Asymptotic properties of solutions to difference equations of Emden–Fowler type

Asymptotic properties of solutions to difference equations of Emden–Fowler type

Janusz Migda

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

Received 24 August 2019, appeared 22 October 2019 Communicated by Stevo Stevi´c

Abstract. We study the higher order difference equations of the following form

∆^mxn=anf(x_σ(n)) +bn.

We are interested in the asymptotic behavior of solutionsx of the above equation. As- suming f is a power type function and∆^myn =bn, we present sufficient conditions that guarantee the existence of a solutionxsuch that

xn =yn+o(n^s),

where s≤ 0 is fixed. We establish also conditions under which for a given solutionx there exists a sequenceysuch that∆^myn =bnandxhas the above asymptotic behavior.

Keywords: Emden–Fowler difference equation, asymptotic behavior.

2010 Mathematics Subject Classification: 39A10, 39A22.

1 Introduction

We use the standard symbol N to denote the set of all positive integers. Analogously R denotes the set of all real numbers. In this paper we assume that

m∈ _N, f :R→_R, σ:N→_N, lim

n→_∞σ(n) =_∞.

We consider difference equations of the form

∆^mx_n=a_nf(x_σ(n)) +b_n (E) where a_n,b_n ∈ _R. We say that a sequence x : N → _R is a solution of (E) if there exists an indexqsuch that (E) is satisfied for anyn≥q.

Asymptotic properties of solutions were investigated by many authors. Some classical results on asymptotic behavior of solutions of differential equations can be found in [8,9,12,

BEmail: migda@amu.edu.pl

14,22–25]. For the case of difference equations, see, e.g., [6,7,15,17,18,26–36]. With respect to dynamic equations on time-scales we refer the reader to [1–4].

The main purpose of this paper is to generalize the following theorem.

Theorem 1.1. Assumeσ:N→_N, y:N→_R,∆^my=b, s∈ (−_∞, 0],λ∈ (0,∞),

nlim→_∞σ(n) =_∞, lim

n→_∞yn= _∞,

∑

n^m−1−s|an|<_∞,

and f is continuous and bounded on [λ,∞). Then there exists a solution x of (E) such that x_n = y_n+_o(n^s).

This theorem follows from [18, Theorem 4.7] or [20, Theorem 4.1]. Some specific cases have been proved in [15, Theorem 5.1], [16, Theorem 5.1], or [19, Theorem 4.1]. The generalization consists in replacing the condition “f is bounded on [λ,∞)” by the condition “f is of power type on[λ,∞)”. The last condition means that there exists a constant µ∈ [0,∞)such that the functiont^−µf(t)is bounded on[λ,∞), i.e. f(t) =O(t^µ). Ifσ(n) =n,µ∈[0,∞), and f(t) =t^µ then equation (E) takes the form

∆^mx_n= a_nx^µ_n+b_n

which is the ”positive part” of discrete Emden–Fowler equation

∆^mx_n =a_n|x_n|^µsgnx_n+b_n.

Asymptotic properties of solutions to discrete Emden–Fowler type equations were investi- gated, for example, in [5,10,11,13,27,35].

The paper is organized as follows. In Section 2, we introduce notation and terminol- ogy. In Section 3, in Theorem3.1, we obtain our main result. In Theorem 3.1 the condition

∑^∞n=1n^m−1−s|a_n|< _∞from Theorem1.1is replaced by a stronger condition

∑

n^α+m−1−s|a_n|<_∞ (1.1) whereαis a non-negative constant dependent on the order of growth of f, the order of growth of y, and the order of growth of σ. In Section 4, in Theorem 4.2, we show that constant αis properly chosen, i.e. that condition (1.1) is not too strong. Section 5 is devoted to the problem of approximation of solutions. In Section 6 we give some remarks and additional results.

2 Preliminaries

We use the symbolR^N to denote the space of all sequences x : N→ _{R. If}x ∈ _R^N, then |x| denotes the sequence defined by|x|(n) =|xn|. Moreover

kxk=sup

n∈_N

|x_n|, c₀={z :N→_R: lim

n→_∞z_n=0}, A(m):=

(

x∈_R^N:

∑

n^m−1|x_n|<_∞ )

, r^m : A(m)→c₀, r^m(x)(n) =

∑

k−n+m−1 m−₁

x_k =

∑

m+k−1 m−₁

x_n+k.

