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Symmetry 2018, 10(6), 207; https://doi.org/10.3390/sym10060207

Article
Nonoscillatory Solutions to Second-Order Neutral Difference Equations
1
Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland
2
Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
*
Author to whom correspondence should be addressed.
Received: 30 April 2018 / Accepted: 5 June 2018 / Published: 8 June 2018

## Abstract

:
We study asymptotic behavior of nonoscillatory solutions to second-order neutral difference equation of the form: $Δ ( r n Δ ( x n + p n x n − τ ) ) = a n f ( n , x n ) + b n .$ The obtained results are based on the discrete Bihari type lemma and a Stolz type lemma.
Keywords:
second-order difference equation; asymptotic behavior; nonoscillatory solution; quasi-difference

## 1. Introduction

We are concerned with the following nonlinear second-order difference equations
$Δ ( r n Δ ( x n + p n x n − τ ) ) = a n f ( n , x n ) + b n ,$
where
$τ ∈ N , r n , a n , b n , p n ∈ R , f : N × R → R , r n > 0 , p n → λ ∈ R .$
Here $N$, $R$ denote the set of nonnegative integers and all real numbers, respectively. By a solution of Equation (1), we mean a sequence x which satisfies Equation (1) for all large n. A solution x is said to be nonoscillatory if it is eventually positive or eventually negative; otherwise, it is called oscillatory.
In the sequel, we will use the following notation:
$r n ∗ = ∑ i = 1 n − 1 1 r n ,$
by convention $r 1 ∗ = 0$.
The second-order difference equations have been a subject of numerous studies. In particular, investigation of neutral difference equations is important since such equations have applications in various problems of physics, biology, and economics. Recently, there have been many papers devoted to the oscillation of solutions to equations of the type defined by Equation (1) (see, for example, [1,2,3,4,5,6,7,8] and the references cited therein). In comparison with oscillation, there are not as many results on the nonoscillation of these equations.
The asymptotic behavior of solutions of Equation (1) in the case $p n ≡ 0$ has been studied for several decades by many authors ([5,9,10,11,12,13,14]), while some generalizations on time-scale variants of the equation have been studied in [15,16,17]. However, there are relatively few works devoted to the study of the asymptotic behavior of nonoscillatory solutions expressed by Equation (1) when $p n ≢ 0$. In 2003, using the Leray–Schauder theorem, Agarwal et al. [18] obtained sufficient conditions for the existence of nonoscillatory solutions for the discrete equation
$Δ ( r n Δ x n + p x n − k ) + F ( n + 1 , x n + 1 − σ ) = 0 .$
Liu et al. in [19], proved the existence of uncountably many bounded nonoscillatory solutions to the problem
$Δ ( r n Δ x n + p x n − k ) + f ( n , x n − d 1 n , ⋯ , x n − d k n ) = c n ,$
using Banach’s fixed point theorem, under the Lipschitz continuity condition. Galewski et al. [20] studied the existence of a bounded solution to the more general equation
$Δ ( r n Δ x n + p n x n − k ) γ + q n x n α + a n f ( n , x n + 1 ) = 0 ,$
using the techniques connected with the measure of noncompactness. Some sufficient conditions for the existence of a nonoscillatory solution to the equation
$Δ ( r n Δ ( x n + p x n − τ ) ) + a n f ( x n − k ) − b n x n − l = 0 ,$
for $p ≠ − 1$ were obtained by Tian et al. in [21]. Moreover, for classification of nonoscillatory solutions to equations of the type defined by Equation (1), see [22,23,24,25,26].
In [27], the following equation was considered:
$Δ 2 x n + p x n − τ = a n f ( n , x n ) + b n .$
The results obtained in [27] were extended to higher-order equations in [28]. In this paper, we present generalizations in a different direction, namely to difference equations with quasi-difference of the type defined by Equation (1). In Theorem 1, using the discrete Bihari type lemma and discrete L’Hospital’s type lemma, we obtain sufficient conditions, under which all nonoscillatory solutions of Equation (1) have the property
$x n = c r n ∗ + o ( r n ∗ ) .$
Moreover, in Theorem 2, we show that, under some additional conditions, all nonoscillatory solutions of Equation (1) have the property
$x n = c r n ∗ + d + o ( 1 ) .$
The results are new even for linear equations of the type defined by Equation (1) and when $p n = 0$. We also present applications of the obtained results to some special cases of Equation (1).

