Abstract
In this paper the nonlinear secondorder neutral difference equation of the following form: \(\Delta ( a_{n}\Delta(x_{n}p_{n}x_{n1}) ) + q_{n}f(x_{n\tau})=0\) is considered. By suitable substitution the above equation is transformed into a new one, which is a thirdorder nonneutral difference equation. Using results obtained for the new equation, the asymptotic properties of the neutral difference equation are studied. Some classification of nonoscillatory solutions is presented, as well as an estimation of the solutions. Finally, we present necessary and sufficient conditions for the existence of solutions to both considered equations being asymptotically equivalent to the given sequences.
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1 Introduction
In this paper we consider the difference equation in the following form:
where Δ is the forward difference operator defined by \(\Delta y_{n} = y_{n+1}  y_{n}\), \((a_{n})\), \((p_{n})\), \((q_{n})\) are sequences of positive real numbers, τ is a nonnegative integer, and the function \(f\colon\mathbb{N}\to\mathbb{R}\). Here \(\mathbb{R}\) is the set of real numbers \(\mathbb{N} = \{1,2,\ldots\}\), and \(\mathbb{N}_{k} = \{ k,k+1,k+2,\ldots\}\), \(k \in\mathbb{N}\).
By a solution to (1) we mean a sequence \((x_{n})\) which satisfies (1) for n sufficiently large. We consider only solutions which are nontrivial for all large n. A solution to (1) is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called oscillatory.
Let us denote
This implies that \(x_{n}p_{n}x_{n1}= (\Delta y_{n})\prod_{i=1}^{n}p_{i}\). Substitution of (2) transforms (1) into the following:
Setting
and assuming that
in (3), we get the thirdorder nonlinear difference equation of the following form:
where
By virtue of (4), the positivity of terms of the sequence \((p_{n})\) implies the positivity of terms of the sequence \((b_{n})\). Note that \(f(xy)=f(x)f(y)\) is satisfied for all power functions. Hence, by (5) and (7), if \(f(x)=x^{\gamma}\), where γ is a positive constant, then \(g=f\) and \(b_{n}^{*}=b_{n\tau}^{\gamma}\) for all \(n \in\mathbb{N}_{\tau}\). If f is not a power function, in some cases we can find the function g assumed by (5). For example, for \(f(x)=x^{3} 2^{x1}\) and \(b_{n} \equiv b \in\mathbb{R}\) we have \(b_{n}^{*}=\frac{1}{2}b^{3}\) and \(g(x)=(2^{b})^{x} x^{3}\).
Neutral type difference equations have been widely studied in the literature. Some recent results on the asymptotic behavior of secondorder neutral difference equations can be found, for example, in [1–7]. The higherorder neutral difference equations were studied in [8–13].
For results concerning the oscillatory and asymptotic behavior of the thirdorder difference equation we refer to [14, 15], for equations with quasidifferences to [16–19], and to the references cited therein. Many results on the oscillation of second and thirdorder functional differential and difference equations can also be found in [20].
The purpose of this paper is to study the asymptotic properties of the neutral difference equation (1). Transforming the considered equation into a new one, which is a thirdorder difference equation of type (6), we get various results concerning the asymptotic behavior of solutions to this equation. These results are then used to establish some properties of the solutions to (1). In particular, we obtain necessary and sufficient conditions for the existence of solutions asymptotically equivalent to the given sequences.
Fourthorder nonneutral difference equations with one quasidifference, by the techniques here used, were studied in [21–23]. Some generalizations of the results presented in these papers were published in [24, 25]. Even so, there is not a full analogy to the results since the Kneser type classification of the nonoscillatory solution is different for odd or evenorder equations, and of neutral or nonneutral type as well.
Throughout the rest of our investigations, one or several of the following assumptions will be imposed:
Notice that, by virtue of (5), the positivity of the sequence \((b_{n})\) implies that conditions (H3) and (H4) hold also for the function g.
The following definitions and theorems will be used in the sequel.