It is not difficult to see that A(m)is a linear subspace of c₀andr^m is a linear operator. We will use the following two lemmas.

Lemma 2.1. Assume s∈ (−_{∞, 0}], p∈_{N, and∑}^∞_n=1n^m−1−s|u_n|<_{∞. Then} r^m(|u|)(p)≤

∑

k^m−1|u_k|, r^m(u)(n) =o(n^s), and ∆^m(r^m(u))(p) = (−1)^mup. Proof. This lemma follows from [17, Lemma 4.1 and Lemma 4.2].

Let us recall that a topological spaceXhas a fixed point property if every continuous map U: X→Xhas a fixed point.

Lemma 2.2 ([17, Lemma 4.7]). Let y be an arbitrary real sequence and letγ be a sequence which is positive and convergent to zero. Then the space {u∈ _R^N _: |u−y| ≤γ}with respect to the uniform convergence topology has a fixed point property.

3 Solutions with prescribed asymptotic behavior

In this section, in Theorem3.1, we present our main result. Next, in Corollary3.5we establish conditions under which there exist asymptotically polynomial solutions of equation (E).

Theorem 3.1. Assume y:N→_R,∆^my= b, lim

n→_∞y_n=_{∞, s} ∈(−_{∞, 0}],λ∈(0,∞), µ∈[0,∞), τ,ω,∈(0,∞), f(t) =O(t^µ), yn=O(n^τ), σ(n) =O(n^ω),

∑

n^{µτω+m−1−s}|a_n|<_∞, (3.1) and f is continuous on[λ,∞). Then there exists a solution x of (E)such that

x_n=y_n+o(n^s). Proof. There exist positive constantsL,Qsuch that

|yn| ≤Qn^τ, σ(n)≤ Ln^ω

forn∈N. We may assume that there exists a positive constantKsuch that

|f(t)| ≤Kt^µ fort ∈[λ,∞). Define a constantM and sequencesα, ρby

M= (2QL^τ)^µK, α_n =n^µτω|a_n|, ρ=r^m(α). (3.2) Then, by the conditions of the theorem and Lemma 2.1, we have α ∈ A(m) and ρ_n = o(1). There exists an index p₁∈ _Nsuch that

Mρ_n≤1, and y_n≥λ+1 forn≥ p₁. Let

S={x∈_R^N: |x−y| ≤ Mρ and xn=yn for n< p₁}.

There exists an index p2such that p2 ≥ p₁ andσ(n)≥ p₁forn≥ p2. If x∈S,n≥ p₁, then

|x_n−y_n| ≤Mρ_n≤_{1 and} y_n ≥λ+_1.

Hencex_n ≥y_n−1≥λ+1−1= λ. Therefore

x_σ(n) ≥λ (3.3)

for anyx ∈Sand anyn≥ p₂. Let x∈S. Ifn≥ p₂, then

|f(x_σ(n))| ≤Kx^µ

σ(n)= K(x_σ(n)−y_σ(n)+y_σ(n))^µ≤K(|x_σ(n)−y_σ(n)|+|y_σ(n)|)^µ

≤ K(Mρ_σ(_n)+|y_σ(_n)|)^µ≤K(1+|y_σ(_n)|)^µ ≤K(2|y_σ(_n)|)^µ

≤ K2^µ(Qσ(n)^τ)^µ≤K2^µQ^µ(Ln^ω)^τµ =K2^µQ^µL^τµn^µτω. Hence

|f(x_σ(n))| ≤ Mn^µτω (3.4)

for anyx ∈Sand anyn≥ p₂. Let x∈_R^N, forn∈_Nlet

x^∗_n=anf(x_σ(n)). (3.5)

Letx∈ S. By (3.4), we have

|x^∗_n| ≤ |a_n||f(x_σ(n))| ≤Mn^µτω|a_n|= Mα_n

forn≥ p₂. By (3.1) and (3.2), we getx^∗ ∈A(m). Hence we may define a sequenceU(x)by U(x)(n) =

(y_n forn< p₂

y_n+ (−1)^mr^m(x^∗)(n) forn≥ p₂. (3.6) Ifn≥ p2, then

|U(x)(n)−y_n|=|r^m(x^∗)(n)| ≤r^m(|x^∗|)(n)≤ Mr^m(α)(n) =Mρ_n.

