δ(x,ε) implies |Fx−Fz| < ε. If z ∈ B, kz−xk<δ(x,ε), andn≥ p, then, we obtain

|H(x)(n)−H(z)(n)|= |r_n^m(aF(x))−r^m_n(aF(z))|

≤ r^m_n(|a||Fx−Fz|)≤r^m_n(ε|a|)≤ εr^m₁|a|= M⁻¹ρ₁ε.

Hence kHx−Hzk ≤ M⁻¹ρ₁ε. Therefore H is continuous. By Theorem 2.2, there exists an x∈ Bsuch thatHx= x. Thenx_n= y_n+ (−1)^mr^m_naF(x)forn≥ p. Hence

x−y−(−₁)^mr^maFx ∈_Fin(p)_. _(4.1) Using Lemma2.3we have

∆^m((−₁)^mr^maF(x)) =aF(x)_, _and _∆^m_Fin(p) =_Fin(p)_. _(4.2) Using (4.1), (4.2), and the equality∆^my=b, we get

∆^mx−aF(x)−b∈_Fin(p)_.

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Hence x∈Sol_p(E). Moreover, we have(−1)^maFx∈O(a). Hence, by (4.1), y∈ x+r^mO(a) +Fin(p).

By Lemma2.3we get

r^mO(a) +Fin(p) =r^mO(a) +r^mFin(p) =r^m(O(a) +Fin(p))

=r^mO(a)⊂r^mA⊂Z.

Corollary 4.2. Assume(A,Z)is an evanescent m-pair, a∈A, p∈_N, and W is an(F,a,p)-regular subset ofSQ. Then

W∩_∆⁻^mb⊂Sol_p(E) +Z.

Proof. The assertion is an immediate consequence of Theorem4.1.

Corollary 4.3. Assume(A,Z)is an evanescent m-pair, a∈ A, and W is an F-regular subset ofSQ.

Then

W∩_∆⁻^mb⊂_Sol_∞(_E) +Z. (4.3)

Proof. Lety∈W∩_∆⁻^mb. Choose a positive constant cand an indexqsuch thatFis paracontinuous and bounded on B^∗(_y,_c,_q)_{. Let}

M=kF|B^∗(y,c,q)k.

Sincer^m|a| ∈o(1), there exists an indexk ≥qsuch that Mr^m_k |a| ≤c. Since the sequencer^m|a| is nonincreasing, we have

B^∗(y,Mr^m|a|,k)⊂B^∗(y,c,q).

By Theorem4.1,y ∈Sol_k(_E) +_Z⊂Sol_∞(_E) +_Zand we obtain (4.3).

Example 4.4. Assumeg:R→_R is positive, continuous, and nondecreasing,

f :N×_R→_R, f(n,t) =g(t), F: SQ→SQ, F(x)(n) = f(n,x_n). Moreover, assumea ∈A(m), p∈_N, M,α,λ∈_R, g(λ) =M,λ>α, and

∑

∞ n=p

n^m⁻¹|an| ≤ ^λ−α M

Then the setW = {y ∈SQ : y(_N)⊂(−_∞,α)}is(_F,_a,_p)_-regular.

Justification. By Lemma2.3

r^m_n|a| ≤r^m_p|a| ≤

∑

^∞

n=p

n^m⁻¹|an| ≤ ^λ−α M for any n≥ p. Lety∈W and let

B=B^∗(y,Mr^m|a|,p).

If x∈B, then|x_n−y_n| ≤Mr^m_p|a|_{for any} n. Hencex_n≤y_n+Mr^m_p|a|_and

|F(x)(n)|=|f(n,x_n)|= g(x_n)≤g(y_n+Mr^m_p|a|)≤g(α+λ−α) =g(λ) = M

for anyx ∈ Band anyn ∈_{N. Hence}kF|Bk ≤ M. By Example3.4, Fis paracontinuous onB.

ThereforeW is(F,a,p)_-regular.

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The following theorem is a consequence of Lemma2.4. Note that in this theorem we do not assume that a pair(A,Z)is evanescent.

Theorem 4.5. Assume(A,Z)is an m-pair, a ∈A, and W is an F-ordinary subset ofSQ. Then

W∩_Sol_∞(_E)⊂ _∆⁻^mb+Z. (4.4)

Proof. Ifx ∈Sol_∞(E)andFx ∈O(1), then

∆^mx∈aF(x) +b+Fin⊂O(a) +b+Fin=O(a) +b.