## 2. Main Results

For the proof of the main results, we will need some auxiliary lemmas.
Lemma 1.
Assume $x , p , z$ are real sequences, x is bounded, $τ ∈ N$,
$z n = x n + p n x n − τ ,$
for $n ≥ τ$, $p n → λ ∈ R$, $| λ | ≠ 1$, and $z n → α ∈ R$. Then x is convergent and
$lim n → ∞ x n = α 1 + λ .$
Proof.
The assertion is a consequence of ([27], Lemma 1). ☐
Remark 1.
Lemma 1 was essentially proved in Lemma 1 in [29], where the case of complex sequences was studied in detail for the case of constant sequence $p n$. For the case of sequences in Banach spaces, see Lemma 1 in [30].
The following lemma is a discrete version of Bihari type lemma.
Lemma 2.
Assume $a , w$ are real sequences, $n 0 ∈ N$, $g : [ 0 , ∞ ) → [ 0 , ∞ )$, $λ ∈ [ 0 , ∞ )$,
$∑ n = 1 ∞ | a k | < ∞ , g ( λ ) > 0 , ∫ λ ∞ d s g ( s ) = ∞ , | w n | ≤ λ + ∑ k = n 0 n − 1 | a k | g ( | w k | ) ,$
for $n ≥ n 0$, and g is nondecreasing. Then the sequence w is bounded.
Proof.
The assertion is a consequence of ([28], Lemma 4.1). ☐
In the proof of Theorem 1, we will use the following Stolz-type lemma, which should be a folklore one, but it is difficult to find a specific reference in the literature. Because of this, for the completeness and benefit of the reader, we will provide a proof of the lemma.
Lemma 3.
Assume $x , y$ are real sequences, y is bounded and eventually strictly monotonic, and the sequence $( Δ x n / Δ y n )$ is convergent. Then the sequence x is convergent. Moreover, if $lim n → ∞ y n ≠ 0$, then the sequence $( x n / y n )$ is convergent.
Proof.
First assume that the sequence y is eventually increasing. Let $ε > 0$ and
$L = lim n → ∞ Δ x n Δ y n .$
Choose an index k such that
$L − ε ≤ Δ x n Δ y n ≤ L + ε and Δ y n > 0 ,$
for $n ≥ k$. Then
$( L − ε ) Δ y n ≤ Δ x n ≤ ( L + ε ) Δ y n ,$
for $n ≥ k$. Summing from k to $n − 1$, we obtain
$( L − ε ) ( y n − y k ) ≤ x n − x k ≤ ( L + ε ) ( y n − y k ) .$
Since y is bounded, there exists a positive constant S such that $| y n − y m | ≤ S$ for any $n , m$. Therefore, we have
$L y n − ε S − L y k + x k ≤ x n ≤ L y n + ε S − L y k + x k ,$
for any $n ≥ k$. Let
$U = L lim n → ∞ y n .$
Choose an index $q ≥ k$ such that $U − ε ≤ L y n ≤ U + ε$ for $n ≥ q$. Let $T = x k − L y k$. Then, using Equations (3) and (4), we have
$U − ε − ε S + T ≤ x n ≤ U + ε + ε S + T ,$
for any $n ≥ q$. Hence, $| x n − x m | ≤ 2 ε ( S + 1 )$ for any $n , m ≥ q$. Therefore, the sequence x is convergent. If y is eventually decreasing, then the proof of convergence of x is analogous. The last part of the lemma is now obvious. ☐
Remark 2.
The following simple example shows that, in Lemma 3, the limit of $x n / y n$ can be different than the limit of $Δ x n / Δ y n$. Let
$x n = 2 − 1 n , y n = 1 − 1 n ,$
then the sequence y is bounded, increasing and
$lim n → ∞ x n y n = 2 ≠ lim n → ∞ Δ x n Δ y n = 1 .$
The next lemma will be used in the proof of Corollary 1. This lemma is probably known, but for the convenience of the reader, we give a proof.
Lemma 4.
Assume $σ ∈ ( 0 , ∞ )$ and $r n = n 1 − σ$. Then
$r n ∗ = σ − 1 n σ + o ( n σ ) .$
Proof.
By Theorem 2.2 in [31], we have
$Δ n σ = σ n σ − 1 + o ( n σ − 1 ) .$
Since $Δ r n ∗ = r n − 1 = n σ − 1$, we have
$Δ r n ∗ Δ n σ = n σ − 1 σ n σ − 1 + o ( n σ − 1 ) = 1 σ + o ( 1 ) → 1 σ .$