ThereforeU(S)⊂ S. We will show that the mapUis continuous. Assumeε is a positive real number. By (3.1),

∑

n^m−1n^µτω|an|< _∞.

Select an indexq≥ p₂ and a positive constantγsuch that 2M

∑

n^µτω+m−1|an|<ε and γ

∑

n^m−1|an|<ε. (3.7) Let

x∈ S, η=1+max(x_σ(1), . . . ,x_σ(q)), W = [λ,η].

Since the function f is uniformly continuous onW, there exists a numberδ∈ (0, 1)such that

|f(r)− f(t)| ≤γ

for anyr,t∈W such that|r−t|< δ. Select a sequencez∈ Ssuch thatkz−xk<δ. Then kU(x)−U(z)k= sup

n≥p2

|r^m(x^∗)(n)−r^m(z^∗)(n)|= sup

n≥p2

|r^m(x^∗−z^∗)(n)|

≤ sup

n≥p₂

r^m(|x^∗−z^∗|)(n) =r^m(|x^∗−z^∗|)(p₂)≤

∑

n=p2

n^m−1|x^∗_n−z^∗_n|

≤

∑

n^m−1|an||f(x_σ(n))− f(z_σ(n))|+

∑

n^m−1|an||f(x_σ(n))− f(z_σ(n))|

≤

∑

n^m−1|an|γ+

∑

n^m−1|an|2Mn^µτω <2ε.

HenceU is continuous and, by Lemma2.2, there exists a sequence x ∈Ssuch thatU(x) =x.

Then, by (3.6),

x_n= y_n+ (−1)^mr^m(x^∗)(n)

forn≥ p₂. Hence, using Lemma2.1and (3.5), forn≥ p₂, we obtain

∆^mx_n= _∆^my_n+ (−1)^m_∆^m(r^m(x^∗))(n) =b_n+x^∗_n= a_nf(x_σ(n)) +b_n.

Moreover, since x ∈ S, we get|x_n−y_n| ≤ Mρ_neventually. By Lemma2.1, ρ_n = _o(n^s)_{. Hence} xn−yn=o(n^s)and we obtain

xn=yn+o(n^s).

If the sequence b is “sufficiently small”, then in Theorem 3.1, in place of a solution y of the equation ∆^myn = bn we can take a polynomial sequence. More precisely, we have the following result.

Corollary 3.2. Assumeϕnis a polynomial sequence, lim

n→_∞ϕn=_{∞, s}∈(−_{∞, 0}],

µ∈[_0,∞)_, ω∈ (_0,∞)_, τ∈(_0,m)_, f(t) =_O(t^µ)_, ϕ_n=_O(n^τ)_, σ(n) =_O(n^ω)_,

∑

n^{µτω+m−1−s}|an|< _∞,

∑

n^m−1−s|bn|<_∞, (3.8) λ∈ (_0,∞), and f is continuous on[_λ,∞). Then there exists a solution x of (E)such that

xn = ϕn+o(n^s).

Proof. By [15, Lemma 2.3], there exists a sequencewsuch thatw_n= _o(n^s)_and∆^mw =b. Let y= ϕ+w. Then ∆^my= _∆^mϕ+_∆^mw =band limn→_∞yn =∞. By Theorem3.1 there exists a solution xof (E) such thatx_n =y_n+o(n^s). Hence

x_n= ϕ_n+w_n+o(n^s) =ϕ_n+o(n^s).

In the case of the classical Emden–Fowler discrete equation we obtain the following corol- lary.

Corollary 3.3. Assumeµ∈[0,∞), s∈ (−_{∞, 0}],

∑

n^{(µ+1)(m−1)−s}|a_n|<_∞, and

∑

n^m−1−s|b_n|<_∞. (3.9) Then for any polynomial sequence ϕ_n such thatdegϕ< m, and lim

n→_∞ϕ_n = _∞, there exists a solution x of the equation

∆^mx_n=a_nx^µ_n+b_n (3.10)

such that xn= ϕn+o(n^s).