Hence, using Lemma2.4, we obtain (4.4).

Lemma 4.6. Assume X is a linear space, W,S,Y ⊂ X, Z is a linear subspace of X, W is Z-invariant and W∩S⊂Y+Z. Then W∩S⊂W∩Y+Z.

Proof. Let w ∈ W∩S. By assumption, w = y+z for some y ∈ Y and z ∈ Z. Since W is Z-invariant, we havey=w−z∈W. Hencey ∈W∩Yand we obtain

w=y+z∈W∩Y+Z.

Theorem 4.7. Assume(A,Z)is an evanescent m-pair, a ∈ A, and W is a Z-invariant subset ofSQ.

Then

(a) if W is F-regular, then

W∩Sol_∞(E) +Z=W∩_∆⁻^mb+Z, (b) if W is F-optimal, then

W∩Sol(E) +Z=W∩Sol_∞(E) +Z=W∩_∆⁻^mb+Z.

Proof. AssumeW isF-regular. ThenW isF-ordinary and, by Theorem4.5,

W∩_Sol_∞(_E)⊂ _∆⁻^mb+Z. (4.5)

By Corollary4.3, we have

W∩_∆⁻^mb⊂Sol_∞(E) +Z. (4.6)

Using (4.6), (4.5), and Lemma4.6we obtain (a).

AssumeW isF-optimal. Then,W isF-ordinary and, using Theorem4.5,

W∩Sol(E)⊂W∩Sol_∞(E)⊂ _∆⁻^mb+Z. (4.7) Assumey∈W∩_∆⁻^mb. Let M=kF|Reg(y)k. Sincea∈ A⊂A(m), we have

∑

∞ n=1

n^m⁻¹|a_n|<_∞.

Choose a positivecsuch that

M

∑

∞ n=1

n^m⁻¹|a_n|<c.

Then

B^∗(y,Mr^m|a|_{, 1})⊂_B^∗(y,c, 1)⊂y+_O(₁) =_Reg(y)_.

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By Theorem4.1,

y∈Sol₁(E) +Z=Sol(E) +Z.

Hence

W∩_∆⁻^mb⊂_Sol(_E) +Z. (4.8)

Using (4.8), (4.7), and Lemma4.6we obtain (b).

Let X⊂ SQ. We say that an operator H : X→ SQ is unboundedat a point p ∈ [−_∞,∞]if there exists a sequencex ∈Xand an increasing sequence α:N→_Nsuch that

nlim→_∞x(α(n)) = p and lim

n→_∞|H(x)(α(n))|=_∞.

Let

U(H) ={p ∈[−_∞,_∞]: His unbounded at p}. For x∈SQ let

L(x) ={p∈[−_∞,_∞]: pis a limit point ofx}. The following theorem extends [22, Theorem 4.2].

Theorem 4.8. 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) = ∅. If the sequence F(x)is unbounded from above, then there exists an increasing sequenceβ:N→_Nsuch that

nlim→_∞F(x)(β(n)) =_∞. (4.9) Lety =_x◦βand let p∈L(_y). There exists an increasing sequence γ:N→_Nsuch that

nlim→_∞y(γ(n)) =p. (4.10)

Letα= β◦γ. Then

x(α(n)) = (x◦β◦γ)(n) = (x◦β)(γ(n)) =y(γ(n)). Hence, by (4.10), limn→_∞x(α(n)) =p. Moreover, using (4.9), we get

nlim→_∞F(x)(α(n)) = lim

n→_∞F(x)(β(γ(n))) =_∞.

Therefore p ∈U(F). Sincey is a subsequence ofx, we have p∈L(x). Thus p∈ L(x)∩U(F).

Analogously, if the sequenceF(x)is unbounded from below, then L(x)∩U(F)6=_∅. Therefore F(x)is bounded. Sincex∈ Sol_∞(E), we have

∆^mx ∈aF(x) +b+Fin⊂aO(1) +Fin+b=O(a) +b.

By Lemma2.4(b) we getx∈_∆⁻^mb+Z.