By the Stolz–Cesaro theorem,
$r n ∗ n σ → 1 σ ⇒ r n ∗ n σ = σ − 1 + o ( 1 ) ,$
and we obtain Equation (5). ☐
Theorem 1.
Assume $g : [ 0 , ∞ ) → [ 0 , ∞ )$ is nondecreasing, $α ∈ ( 0 , ∞ )$, $g ( α ) > 0$,
$∑ n = 1 ∞ | a n | < ∞ , ∑ n = 1 ∞ | b n | < ∞ ,$
$p n ≥ 0 , p n → λ ∈ R , r n ∗ r n + 1 ∗ → μ ∈ R , λ ≠ 1 ≠ λ μ τ ,$
$∫ α ∞ d s g ( s ) = ∞ , | f ( n , u ) | ≤ g | u | r n ∗ for ( n , u ) ∈ N × R .$
Then every nonoscillatory solution x of Equation (1) has the property
$x n = c r n ∗ + o ( r n ∗ ) ,$
where c is a real constant.
Proof.
Let x be a nonoscillatory solution of Equation (1). Then there is an index $n 0$, such that $x n > 0$ for any $n ≥ n 0$ or $x n < 0$ for any $n ≥ n 0$. Set
$z n = x n + p n x n − τ .$
Then
$| x n | < | z n | ,$
for $n ≥ n 1 = n 0 + τ$, and Equation (1) takes the form
$Δ ( r n Δ z n ) = a n f ( n , x n ) + b n .$
Let us denote $z n 1 = c 0$ and $r n 1 Δ z n 1 = c 1$. Summing the above equation from $n 1$ to $n − 1$, we obtain
$r n Δ z n = c 1 + ∑ j = n 1 n − 1 a j f ( j , x j ) + ∑ j = n 1 n − 1 b j .$
Dividing both sides of Equation (11) by $r n$ and summing again, we have
$z n = c 0 + c 1 ∑ i = n 1 n − 1 1 r i + ∑ i = n 1 n − 1 1 r i ∑ j = n 1 i − 1 a j f ( j , x j ) + ∑ i = n 1 n − 1 1 r i ∑ j = n 1 i − 1 b j .$
Hence, using Equation (2), we have
$| z n | ≤ | c 0 | + | c 1 | r n ∗ + ∑ i = n 1 n − 1 1 r i ∑ j = n 1 i − 1 | a j | | f ( j , x j ) | + ∑ i = n 1 n − 1 1 r i ∑ j = n 1 i − 1 | b j | .$
Changing the order of summation, we obtain
$| z n | ≤ | c 0 | + | c 1 | r n ∗ + ∑ i = n 1 n − 1 | a i | | f ( i , x i ) | ∑ j = i n − 1 1 r j + ∑ i = n 1 n − 1 | b j | ∑ j = i n − 1 1 r i ≤ | c 0 | + | c 1 | r n ∗ + r n ∗ ∑ i = n 1 n − 1 | a i | | f ( i , x i ) | + r n ∗ ∑ i = n 1 n − 1 | b j | .$
Hence, by Equation (8),
$| z n | r n ∗ ≤ | c 0 | r n 0 ∗ + | c 1 | + ∑ i = n 1 n − 1 | a i | | f ( i , x i ) | + ∑ i = n 1 n − 1 | b j | ≤ d 1 + ∑ i = n 1 n − 1 | a i | g | x i | r i ∗ ≤ d 1 + ∑ i = n 1 n − 1 | a i | g | z i | r i ∗ ,$
where $d 1$ is an appropriate constant. Therefore, by Lemma 2, there exists a constant K such that
$| z n | r n ∗ ≤ K ,$
for any $n ≥ n 1 + τ$. On the other hand, we have
$∑ i = n 1 n − 1 | a i | | f ( i , x i ) | ≤ ∑ i = n 1 n − 1 | a i | g | x i | r i ∗ ≤ ∑ i = n 1 n − 1 | a i | g | z i | r i ∗ ≤ g ( K ) ∑ i = n 1 n − 1 | a i | .$
Therefore, the series $∑ i = n 1 ∞ a i f ( i , x i )$ is absolutely convergent. Thus, by Equations (11) and (6), we see that the sequence $( r n Δ z n )$ is convergent. Note that $Δ r n ∗ = r n − 1$. Hence,
$Δ z n Δ r n ∗ = r n Δ z n .$
If the sequence $( r n ∗ )$ is unbounded, then by the Stolz–Cesaro Theorem we have
$lim n → ∞ z n r n ∗ = lim n → ∞ Δ z n Δ r n ∗ = lim n → ∞ r n Δ z n .$
If the sequence $( r n ∗ )$ is bounded, then by Lemma 3 the sequence $( z n / r n ∗ )$ is convergent. Now,
$w n = z n r n ∗ , y n = x n r n ∗ , u n = p n r n − τ ∗ r n ∗ .$
Then, Equation (9) implies
$w n = y n + u n y n − τ .$
Using Equations (10) and (12), we have
$| y n | = | x n | r n ∗ ≤ | z n | r n ∗ ≤ K .$
It is easy to see that the assumption
$r n ∗ r n + 1 ∗ → μ$
implies
$r n − τ ∗ r n ∗ → μ τ .$
Hence, by Equation (13), $u n → λ μ τ ≠ 1$. By Lemma 1, we have
$lim n → ∞ x n r n ∗ = lim n → ∞ y n = lim n → ∞ w n 1 + λ μ τ = c .$
Therefore,