Proof. Let τ = m−1 and ω = 1. Applying Corollary 3.2 to equation (3.10) we get the result.

In particular, in the linear case, we obtain the following corollary.

Corollary 3.4. Assume s∈(−_{∞, 0}],

∑

n^2m−2−s|a_n|<_∞, and

∑

n^m−1−s|b_n|<_∞.

Then for any polynomial sequence ϕn, such thatdegϕ<m, and lim

n→_∞ϕn= ∞, there exists a solution x of the equation∆^mxn= anxn+bnsuch that xn = ϕn+o(n^s).

The proof of the next theorem is similar to the proof of Theorem 3.1, therefore it will be omitted.

Theorem 3.5. Assume y:N→_R,∆^my=b, lim

n→_∞y_n= −_∞, s ∈(−_∞, 0],λ∈(−_∞, 0), µ∈[0,∞), τ,ω,∈(0,∞), yn=O(n^τ), σ(n) =O(n^ω),

∑

n^{µτω+m−1−s}|an|<_∞, the function f is continuous on (−_∞,λ], and the function |t|^−µf(t) is bounded on (−_∞,λ]. Then there exists a solution x of (E)such that xn=yn+o(n^s).

Of course, Corollary3.2, Corollary 3.3, and Corollary3.4can be reformulated and proven in a similar way. We leave it to the reader.

4 Necessary conditions

In this section we show that the condition of strong convergence of the sequenceanin Theorem 3.1is not too strong. More precisely, we present conditions under which the condition (3.8) in Corollary3.2is necessary for the existence of asymptotically polynomial solution of equation (E). We present the main results in Theorem4.2(in the cases=0) and in Theorem4.6(in the cases<0).

Lemma 4.1. If zn =o(1)and the sequence∆^mznis nonoscillatory, then

∑

n^m−1|_∆^mz_n|< _∞.

Proof. This statement is a consequence of [17, Lemma 4.1 (f) and (b)].

Theorem 4.2. Assumeω,K,L∈(0,∞),µ∈[0,∞),

σ(n)≥ Ln^ω, an≥0, bn≥0 for large n, f(t)≥Kt^µ for large t, ϕis a polynomial sequence,degϕ= τ < m, lim

n→_∞ϕ_n = ∞, and a solution x of (E)exists such that x_n= ϕ_n+o(1). Then

∑

n^µτω+m−1|a_n|<_∞ and

∑

n^m−1|b_n|< _∞.

Proof. There exists a positive numberεsuch thatx_n ≥εn^τ eventually. Select p₁∈ _Nsuch that a_n ≥_0, b_n≥_0, σ(n)≥ Ln^ω, x_n ≥εn^τ

forn≥ p₁. Select p₂≥ p₁such thatσ(n)≥ p₁ forn≥ p₂. Forn∈_Nlet u_n= a_nf(x_σ(n)) +b_n.

Then

u_n=_∆^mx_n= _∆^m(ϕ_n+_o(₁)) =_∆^m(_o(₁)) andu_n is nonoscillatory. Hence by Lemma4.1, we get

∑

n^m−1|un|< _∞. (4.1)

Sinceanf(x_σ(n))≥0 for largen, we havebn≤ unfor largen. By (4.1) we obtain

∑

n^m−1|bn|<_∞.

Ifn≥ p₂, then

f(x_σ(n))≥Kx_σ^µ_(n)≥K(εσ(n)^τ)^µ ≥K(ε(Ln^ω)^τ)^µ=Kε^µL^τµn^ωτµ. Hence there exists a positive constant δsuch that

f(x_σ(n))≥δn^µτω forn≥ p2. Ifn≥ p2, then we get

n^µτω|an| ≤δ⁻¹|an|f(x_σ(n))≤δ⁻¹|un|. Now, by (4.1), we obtain

∑

n^µτω+m−1|an|<_∞.

Using Theorem4.2to the case of classical discrete Emden–Fowler equation we get:

Corollary 4.3. Assumeµ∈ [_0,∞), a_n ≥0, b_n ≥0eventually, ϕis a polynomial sequence of degree m−1,limn→_∞ϕn=∞, and there exists a solution x of the equation

∆^mxn=anx^µ_n+bn

such that xn= ϕn+o(1). Then

∑

n^{(µ+1)(m−1)}|a_n|<_∞ and

∑

n^m−1|b_n|<_∞.