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5 Solutions of functional equations

For a subsetVofRwe denote byV the closure ofV in the extended line[−_∞,_∞]. Theorem 5.1. Assume(A,Z)is an evanescent m-pair, a∈ A, k,p∈_{N, c}>0,

f :N×_R^k →_R, σ₁, . . . ,σ_k :N→_N, σ_i(n)→_∞ for i =1, . . . ,k, V ⊂_R, W ={x∈SQ : L(x)⊂V}, U=_N(p)×B^∗(V,c)^k, and f is continuous. Then W iso(1)-invariant and

(a) if f is bounded on U, then for any Z-invariant subset Q of W we have

Q∩_∆⁻^mb+Z=Q∩Sol_∞(E1)+Z; (5.1) (b) if f is bounded, then for any Z-invariant subset Q ofSQwe have

Q∩Sol(E1) +Z=Q∩Sol_∞(E1) +Z=Q∩_∆⁻^mb+Z. (5.2) Proof. Ify∈SQ andz∈o(1), then L(y+z) =L(y). Hence the setW is o(1)-invariant. Let

F: SQ→SQ, F(x)(n) = f(n,x_σ₁₍_n₎, . . . ,x_σ_k₍_n₎).

Assume f is bounded onU. Choosey∈W andε∈(0,c/2). It is easy to see that the set y(_N)\B^∗(V,ε)

is finite. Hence there exists an index p₁ ≥ psuch that y_n ∈ B^∗(V,ε)for any n ≥ p₁. Choose q∈_Nsuch thatσ_i(n)≥ p₁ for anyn≥qand anyi∈ {1, . . . ,k}. Let

q₁ =_max

k

[

i=1

σ_i(_N(_1,q))_, Y= y(_N(_1,q₁))_, C=_N(_1,q)×B^∗(Y,ε)^k_. ThenCis compact and f is bounded onC. Define

M₁ =kf|Uk and M₂ =kf|Ck.

Letx∈B^∗(y,ε,q). Then|x_n−y_n| ≤εfor any n. Ifn≥qandi∈_N(1,k), thenσ_i(n)≥ p₁ and y_σ_i₍_n₎∈_B^∗(V,ε)_.

Hence there existsu∈ Vsuch that|u−y_σ_i₍_n₎| ≤ε. Then

|x_σ_i(n)−u| ≤ |x_σ_i(n)−y_σ_i(n)|+|y_σ_i(n)−u| ≤2ε<c.

Hence(x_σ₁₍_n₎, . . . ,x_σ_k₍_n₎)∈B^∗(V,c)^k and forn≥qwe get

|F(x)(n)|=|f(n,x_σ₁₍_n₎, . . . ,x_σ_k₍_n₎)| ≤ M₁. Ifn≤qandi∈_N(1,k), then

x_σ_i₍_n₎ ∈_B^∗(y_σ_i₍_n₎,ε)⊂_B^∗(Y,ε)

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Hence

(n,x_σ₁(n), . . . ,x_σ_k(n))∈ C and |F(x)(n)| ≤ M2.

ThereforeFis bounded on B^∗(y,ε,q). By Example3.4,Fis paracontinuous on B^∗(y,ε,q). Thus W is F-regular. Hence any subsetQofW isF-regular. If, moreover Qis Z-invariant, then, by Theorem4.7(a), we obtain (5.1). Now, assume f is bounded onN×_R^kandQis aZ-invariant subset of SQ. Then we can takeV=_R,W =SQ, and by (5.1), we obtain

Q∩Sol(E1) +Z⊂ Q∩Sol_∞(E1) +Z= Q∩_∆⁻^mb+Z. (5.3) Assumey∈ Q∩_∆⁻^mb. Let M =kfk. ThenkFk ≤M and

kF|B^∗(y,Mr^m|a|, 1)k ≤ M.

By Example3.4,F is paracontinuous on B^∗(y,Mr^m|a|, 1). Hence, by Theorem4.1, y∈Sol₁(E1) +Z=Sol(E1) +Z.

Therefore

Q∩_∆⁻^mb⊂Sol(E1) +Z. (5.4)

Using (5.3), (5.4) and Lemma4.6we obtain (5.2).