$x n r n ∗ = c + o ( 1 ) ⇒ x n = c r n ∗ + o ( r n ∗ ) .$
☐
Theorem 1 extends Theorem 1 in [27].
Note that checking the assumption $r n ∗ r n + 1 ∗ → μ ∈ R$ of Theorem 1 may be difficult, so the following result can be useful.
Lemma 5.
Assume at least one of the following conditions holds
$( a ) r n ∗ = ( 1 ) , ( b ) r n − 1 = ( 1 ) , ( c ) r n + 1 r n → 1 .$
Then
$r n ∗ r n + 1 ∗ → 1 .$
Proof.
(a) Assume $r n ∗ = ( 1 )$. Since the sequence $( r n ∗ )$ is positive and increasing, there exists a limit $lim n → ∞ r n ∗ = ω ∈ ( 0 , ∞ )$. Then $lim n → ∞ r n + 1 ∗ = ω$ and we have Equation (14).
Now, assume that the sequence $r ∗$ is unbounded. Then $r n ∗ → ∞$.
(b) If the sequence $( r n − 1 )$ is bounded, then
$r n ∗ r n + 1 ∗ = r n ∗ r n ∗ + 1 r n = 1 1 + 1 r n r n ∗ → 1 1 + 0 .$
(c) Note that
$Δ r n ∗ Δ r n + 1 ∗ = 1 r n 1 r n + 1 = r n + 1 r n .$
Hence, by the Stolz–Cesaro theorem, (c) implies Equation (14). ☐
Note that, if r is a potential sequence, i.e., $r n = n ω$, where $ω$ is a fixed real number, then $r n + 1 / r n → 1$. In this case, from Theorem 1, we have the following corollary.
Corollary 1.
Assume $σ ∈ ( 0 , ∞ )$, $r n = n 1 − σ$, $h : [ 0 , ∞ ) → [ 0 , ∞ )$ is nondecreasing, $α ∈ ( 0 , ∞ )$, $h ( α ) > 0$,
$p n ≥ 0 , p n → λ ∈ R , λ ≠ 1 , ∑ n = 1 ∞ | a n | < ∞ , ∑ n = 1 ∞ | b n | < ∞ ,$
$∫ α ∞ d s h ( s ) = ∞ , | f ( n , u ) | ≤ h | u | n σ for ( n , u ) ∈ N × R .$
Then every nonoscillatory solution x of Equation (1) has the property
$x n = c n σ + o ( n σ ) ,$
where c is a real constant.
Proof.
By Lemma 5(c) we have
$r n ∗ r n + 1 ∗ → 1 .$
Let $α = σ − 1$. Then $α > 0$ and, by Lemma 4,
$r n ∗ = α n σ + o ( n σ ) = n σ ( α + o ( 1 ) ) = n σ ( 1 ) = ( n σ ) .$
Choose a positive constant L such that for any n we have
$r n ∗ ≤ L n σ .$
Define a function $g : [ 0 , ∞ ) → [ 0 , ∞ )$ by $g ( s ) = h ( L s )$. Then g is nondecreasing and
$∫ α L ∞ d s g ( s ) = ∞ .$
Moreover, for any $( n , u ) ∈ N × R$, we have
$| f ( n , u ) | ≤ h | u | n σ ≤ h L | u | r n ∗ = g | u | r n ∗ .$
Let x be a nonoscillatory solution of Equation (1). By Theorem 1, there exists a constant $c ′$ such that $x n = c ′ r n ∗ + o ( r n ∗ )$. Hence,
$x n = c ′ α n σ + o ( ( n σ ) ) = c n σ + o ( n σ ) .$
☐
Theorem 1, applied to the linear equation
$Δ ( r n Δ ( x n + p n x n − τ ) ) = q n x n ,$
Corollary 2.
Assume that $p n ≥ 0 , p n → λ ∈ R , r n ∗ r n + 1 ∗ → μ ∈ R , λ ≠ 1 ≠ λ μ τ ,$ and
$∑ n = 1 ∞ r n ∗ | q n | < ∞ .$
Then every nonoscillatory solution $( x n )$ of Equation (15) has the asymptotic property
$x n = c r n ∗ + o ( r n ∗ ) ,$
where c is a real constant.
Proof.
We get the conclusion of Corollary 2 by applying Theorem 1 with
$a n = r n ∗ q n , f ( n , u ) = u r n ∗ and g ( u ) = u .$
☐
Applying Theorem 1 to nonlinear difference equation of the form
$Δ ( r n Δ ( x n + p n x n − τ ) ) = q n x n α , 0 < α < 1 ,$
where $( p n )$, $( q n )$ are sequences of real numbers and $τ$ is a nonnegative integer, we have the following corollary.
Corollary 3.
Assume that $p n ≥ 0 , p n → λ ∈ R , r n ∗ r n + 1 ∗ → μ ∈ R , λ ≠ 1 ≠ λ μ τ ,$ and
$∑ n = 1 ∞ ( r n ∗ ) α | q n | < ∞ .$
Then every nonoscillatory solution $( x n )$ of Equation (16) has the property $x n = c r n ∗ + o ( r n ∗ ) ,$ where c is a real number.
Proof.