The cases<0 is, unexpectedly, more difficult. In this case we get a slightly weaker result.

First, we will prove two lemmas.

Lemma 4.4. Assume s∈ (−_{∞, 0}]and zn=o(n^s), then for everyε>0the series

∑

∆z_n n^s+ε is convergent.

Proof. Let ε be a positive real number and let t = −s−ε. By [15, Theorem 2.2], we have

∆n^t=O(n^t−1). Hence

n^t∆zn =n^tz_n+1−n^tzn=n^tz_n+1−(n+1)^tz_n+1+ (n+1)^tz_n+1−n^tzn

=−z_n+1∆n^t+_∆(n^tz_n) =_∆(n^tz_n) +z_n+1O(n^t−1) =_∆(n^tz_n) +z_n+1n^t−1O(1). Moreover

z_n+1n^t−1= z_n+1n^−s−εn⁻¹ = ^zn+1 (n+1)^s

n+1 n

s

1

n^1+ε =o(1)O(1) ¹ n^1+ε =O

1 n^1+ε

. Therefore the series

∑

z_n+1n^t−1O(1) =

∑

O 1

n^1+ε

is convergent. Sincen^tz_n =o(1), the series

∑

∆(n^tzn)

is convergent, too. Thus we obtain the convergence of the series

∑

∆zn

n^s+ε =

∑

n^t∆zn =

∑

∆(n^tz_n) +

∑

z_n+1n^t−1O(1).

Lemma 4.5. Assume m ∈_N, s ∈(−_{∞, 0}), z_n =_o(n^s), and the sequence∆^mz_nis nonoscillatory for large n. Then for everyε>0the series

∑

∆^mz_n n^s+ε−m+1 is convergent.

Proof. Induction on m. By Lemma 4.4, the assertion is true for m = 1. Assume it is true for some m ≥ 1 and the sequence ∆^m+1zn is nonoscillatory for large n. Moreover, assume that

∆^m+1z_n ≥0 for largen. Let

yn =_∆^mzn.

Theny_n = o(1)and ∆yn ≥ 0 for largen. Hence ∆^mz_n = y_n ≤ 0 for largen. Obviously, we may assume, that

∆^mzn=yn<0 (4.2)

eventually. Select anε>0 and let

λ∈(0,ε)∩(0,−s). (4.3)

By inductive hypothesis the series

∑

y_n n^s+λ−m+1 is convergent. Let

t =m−₁−s−_λ.

By (4.3),t>0. Using de l’Hospital Theorem we obtain the following known limit

nlim→_∞

∆n^t+1

n^t = lim

n→_∞

(n+1)^t+1−n^t+1

n⁻¹n^t+1 = lim

n→_∞

(1+n⁻¹)^t+1−1 n⁻¹

= lim

n→_∞

(t+1)(1+n⁻¹)^t(−n⁻²)

−n⁻² = t+1.

Hence, by the Cesàro–Stolz theorem, we obtain

nlim→_∞

1^t+2^t+· · ·+n^t

n^t+1 = lim

n→_∞

(n+1)^t

∆n^t+1 = lim

n→_∞

n+1 n

t

n^t

∆n^t+1 = ¹

t+1. (4.4) Letu₀=0,c₁ =0,S=_∑^∞_k=1k^ty_k. Forn≥1 let

u_n =

∑

k^ty_k, b_n=y⁻_n¹, c_n+1=

∑

(b_k+1−b_k)u_k. Then

nlim→_∞

c_n+1−cn

b_n+1−b_n = lim

n→_∞

(b_n+1−bn)un

b_n+1−b_n = lim

n→_∞u_n=S. (4.5)

Note that lim_n→_∞y_n = 0. Moreover, by assumption ∆yn ≥ 0 eventually. By (4.2), y_n < 0 eventually. Hence the sequenceb_nis eventually monotonic and lim_n→_∞b_n =−_∞. Using (4.5) and Cesàro–Stolz theorem we get

nlim→_∞y_nc_n = lim

n→_∞

c_n bn

= S.