Theorem 5.2. Assume(A,Z)is an evanescent m-pair, a∈ A, k,p ∈_N, c>0, V0,V₁, . . . ,V_k ⊂_R, U=_N(p)×B^∗(V0,c)× · · · ×B^∗(V_k,c),

W =x∈SQ : L(_∆ⁱx)⊂V_i for i=0, 1, . . . ,k , and f :N×_R^k⁺¹ →_Ris continuous. Then W iso(1)-invariant and

(a) if f is bounded on U, then for any Z-invariant subset Q of W we have

Q∩_∆⁻^mb+Z=Q∩Sol_∞(E2)+Z; (5.5) (b) if f is bounded, then for any Z-invariant subset Q ofSQwe have

Q∩_Sol(_E2) +Z=Q∩_Sol_∞(_E2) +Z=Q∩_∆⁻^mb+Z. (5.6) Proof. ObviouslyW is o(1)-invariant. Let

F: SQ→SQ, F(x)(n) = f(n,xn,∆xn,∆²xn, . . . ,∆^kxn).

Assume f is bounded onU. Choosey∈ W andε ∈ (0,c/2^k⁺¹). It is easy to see that, for any i∈_N(0,k), the set

∆ⁱy(_N)\B^∗(V_i,ε)

is finite. Hence there exists an index psuch that ∆ⁱy_n ∈ B^∗(V_i,ε)for anyi∈ _N(0,k)and any n≥ p. Let M= kF|Uk_,x ∈_B^∗(y,ε,p)_andn≥ p. Ifi∈_N(_0,k)_{, then}

|_∆ⁱxn−_∆ⁱyn| ≤2ⁱε

and there exists a point u_i ∈V_i such that|_∆ⁱy_n−u_i|<ε. Hence

∆ⁱx_n∈_B^∗(V_i, 2ⁱ⁺¹ε)⊂_B^∗(V_i,c)_.

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Therefore |F(x)(n)| ≤ M for any x ∈ B^∗(y,ε,p) and any n ≥ p. Thus F is bounded on B^∗(y,ε,p). By Example3.5, F is paracontinuous on B^∗(y,ε,p). HenceW isF-regular. Assume Qis a Z-invariant subset ofW. Then Qis F-regular and, by Theorem 4.7(a), we obtain (5.5).

Now, assume f is bounded andQis aZ-invariant subset of SQ. Then we can take V₀ =V₁=· · · =V_k =_R, W =SQ,

and, by (5.5) we obtain

Q∩Sol(E2) +Z⊂Q∩Sol_∞(E2) +Z=Q∩_∆⁻^mb+Z. (5.7) Assumey∈ Q∩_∆⁻^mb. Let M=kfk. ThenkFk ≤ Mand

kF|B^∗(y,Mr^m|a|, 1)k ≤M.

By Example3.4, Fis paracontinuous on B^∗(y,Mr^m|a|, 1). Hence, by Theorem4.1, y∈Sol1(_E2) +_Z=_Sol(_E2) +_Z.

Therefore

Q∩_∆⁻^mb⊂_Sol(_E2) +Z. (5.8)

Using (5.7), (5.8) and Lemma4.6we obtain (5.6).

References

[1] C. Avramescu, Evanescent solutions for linear ordinary differential equations, Electron.

J. Qual. Theory Differ. Equ.2002, No. 9, 1–12.MR1906597

[2] M. Bohner, S. Stevi ´c, Asymptotic behavior of second-order dynamic equations, Appl.

Math. Comput.188(2007), No. 2, 1503–1512.MR2335647

[3] M. Bohner, S. Stevi ´c, Trench’s perturbation theorem for dynamics equations, Discrete Dyn. Nat. Soc.2007, Article ID 75672, 11 pp. MR2375475;url

[4] S. S. Cheng, W. T. Patula, An existence theorem for a nonlinear difference equation, Nonlinear Anal.20(1993), No. 3, 193–203.MR1202198

[5] J. Diblík, A criterion of asymptotic convergence for a class of nonlinear differential equations with delay,Nonlinear Anal.47(2001), No. 6, 4095–4106.MR1972350

[6] A. Drozdowicz, J. Popenda, Asymptotic behavior of the solutions of the second order difference equations,Proc. Amer. Math. Soc.99(1987), No. 1, 135–140.MR0866443

[7] M. Ehrnström, Linear asymptotic behaviour of second order ordinary differential equations,Glasgow Math. J.49(2007), 105–120.MR2337871

[8] A. Gleska, J. Werbowski, Comparison theorems for the asymptotic behavior of solutions of nonlinear difference equations, J. Math. Anal. Appl. 226(1998), No. 2, 456–465.