The conclusion follows from Theorem 1 with $a n = r n ∗ α q n ,$ $f ( n , u ) = u α r n ∗ α$ and $g ( u ) = u α$. ☐
Example 1.
Consider the difference equation
$Δ n ( n + 1 ) Δ x n + 2 n + 1 n x n − 1 = − 12 ( n − 1 ) ( n − 2 ) x n .$
Here, $r n = n ( n + 1 )$, $p n = 2 n + 1 n$, $τ = 1$, and $q n = − 12 ( n − 1 ) ( n − 2 )$. Hence,
$r n ∗ = 1 − 1 n , r n ∗ r n + 1 ∗ → 1 , ∑ n = 1 ∞ r n ∗ | q n | < ∞ .$
Therefore, all assumptions of Corollary 2 are satisfied. It is not difficult to check that the sequence $x n = 1 − 2 n$ is a solution of Equation (17) with the property
$x n = 1 − 1 n − 1 n = r n ∗ + o ( r n ∗ ) .$
Next, we give sufficient conditions under which all nonoscillatory solutions of Equation (1) have the property $x n = c r n ∗ + d + o ( 1 ) .$
Theorem 2.
Assume $g : [ 0 , ∞ ) → [ 0 , ∞ )$ is nondecreasing,
$∑ n = 1 ∞ n | a n | < ∞ , ∑ n = 1 ∞ n | b n | < ∞ ,$
$p n ≥ 0 , p n → λ ∈ R , λ ≠ 1 , p n − λ = o 1 r n ∗ , r n − 1 → ρ ∈ R ,$
$∫ 0 ∞ d s g ( s ) = ∞ , | f ( n , u ) | ≤ g | u | r n ∗ for ( n , u ) ∈ N × R .$
Then every nonoscillatory solution x of Equation (1) has the property
$x n = c r n ∗ + d + o ( 1 ) ,$
where $c , d$ are real constants.
Proof.
Note that all assumptions of Theorem 1 are satisfied. Let x be a nonoscillatory solution of Equation (1) and let z be defined by Equation (9). As in the proof of Theorem 1, there exists a constant K such that
$| x n | r n ∗ ≤ | z n | r n ∗ ≤ K ,$
for $n ≥ n 1 = n 0 + τ$. Hence,
$| f ( n , x n ) | ≤ g | x n | r n ∗ ≤ g ( K ) ,$
for any $n ≥ n 1$. Therefore, by Equations (1) and (18), the series
$∑ n = 1 ∞ n | Δ ( r n Δ z n ) | ,$
is convergent. Choose a constant L such that $r n − 1 ≤ L$ for any n. Then
$∑ n = 1 ∞ 1 r n ∑ j = n ∞ | Δ ( r j Δ z j ) | ≤ L ∑ n = 1 ∞ ∑ j = n ∞ | Δ ( r j Δ z j ) | = L ∑ n = 1 ∞ n | Δ ( r n Δ z n ) | < ∞ .$
Define a sequence U by
$U n = ∑ j = n ∞ 1 r j ∑ i = j ∞ Δ ( r i Δ z i ) .$
Then
$U n = o ( 1 ) ,$
and
$Δ ( r n Δ U n ) = − Δ r n 1 r n ∑ i = n ∞ Δ ( r i Δ z i ) = − Δ ∑ i = n ∞ Δ ( r i Δ z i ) = Δ ( r n Δ z n ) .$
Define a sequence W by
$W n = z n − U n .$
Using Equation (20), we have
$Δ ( r n Δ W n ) = Δ ( r n Δ z n ) − Δ ( r n Δ U n ) = Δ ( r n Δ z n ) − Δ ( r n Δ z n ) = 0 .$
Hence, there exists a constant P such that $r n Δ W n = P$ for any n. Summing the equality
$Δ W n = P r n ,$
from 1 to $n − 1$, we obtain
$W n = Q + P ∑ k = 1 n − 1 1 r k = P r n ∗ + Q ,$
where $Q = W 1$. Using Equations (21), (22), and (19), we obtain
$z n = W n + U n = P r n ∗ + Q + o ( 1 ) .$
Let $z ′$ be a sequence defined by
$z n ′ = x n + λ x n − τ ,$
for $n > τ$. Then
$z n ′ = z n + ( λ − p n ) x n − τ .$
By Theorem 1, we have
$lim n → ∞ x n − τ r n − τ ∗ = lim n → ∞ x n r n ∗ ∈ R .$
Moreover, since the sequence $r − 1$ is convergent, the sequence
$r n ∗ r n + 1 ∗ = r n ∗ r n ∗ + 1 r n = 1 1 + 1 r n r n ∗$
is convergent, too. Hence, the sequence
$r n − τ ∗ r n ∗$
is convergent. Therefore,
$( λ − p n ) x n − τ = o 1 r n ∗ x n − τ = o ( 1 ) r n ∗ x n − τ = o ( 1 ) x n − τ r n − τ ∗ r n − τ ∗ r n ∗ = o ( 1 ) .$
Thus, by Equations (22) and (24),
$z n ′ = z n + o ( 1 ) = P r n ∗ + Q + o ( 1 ) .$
Let
$u n = x n − P 1 + λ r n ∗ .$
Then