Sinceb_k(u_k−u_k−1) =k^t, we have y_n

∑

k^t= y_n

∑

b_k(u_k−u_k−1)

= y_n(b₁(u₁−u₀) +b₂(u₂−u₁) +· · ·+b_n(u_n−u_n−1))

= y_n(−b₁u₀−(b₂−b₁)u₁−(b₃−b₁)u₂+· · · −(b_n−b_n−1)u_n−1+b_nu_n)

= y_nb_nu_n−y_nc_n=u_n−y_nc_n. Hence

nlim→_∞yn

∑

k^t= S−S=0.

Therefore, by (4.4) we get

nlim→_∞n^t+1y_n= _lim

n→_∞

n^t+1

∑ⁿk=₁k^t

∑

k^t

!

y_n= (t+₁)₀=_0.

Thus y_n=o(n^−t−1) =o(n^s+λ−m)and, by Lemma4.4, the series

∑

∆^m+1z_n n^s+ε−m =

∑

∆yn

n^{s+λ−m+(ε−λ)} is convergent.

Using Lemma4.5in place of Lemma4.1in the proof of Theorem4.2we obtain the follow- ing result.

Theorem 4.6. Assumeω,K,L∈(0,∞),µ∈[0,∞), s ∈(−_{∞, 0}]

σ(n)≥ Ln^ω, a_n≥0, b_n≥0 for large n, f(t)≥Kt^µ for large t, ϕis a polynomial sequence,degϕ = τ< m, lim

n→_∞ϕn = ∞, and there exists a solution x of (E) such that xn= ϕ_n+o(n^s). Then for anyε>0we have

∑

n^{µτω+m−1−s−ε}|a_n|<_∞ and

∑

n^{m−1−s−ε}|b_n|<_∞. (4.6) Remark 4.7. The problem of whether condition (4.6) in Theorem4.6 can be replaced by

∑

n^{µτω+m−1−s}|a_n|<_∞ and

∑

n^m−1−s|b_n|<_∞.

remains open.

5 Approximation of solutions

In this section we establish conditions under which a given solutionx of (E) can be approxi- mated by solutions of the equation ∆^myn = bn. More precisely, we present conditions under which for a given solution x of (E) and a given nonpositive real number s there exists a se- quenceysuch that∆^my_n =b_n andx_n =y_n+_o(n^s)_.

Lemma 5.1. Assume b,x,u:N→_{R, s}∈(−_{∞, 0}],

∆^mx_n =O(u_n) +b_n, and

∑

n^m−1−s|u_n|<_∞.

Then there exists a sequence y such that∆^my_n =b_nand x_n=y_n+o(n^s). Proof. This statement follows from [17, Lemma 3.11 (a)].

Theorem 5.2. Assume s∈ (−_{∞, 0}],α∈ [_0,∞), and

∑

n^α+m−1−s|an|< _∞.

Then for any solution x of (E)such that f(x_σ(n)) =O(n^α)there exists a sequence y such that∆^my_n= bn for any n and xn=yn+o(n^s).

Proof. Assumexis a solution of (E) such thatf(x_σ(n)) =O(n^α). There exists a positive constant Msuch that the condition|f(x_σ(_n))| ≤ Mn^α is satisfied for anyn. Define a sequenceu by the formula

u_n= a_nf(x_σ(n)).

Then ∞

n

∑

n^m−1−s|u_n| ≤M

∑

n^α+m−1−s|a_n|< _∞.

Since x is a solution of (E), we have ∆^mx_n = u_n+b_n eventually. Hence, ∆^mx_n = O(u_n) +b_n and, by Lemma5.1, there exists a sequencey such that∆^my_n=b_nandx_n=y_n+_o(n^s)_.

Corollary 5.3. Assume s∈ (−_{∞, 0}],µ∈[0,∞),τ,ω,∈ (0,∞), f(t) =O(t^µ), σ(n) =O(n^ω), and

∑

n^{µτω+m−1−s}|a_n|<_∞.

Then for any solution x of (E)such thatlim_n→_∞x_n =_∞and x_n =O(n^τ)there exists a solution y of the equation∆^my_n=b_nsuch that x_n=y_n+_o(n^s).

Proof. It is easy to see that f(x_σ(n)) =O(n^µτω). Takingα=µτωin Theorem5.2we obtain the result.