MR1650201

[9] T. G. Hallam, Asymptotic behavior of the solutions of an nth order nonhomogeneous ordinary differential equation,Trans. Amer. Math. Soc.122(1966), 177–194.MR0188562

(15)

[10] M. Hasanbulli, Y. V. Rogovchenko, Asymptotic behavior of nonoscillatory solutions to n-th order nonlinear neutral differential equations, Nonlinear Anal. 69(2008), 1208–1218.

MR2426686

[11] J. W. Hooker, W. T. Patula, A second-order nonlinear difference equation: oscillation and asymptotic behavior,J. Math. Anal. Appl.91(1983), No. 1, 9–29.MR0688528

[12] Q. Kong, Asymptotic behavior of a class of nonlinear differential equations ofnth order, Proc. Amer. Math. Soc.103(1988), 831–838.MR0947667

[13] T. Kusano, W. F. Trench, Global existence theorems for solutions of nonlinear differential equations with prescribed asymptotic behavior, J. London Math. Soc. 31(1985), 478–486.

MR812777

[14] O. Lipovan, On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations,Glasgow Math. J.45(2003), 179–187.MR1973349

[15] M. Migda, J. Migda, On the asymptotic behavior of solutions of higher order nonlinear difference equations,Nonlinear Anal.47(2001), 4687–4695.MR1975862

[16] J. Migda, Asymptotic properties of solutions of nonautonomous difference equations, Arch. Math. (Brno)46(2010), 1–11.MR2644450

[17] J. Migda, Asymptotically polynomial solutions of difference equations, Adv. Difference Equ.92(2013), 16 pp.MR3053806;url

[18] J. Migda, Approximative solutions of difference equations,Electron. J. Qual. Theory Differ.

Equ.2014, No. 13, 1–26.MR3183611

[19] J. Migda, Approximative full solutions of difference equations, Int. J. Difference Equ.

9(2014), 111–121.

[20] J. Migda, Iterated remainder operator, tests for multiple convergence of series and solutions of difference equations,Adv. Difference Equ.2014, 2014:189, 18 pp.MR3357342 [21] J. Migda, Regional topology and approximative solutions of difference and differential

equations,Tatra Mt. Math. Publ.63(2015), 183–203.MR3411445

[22] J. Migda, Qualitative approximation of solutions to difference equations,Electron. J. Qual.

Theory Differ. Equ.2015, No. 32, 1–26.MR3353203

[23] J. Migda, Approximative solutions to difference equations of neutral type, Appl. Math.

Comput.268(2015), 763–774.MR3399460

[24] A. B. Mingarelli, K. Sadarangani, Asymptotic solutions of forced nonlinear second order differential equations and their extensions, Electron. J. Differential Equations 2007, No. 40, 1–40.MR2299594

[25] O. G. Mustafa, Y. V. Rogovchenko, Asymptotic integration of a class of nonlinear differential equations,Appl. Math. Lett.19(2006), 849–853.MR2240473

[26] Ch. G. Philos, I. K. Purnaras, P. Ch. Tsamatos, Asymptotic to polynomials solutions for nonlinear differential equations,Nonlinear Anal.59(2004), 1157–1179.MR2098511

(16)

[27] C. G. Philos, P. C. Tsamatosa, Solutions approaching polynomials at infinity to nonlinear ordinary differential equations,Electron. J. Differential Equations2005, No. 79, 1–25.

MR2162240

[28] J. Popenda, Asymptotic properties of solutions of difference equations,Proc. Indian Acad.

Sci. Math. Sci.95(1986), No. 2, 141–153.MR0913886

[29] S. Stevi ´c, Asymptotic behaviour of second-order difference equation, ANZIAM J.

46(2004), No. 1, 157–170.MR2075520

[30] S. Stevi ´c, Growth estimates for solutions of nonlinear second-order difference equations, ANZIAM J.46(2005), No. 3, 439–448.MR2124934

[31] E. Thandapani, R. Arul, J. R. Graef, P. W. Spikes, Asymptotic behavior of solutions of second order difference equations with summable coefficients, Bull. Inst. Math. Acad.

Sinica27(1999), 1–22.MR1681601

[32] W. F. Trench, Asymptotic behavior of solutions of a linear second order difference equation,J. Comput. Appl. Math.41(1992), 95–103.MR1181711

[33] A. Zafer, Oscillatory and asymptotic behavior of higher order difference equations, Math. Comput. Modelling21(1995), No. 4, 43–50.MR1317929