We say that the sequence \((u_{n})\) is asymptotically constant if this sequence has a nonzero limit, and we say that it is an asymptotically zero sequence if the limit of this sequence equals zero. We say that the sequence \((u_{n})\) is asymptotically equivalent to \((v_{n})\) if \((\frac {u_{n}}{v_{n}})\) has a nonzero limit. In the present paper, we study the three types of solutions: asymptotically zero solutions, asymptotically constant solutions, and unbounded solutions. It is called a trichotomy of nonoscillatory solutions.
Definition 1
(Uniformly Cauchy subset [26])
A subset S of the Banach space B is said to be uniformly Cauchy if for every \(\varepsilon>0\) there exists a positive integer N such that \(\vert x_{i}x_{j} \vert<\varepsilon\) whenever \(i,j >N\) for any \((x_{n})\in B\).
Lemma 1
(ArzelaAscoli’s theorem [26])
Each bounded and uniformly Cauchy subset of B is relatively compact.
Theorem 1
(Schauder theorem [27])
Let S be a nonempty, closed, and convex subset of a Banach space B and \(T \colon S \to S\) be a continuous mapping such that \(T(S)\) is a relatively compact subset of B. Then T has at least one fixed point in S.
The following theorem of StolzCesáro is a discrete analog of l’Hospital’s rule.
Theorem 2
(StolzCesáro theorem [28])
Let \((u_{n})\), \((v_{n})\) be two sequences of real numbers. Assume that \((v_{n})\) is a strictly monotone and divergent sequence, and the following limit exists: \(\lim_{n \to\infty}\frac{\Delta u_{n}}{\Delta v_{n}}=g\). Then
We introduce the following notation:
2 Existence of nonoscillatory solutions
In this section, we obtain necessary and sufficient conditions for the existence of nonoscillatory solutions to (1) with certain asymptotic properties. We start with the following lemmas.
Lemma 2
Condition (H2) implies that
where \((b_{n})\) is defined by (4).
Proof
Condition (H2) implies that \(\prod^{n}_{i=1} p_{i} \leq C_{0} n\), where \(C_{0}\) is a positive constant. It follows that \(\prod^{n}_{i=1} p_{i}^{1} \geq\frac{1}{C_{0}n}\). Using the notation of (4), the above inequality takes the form \(\frac{1}{b_{n}} \geq\frac {1}{C_{0}n}\). Since the series \(\sum^{\infty}_{n=1} \frac{1}{n}\) diverges, condition (9) is satisfied. □
Remark 1
Condition (H1) and (9) imply that
where \((Q_{n})\) is defined by (8).
Lemma 3
Assume that (H1), (H2), and the following conditions:
are satisfied. Let \((y_{n})\) be an eventually positive solution to (6). Then exactly one of the following statements holds:
for all sufficiently large n.
Proof
The proof is obvious and hence omitted. □
Lemma 4
Assume that (H1)(H4) hold. If \((x_{n})\) is an eventually positive solution to (1), then exactly one of the following cases holds:

(I)
$$\lim_{n \to\infty}\frac{x_{n}}{b_{n}}=0; $$

(II)
there exist positive constants \(C_{1}\), \(C_{2}\), and a positive integer \(n_{0}\) such that
$$ C_{1} b_{n} \leq x_{n} \leq C_{2} b_{n} Q_{n+1} \quad \textit{for } n\geq n_{0}, $$(11)where \((b_{n})\) is defined by (4) and \((Q_{n})\) is defined by (8).
Proof
Let \((y_{n})\) be an eventually positive solution to (6). Then, by Lemma 3, we have two possibilities:
or there exists a positive constant \(C_{1}\) such that \(y_{n} \geq C_{1}\).
If \(\lim_{n \to\infty} y_{n}=0\), then condition (I) is satisfied.
Assuming that \(y_{n+1} \geq C_{1}\) and using substitution (2), we obtain
Thus, inequality \(C_{1} b_{n} \leq x_{n}\) from (11) is satisfied.
Next, we prove that in case (II) the inequality \(x_{n} \leq C_{2} b_{n} Q_{n+1}\) is also satisfied. Since (H3) is satisfied for the function g, from the point of view of (6), there exists \(n_{1}\) such that
By Lemma 2, if (H1) and (H2) are satisfied, then there exists \(n_{2} \geq n_{1}\) such that
Summing inequality (12) from \(n_{2}\) to \(n1\), we get
where \(A_{1} = a_{n_{2}} \Delta(b_{n_{2}} \Delta y_{n_{2}})\) is a positive constant. Summing again, we have
where \(A_{2} = \max\lbrace0, b_{n_{2}} \Delta y_{n_{2}}\rbrace\) is a nonnegative constant. Therefore
Summing again, we have
where \(A_{3}=y_{n_{2}}\) is a positive constant.