$u n + λ u n − τ = x n − P 1 + λ r n ∗ + λ x n − τ − P λ 1 + λ r n − τ ∗ = x n − P 1 + λ r n ∗ + λ x n − τ − P λ 1 + λ r n ∗ + P λ 1 + λ ( r n ∗ − r n − τ ∗ ) = x n + λ x n − τ − P 1 1 + λ + λ 1 + λ r n ∗ + P λ 1 + λ ( r n ∗ − r n − τ ∗ ) = z n ′ − P r n ∗ + P λ 1 + λ ( r n ∗ − r n − τ ∗ ) .$
Hence, by Equation (25), we have
$u n + λ u n − τ = Q + o ( 1 ) + P λ 1 + λ ( r n ∗ − r n − τ ∗ ) .$
Note that $r n + 1 ∗ − r n ∗ = r n − 1 → ρ$. Similarly, $r n + 2 ∗ − r n + 1 ∗ → ρ$. Hence,
$r n + 2 ∗ − r n ∗ = r n + 2 ∗ − r n + 1 ∗ + r n + 1 ∗ − r n ∗ → 2 ρ .$
Analogously, $r n ∗ − r n − τ ∗ → τ ρ$. Hence, the sequence $( u n + λ u n − τ )$ is convergent and, by Lemma 1, the sequence u is convergent, too. Therefore, by Equation (26),
$x n = P 1 + λ r n ∗ + u n = c r n ∗ + d + o ( 1 )$
where
$c = P 1 + λ , d = lim n → ∞ u n .$
☐
Remark 3.
Observe that, if the sequence $( r n ∗ )$ is bounded, then the conclusion of Theorem 2 follows directly from Theorem 1. Indeed, in this case, we have $r n ∗ = α + o ( 1 )$. Hence, $o ( r n ∗ ) = o ( 1 )$, and we have
$x n = c r n ∗ + o ( r n ∗ ) = c r n ∗ + o ( 1 ) = c r n ∗ + 0 + o ( 1 ) .$
Applying Theorem 2 to a linear equation expressed by (15), we have the following result.
Corollary 4.
Assume that $p n ≥ 0 , p n → λ ∈ R , r n ∗ r n + 1 ∗ → μ ∈ R , λ ≠ 1 ≠ λ μ τ ,$ $p n − λ = o 1 r n ∗ , lim n → ∞ r n − 1 = ρ ∈ R$, and
$∑ n = 1 ∞ n r n ∗ | q n | < ∞ .$
Then every nonoscillatory solution $( x n )$ of Equation (15) has the asymptotic property
$x n = c r n ∗ + d + o ( 1 )$
where $c , d$ are real constants.
Example 2.
Consider the difference equation
$Δ 1 n Δ x n + 2 x n − 1 = 6 n 2 ( n + 1 ) ( n − 1 ) x n .$
Here, $r n = 1 n$, $p n = 2$, $τ = 1$, and $q n = 6 n 2 ( n + 1 ) ( n − 1 )$. Then
$r n ∗ = n ( n − 1 ) 2 , r n ∗ r n + 1 ∗ → 1 , ∑ n = 1 ∞ r n ∗ | q n | < ∞ .$
Note that all assumptions of Corollary 2 are satisfied. One can see that the sequence $x n = n 2 − 2 n$ is a solution of Equation (27) with the property
$x n = 2 r n ∗ + o ( r n ∗ ) .$
Note also that the assumption $∑ n = 1 ∞ n r n ∗ | q n | < ∞$ of Corollary 4 is not satisfied, and the sequence x does not have the property $x n = c r n ∗ + d + o ( 1 ) .$
Applying Theorem 2 to a nonlinear Equation (16), we have the corollary.
Corollary 5.
Assume that $p n ≥ 0 , p n → λ ∈ R , r n ∗ r n + 1 ∗ → μ ∈ R , λ ≠ 1 ≠ λ μ τ ,$ $p n − λ = o 1 r n ∗ , lim n → ∞ r n − 1 = ρ ∈ R$, and
$∑ n = 1 ∞ n ( r n ∗ ) α | q n | < ∞ .$
Then every nonoscillatory solution $( x n )$ of Equation (16) has the property
$x n = c r n ∗ + d + o ( 1 )$
where $c , d$ are real constants.
Example 3.
Let $r n = 2 n$, $p n = 1 2 n$, $τ = 1$, $α = 1 2$, and $q n = 1 2 n + 1 2 + 2 − n$. Then Equation (16) takes the form
$Δ 2 n Δ x n + 1 2 n x n − 1 = 1 2 n + 1 2 + 2 − n x n .$
For this equation, we have
$r n ∗ = 1 − 1 2 n − 1 , μ = r n ∗ r n + 1 ∗ → 1 .$
Then
$∑ n = 1 ∞ n r n ∗ | q n | < ∑ n = 1 ∞ n 2 n + 1 < ∞ .$
Therefore, since all assumptions of Corollary 5 are satisfied, every nonoscillatory solution $( x n )$ of Equation (28) has the property
$x n = c r n ∗ + d + o ( 1 )$
where $c , d$ are real constants. The sequence $x n = 1 − 1 2 n − 1 + 1 + 1 2 n = 2 + 1 2 n$ is one of such solutions.
Remark 4.
This paper is devoted to nonoscillatory solutions. But, in the case $p n ≡ 0$, our results are true for all solutions. This follows from the proofs of Theorems 1 and 2, respectively.