Theorem 5.4. Assume s∈(−_{∞, 0}],α∈[0,∞),

∑

n^α+m−1−s|a_n|<_∞, and

∑

n^m−1−s|b_n|<_∞.

Then for any solution x of (E)such that f(x_σ(n)) =O(n^α)there exists a polynomial sequence ϕsuch thatdegϕ< m and xn = ϕn+o(n^s).

Proof. Assume x is a solution of (E) such that f(x_σ(n)) = O(n^α). Define a sequence u by un= anf(x_σ(n)) +bn. Then∆^mxn=O(un)and

∑

n^m−1−s|un|<_∞.

By Lemma5.1, there exists a solution ϕof the equation∆^mϕn=0 such that x_n = ϕ_n+o(n^s).

6 Remarks and additional results

The convergence of a series can be difficult to verify. In the classical mathematical analysis, various criteria are known to check the convergence of a given series. Some of these criteria have been generalized in [18]. These generalized criteria can be used to check the convergence of series (3.1) or (3.8). In this way we can get a number of new results. Four of them are presented below.

Corollary 6.1. Assume y:N→_R,∆^my =b, lim

n→_∞yn=_{∞, s}∈(−_{∞, 0}],λ∈(0,∞), µ∈[0,∞), τ,ω,∈(0,∞), f(t) =O(t^µ), y_n=O(n^τ), σ(n) =O(n^ω),

lim inf

n→_∞ n

|a_n|

|a_n+1|−1

>µτω+m−s, (6.1)

and f is continuous on[λ,∞). Then there exists a solution x of (E)such that x_n=y_n+o(n^s).

Proof. By [18, Lemma 6.3] we get

∑

n^{µτω+m−1−s}|a_n|<_∞. Hence the result follows from Theorem3.1.

Using [18, Lemma 6.3] and Corollary3.2 we obtain the following result.

Corollary 6.2. Assumeϕ_nis a polynomial sequence, lim

n→_∞ϕ_n= _{∞, s}∈(−_{∞, 0}],

µ∈[0,∞), ω ∈(0,∞), τ∈ (0,m), f(t) =O(t^µ), ϕ_n=O(n^τ), σ(n) =O(n^ω), lim inf

n→_∞ n

|an|

|a_n+1|−1

>µτω+m−s, lim inf

n→_∞ n

|bn|

|b_n+1|−1

>m−s,

λ∈(0,∞), and f is continuous on[λ,∞). Then there exists a solution x of (E)such that x_n= ϕ_n+o(n^s).

Corollary 6.3. Assume s∈(−_{∞, 0}],α∈ [_0,∞), and lim inf

n→_∞ n

|a_n|

|a_n+1| −1

>α+m−s.

Then for any solution x of (E) such that f(x_σ(n)) = O(n^α)there exists a solution y of the equation

∆^my_n= b_nsuch that x_n =y_n+o(n^s).

Proof. This corollary follows from [18, Lemma 6.3] and Theorem5.2.

Using [18, Lemma 6.4] and Corollary3.3 we obtain the following corollary.

Corollary 6.4. Assumeµ∈[0,∞), s ∈(−_{∞, 0}], lim inf

n→_∞ nln |a_n|

|a_n+1| >(µ+1)(m−1) +1−s, and

lim inf

n→_∞ nln |b_n|

|b_n+₁| >m−s.

Then for any polynomial sequence ϕ_n, such thatdegϕ<m, and lim

n→_∞ϕ_n= ∞, there exists a solution x of the equation∆^mx_n= a_nx^µ_n+b_nsuch that x_n= ϕ_n+o(n^s).

We can also receive some new consequences of Theorem5.2.

Corollary 6.5. Assume s∈(−_{∞, 0}],α∈ [0,∞), and lim sup

n→_∞

ln|a_n|

lnn <_s−m−α.

Then for any solution x of (E)such that f(x_σ(n)) =O(n^α)there exists a sequence y such that∆^my_n= b_n for any n and x_n=y_n+o(n^s).

Proof. Using [18, Lemma 6.2] and Theorem5.2we obtain the result.

Corollary 6.6. Assume s∈(−_{∞, 0}],α∈ [_0,∞),µ∈ (_0,∞),β=α/µ, and lim sup

n→_∞

ln|a_n|

lnn <s−m−α.