By (13), it is easy to see that each term on the right side of inequality (14) is less than
From (14), we get
where \(C_{2} = 3\max\lbrace A_{1}, A_{2}, A_{3} \rbrace\). Hence
Using the substitutions (2) and (4), we obtain
By (8), we see that the required inequality is proved. □
As a consequence of Lemma 4 we obtain the following result.
Lemma 5
Assume that (H1), (H2), (H^{∗}3), and (H^{∗}4) hold. If \((y_{n})\) is an eventually positive solution to (6), then

(I)
$$\lim_{n \to\infty}y_{n}=0; $$

(II)
there exist positive constants \(C_{1}\) and \(C_{2}\) such that
$$C_{1} \leq y_{n} \leq C_{2} Q_{n} \quad \textit{for large }n. $$
Before we derive a necessary and sufficient condition for the existence of a solution to (1) that is asymptotically equivalent to \((b_{n})\), the following theorem needs to be proved.
Theorem 3
Let conditions (H1), (H2), (H^{∗}3), (H^{∗}4) be satisfied. Then a necessary condition for (6) to have an asymptotically constant solution is that
Proof
Let \((y_{n})\) be an asymptotically constant solution to (6). Then \((y_{n})\) is a nonoscillatory sequence. Without loss of generality, we assume that \((y_{n})\) is an eventually positive solution. By Lemma 3 it is of type (i) or type (ii). Each solution to type (i) tends to infinity. This implies that \((y_{n})\) is of type (ii).
Let us denote
Then there exist positive constants \(C_{3}\) and \(C_{4}\) such that
By (H^{∗}3) and (H^{∗}4), we see that there exists a positive constant
which means that, for \(y_{n+1\tau} \in[C_{3}, C_{4}]\), we have
Let \(n_{3}\) be so large that (17) and (ii) are satisfied for \(n \geq n_{3}\). Next, we rewrite (6) in the form
Multiplying the above equation by \(\sum_{j={n_{3}}}^{i}\frac {1}{a_{j}}\sum_{k={n_{3}}}^{j}\frac{1}{b_{k}}\) and summing both sides of it from \(i={n_{3}}2\) to \(n2\) we obtain
By (17), the following inequality holds:
By the formula \(\sum_{i=N}^{n2} y_{i} \Delta x_{i} = x_{i} y_{i} \vert _{i=N}^{n1}  \sum_{i=N}^{n2} x_{i+1} \Delta y_{i}\), we get
which tends to \(y_{n_{3}} \alpha\) where α is defined by (16). Since \((y_{n})\) is a decreasing sequence we have \(y_{n_{3}} \alpha>0\). Set \(C_{6}= y_{n_{3}}  \alpha\). From the above, (18), and (19) we get
This means that
The above condition is equivalent to condition (15). □
The next example shows that the condition (15) in Theorem 3 is not a necessary condition for (6) to have an asymptotically zero solution.
Example 1
Let us consider the following equation of the form (6):
Here \(a_{n} \equiv1\), \(b_{n}\equiv1\), \(q_{n}^{*}\equiv\frac{1}{8}\), \(g(x) =x\), and \(\tau=1\). It is easy to see that condition (15) is not satisfied, but the above equation has an asymptotically zero solution \(y_{n}= \frac{1}{2^{n}}\).
Sufficient conditions, under which, for every real constant, there exists a solution to the higherorder difference equation with quasidifferences convergent to this constant are obtained in Theorem 3.3 in [29]. Hence, for (6), we have the following.
Theorem 4
Assume that (H1), (H2), (H^{∗}3), (H^{∗}4) hold and condition (15) is satisfied. Then for every \(c\in\mathbb{R}\) there exists a solution x to (1) such that \(\lim_{n\to\infty}x(n)=c\).