## 3. Conclusions

In this paper, we have presented sufficient conditions, under which all nonoscillatory solutions of Equation (1) have the property $x n = c r n ∗ + o ( r n ∗ )$ or the property $x n = c r n ∗ + d + o ( 1 )$. The presented results are new even for linear equations of the type defined by Equation (1), and in the case when $p n ≡ 0$. The first part of the proof of Theorem 1, based on the summation method and the use of discrete Bihari type lemma, is in principle standard (see [27,28,32,33]). The second part of the proof required a new approach with the use of Lemma 3. The difficulty was choosing appropriate conditions for the sequences p and r. In Theorem 2, this problem was even greater. Our results can be generalized in two directions. First, one can try to get a more accurate approximation of solutions, e.g., with an accuracy of $o ( n s )$, where s is a nonpositive real number. Secondly, one can try to obtain similar results for higher-order equations. This problem is not easy to solve. A comparison between [27] and [28] illustrates the scale of this difficulty.

## Author Contributions

Conceptualization, M.M. and J.M.; Methodology, M.M. and J.M.; Validation, M.M. and J.M.; Formal Analysis, M.M. and J.M.; Resources, M.M.; Writing—Original Draft Preparation, M.M. and J.M.; Writing—Review & Editing, M.M. and J.M.

## Acknowledgments

The first author was supported by the Ministry of Science and Higher Education of Poland (04/43/DSPB/0095).

## Conflicts of Interest

The authors declare that they have no competing interests.