Then for any positive solution x of the discrete Emden–Fowler equation

∆^mx_n= a_nx^µ_n+b_n

such that x_n =_O(n^β)there exists a sequence y such that ∆^my_n=b_nfor any n and x_n =y_n+_o(n^s).

Proof. Ifx_n =O(n^β), thenx^µ_n =O(n^α). Hence the result follows from Corollary6.5.

Corollary 6.7. Assume s∈ (−_{∞, 0}],µ∈[0,∞),τ,ω,∈ (0,∞), f(t) =O(t^µ), σ(n) =O(n^ω), and lim sup

n→_∞

ln|a_n|

lnn <s−m−µτω.

Then for any solution x of (E)such thatlimn→_∞xn =_∞and xn =O(n^τ)there exists a solution y of the equation∆^my_n=b_nsuch that x_n=y_n+o(n^s).

Proof. This corollary follows from [18, Lemma 6.2] and Corollary5.3.

Remark 6.8. In Corollary 3.2, due to assumptions τ ∈ (0,m), lim

n→_∞ϕn = _∞and ϕn = O(n^τ), the degreemof equation (E) fulfills the conditionm> 1. On the other hand, in Theorem3.1, the case ofm=1 is not excluded.

Example 6.9. Assumem=1, f(t) =t²,y_n= n,b_n =1, σ(n) =n,µ=2, τ=1, ω= 1,s =0, and

a_n= ⁿ

(n+1)(n⁴−2n+1)^.

Then ∆yn =bn andµτω+m−1−s =2. Hence, by Theorem3.1 there exists a solutionx of the equation

∆xn= ^nx

2n

(n+1)(n⁴−2n+1)+1

such that x_n =n+o(1). We leave the reader to check that the sequencex_n =n+1/nis such a solution.

Our results in Sections 3 and 4 relate to unbounded solutions. Below we present three facts about bounded solutions.

Theorem 6.10. Assume y is a bounded solution of the equation∆^myn =bn, s∈ (−_{∞, 0}], q∈_N, α∈ (0,∞), U=

[∞ n=q

[y_n−α,y_n+α],

∑

n^m−1−s|a_n|<_∞, and f is continuous on U. Then there exists a solution x of (E)such that x_n=y_n+o(n^s). Proof. The assertion is a consequence of [18, Corollary 4.8].

Corollary 6.11. Assume c∈_{R, s}∈ (−_{∞, 0}], U is a neighborhood of c,

∑

n^m−1−s(|a_n|+|b_n|)<_∞,

and f is continuous on U. Then there exists a solution x of (E)such that x_n=c+o(n^s). Proof. We omit the proof which is analogous to the proof of Corollary3.2.

The next example shows that in the above corollary the continuity of f on U can not be replaced by the continuity at the pointc.

Example 6.12. LetQbe the set of all rational numbers. Assume

m=1, s =0, σ(n) =n, b_n =0, α∈(0, 1), a_n =α²

n, c=1, and f is defined by

f(t) =

(1 fort∈_Q t fort∈/_Q.

Then f is continuous at the point c,

∑

n^m−1−s(|a_n|+|b_n|) =

∑

n=1

α²

n <_∞,

but, by [21, Example 1], there is no solution to equation (E) convergent toc.

Theorem 6.13. Assume a_n ≥0and b_n ≥0eventually, c ∈R, U is a neighborhood of c,δ ∈ (0,∞), f(t)≥δfor any t∈U, and there exists a solution x of (E)such that x_n=c+o(1). Then

∑

n^m−1(|an|+|bn|)< _∞.

Proof. Forn ∈_Nlet

u_n =a_nf(x_σ(n)) +b_n.

Thenu_n=_∆^mx_n= _∆^m(c+_o(₁)) =_∆^m(_o(₁))_andu_nis nonoscillatory. By Lemma4.1

∑

n^m−1|u_n|< _∞.

Since 0≤ b_n ≤ u_n for largen, we get ∑^∞n=1n^m−1|b_n| < _∞. Since f(x_σ(n)) > δ eventually, we get|an| ≤δ⁻¹|an|f(x_σ(n))≤δ⁻¹|un|. Hence∑^∞n=1n^m−1|an|<_∞.

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