## References

1. Jankowski, R.; Schmeidel, E. Almost oscillation criteria for second order neutral difference equation with quasidifferences. Int. J. Differ. Equ. 2014, 9, 77–86. [Google Scholar]
2. Jiang, J. Oscillation of second order nonlinear neutral delay difference equations. Appl. Math. Comput. 2003, 146, 791–801. [Google Scholar] [CrossRef]
3. Saker, S.H. New oscillation criteria for second-order nonlinear neutral delay difference equations. Appl. Math. Comput. 2003, 142, 99–111. [Google Scholar] [CrossRef]
4. Saker, S.H. Oscillation Theory of Delay Differential and Difference Equations: Second and Third Orders; Verlag Dr. Müller: Saarbrücken, Germany, 2010. [Google Scholar]
5. Schmeidel, E.; Zbaszyniak, Z. An application of Darbos fixed point theorem in the investigation of periodicity of solutions of difference equations. Appl. Math. Comput. 2012, 64, 2185–2191. [Google Scholar] [CrossRef]
6. Seghar, D.; Thandapani, E.; Pinelas, S. Oscillation theorems for second order difference equations with negative neutral term. Tamkang J. Math. 2015, 46, 441–451. [Google Scholar]
7. Thandapani, E.; Balasubramanian, V.; Graef, J.R. Oscillation criteria for second-order difference equations with neqative neutral term. Int. J. Pure Appl. Math. 2013, 87, 283–292. [Google Scholar] [CrossRef]
8. Thandapani, E.; Selvarangam, S.; Rama, R.; Madhan, M. Improved oscillation criteria for second order nonlinear delay difference equations with non-positive neutral term. Fasc. Math. 2016, 26, 155–165. [Google Scholar] [CrossRef]
9. Cheng, S.S.; Li, H.J.; Patula, W.T. Bounded and zero convergent solutions of second-order difference equations. J. Math. Anal. Appl. 1989, 141, 463–483. [Google Scholar] [CrossRef]
10. Migda, M. Asymptotic behaviour of solutions of nonlinear difference equations. Fasc. Math. 2001, 31, 57–63. [Google Scholar]
11. Řehák, P. Asymptotic formulae for solutions of linear second order difference equations. J. Differ. Equ. Appl. 2016, 22, 107–139. [Google Scholar] [CrossRef]
12. Stević, S. Asymptotic behaviour of second-order difference equations. ANZIAM J. 2004, 46, 157–170. [Google Scholar] [CrossRef]
13. Stević, S. Growth estimates for solutions of nonlinear second-order difference equations. ANZIAM J. 2005, 46, 439–448. [Google Scholar] [CrossRef]
14. Thandapani, E.; Manuel, M.M.S.; Graef, J.R.; Spikes, P.W. Monotone properties of certain classes of solutions of second-order difference equations. Comput. Math. Appl. 1998, 36, 291–297. [Google Scholar] [CrossRef]
15. Bohner, M.; Stević, S. Asymptotic behavior of second-order dynamic equations. Appl. Math. Comput. 2007, 188, 1503–1512. [Google Scholar] [CrossRef]
16. Bohner, M.; Stević, S. Linear perturbations of a nonoscillatory second-order dynamic equation. J. Differ. Equ. Appl. 2009, 15, 1211–1221. [Google Scholar] [CrossRef]
17. Higgins, R. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete Contin. Dyn. Syst. B 2010, 13, 609–622. [Google Scholar] [CrossRef][Green Version]
18. Agarwal, R.P.; Grace, S.R.; O’Regan, D. Nonoscillatory solutions for discrete equations. Comput. Math. Appl. 2003, 45, 1297–1302. [Google Scholar] [CrossRef]
19. Liu, Z.; Xu, Y.; Kang, S.M. Global solvability for a second order nonlinear neutral delay difference equation. Comput. Math. Appl. 2009, 57, 587–595. [Google Scholar] [CrossRef][Green Version]
20. Galewski, M.; Jankowski, R.; Nockowska-Rosiak, M.; Schmeidel, E. On the existence of bounded solutions for nonlinear second order neutral difference equations. Electron. J. Qual. Theory Differ. Equ. 2014, 72, 1–12. [Google Scholar] [CrossRef]
21. Tian, Y.; Cai, Y.; Li, T. Existence of nonoscillatory solutions to second-order nonlinear neutral difference equations. J. Nonlinear Sci. Appl. 2015, 8, 884–892. [Google Scholar] [CrossRef][Green Version]
22. Bezubik, A.; Migda, M.; Nockowska-Rosiak, M.; Schmeidel, E. Trichotomy of nonoscillatory solutions to second-order neutral difference equation with quasi-difference. Adv. Differ. Equ. 2015, 2015, 192. [Google Scholar] [CrossRef]
23. Chatzarakis, G.E.; Diblik, J.; Miliaras, G.N.; Stavroulakis, I.P. Classification of neutral difference equations of any order with respect to the asymptotic behavior of their solutions. Appl. Math. Comput. 2014, 228, 77–90. [Google Scholar] [CrossRef]
24. Migda, J. Approximative solutions to difference equations of neutral type. Appl. Math. Comput. 2015, 268, 763–774. [Google Scholar] [CrossRef]
25. Migda, M.; Migda, J. On a class of first order nonlinear difference equations of neutral type. Math. Comput. Model. 2004, 40, 297–306. [Google Scholar] [CrossRef]
26. Zhu, Z.Q.; Wang, G.Q.; Cheng, S.S. A classification scheme for nonoscillatory solutions of a higher order neutral difference equation. Adv. Differ. Equ. 2006, 2006, 47654. [Google Scholar] [CrossRef]
27. Migda, M.; Migda, J. Asymptotic properties of solutions of second-order neutral difference equations. Nonlinear Anal. 2005, 63, e789–e799. [Google Scholar] [CrossRef]
28. Migda, J. Asymptotically polynomial solutions to difference equations of neutral type. Appl. Math. Comput. 2016, 279, 16–27. [Google Scholar] [CrossRef][Green Version]
29. Stević, S. A note on bounded sequences satisfying linear inequalities. Indian J. Math. 2001, 43, 223–230. [Google Scholar]
30. Stević, S. Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos. Chaos Solitons Fractals 2008, 35, 238–245. [Google Scholar] [CrossRef]
31. Migda, J. Asymptotically polynomial solutions of difference equations. Adv. Differ. Equ. 2013, 2013, 92. [Google Scholar] [CrossRef]
32. Thandapani, E.; Arul, R.; Raja, P.S. The asymptotic behavior of nonoscillatory solutions of nonlinear neutral type difference equations. Math. Comput. Model. 2004, 39, 1457–1465. [Google Scholar] [CrossRef]
33. Zafer, A. Oscillatory and asymptotic behavior of higher order difference equations. Math. Comput. Model. 1995, 21, 43–50. [Google Scholar] [CrossRef]