Corollary 1
Let conditions (H1), (H2), (H^{∗}3), (H^{∗}4) be satisfied. Then the condition
implies that (6) has no asymptotically constant solution.
Proof
This corollary follows directly from Theorem 3. □
Theorem 5
If conditions (H1)(H4) are satisfied, then a necessary and sufficient condition for (1) to have a solution \((x_{n})\) asymptotically equivalent to the sequence \((\prod_{i=1}^{n}p_{i} )\) is the condition
Proof
Using the notation of (2), (5), and (7) in condition (15) the conclusion of this theorem follows directly from Theorem 3 and Theorem 4. □
Remark 2
Let the assumptions of Theorem 5 be satisfied. If
then condition (21) is a necessary and sufficient condition for (1) to have an asymptotically zero solution such that \((x_{n}) \sim (\prod_{i=1}^{n}p_{i} )\).
As a consequence of Theorem 5 we get the following result for the EmdenFowler type equation
where \((a_{n})\), \((p_{n})\), \((q_{n})\) are sequences of positive real numbers, τ is a nonnegative integer, and γ is the ratio of odd positive integers.
Corollary 2
Let conditions (H1) and (H2) be satisfied. A necessary and sufficient condition for (22) to have a solution \((x_{n})\) asymptotically equivalent to the sequence \((\prod_{i=1}^{n}p_{i} )\) is the condition
Example 2
Consider the following equation:
Here \(a_{n}= \sqrt{\frac{n+1}{n}}+ \sqrt{\frac{n+2}{n+1}}\), \(p_{n}=\sqrt {\frac{n+1}{n}}\), \(q_{n}=\frac{2}{n(n+1)(n+2)(\sqrt{n} +1)^{3}}\), \(\gamma= 3\), and \(\tau= 1\). All assumptions of Corollary 2 are satisfied. Hence (24) has at least one solution asymptotically equivalent to the sequence \((\prod_{i=1}^{n}p_{i} ) = \sqrt{n+1}\). In fact \(x_{n}= \sqrt{n+1} + 1\) is such solution.
Example 3
Consider the following equation:
Here \(a_{n} \equiv1\), \(p_{n} \equiv1\), \(q_{n}=\frac{1}{2^{\frac{2}{3}n+\frac {10}{3}}(2^{n1}+1)^{\frac{1}{3}}}\), \(\gamma=\frac{1}{3}\), and \(\tau= 2\). It is easy to check that all assumptions of Corollary 2 are satisfied. Hence, (25) has at least one solution asymptotically equivalent to the sequence \((\prod_{i=1}^{n}p_{i} ) \equiv1\). This means that (25) has an asymptotically constant solution. In fact \(x_{n}= 1 + \frac{1}{2^{n+1}}\) is one such solution.
Finally, we present a necessary and sufficient condition for the existence of an asymptotically \((Q_{n})\) solution to (1). We start with the following theorem.
Theorem 6
If conditions (H1), (H2), (H^{∗}3), (H^{∗}4) are satisfied and
then a necessary and sufficient condition for (6) to have a solution \((y_{n})\) satisfying
is that
where C is some nonzero constant.
Proof
Necessity. Let \((y_{n})\) be a nonoscillatory solution to (6) which satisfies (27). Without loss of generality, we may assume that
Then there exist positive constants \(C_{7}\) and \(C_{8}\) such that
Hence
Thus, by (26), we get
where \(C_{9} = C_{7}\) if the function g is nondecreasing and \(C_{9} = C_{8}\) if the function g is nonincreasing.
By (H^{∗}3), we see that \(g(C_{9} Q_{n+1\tau})\) is positive. On the other hand, summing (6) from \(n_{5}=n_{4}+\tau\) to \(n1\), by Lemma 3, we obtain
This implies that
So, by (29), we have
Sufficiency. Let \(C_{10}\) be a given positive constant. Set
From the above, (26) and (H^{∗}4), there exists a maximum of the function g on interval \(I_{n}\), which we denote as the point \(C_{11} Q_{n}\) with \(C_{11} = \frac{C_{10}}{2}\) if the function g is nonincreasing and \(C_{11} = C_{10}\) if the function g is nondecreasing. Thus we get
Assume that (28) holds for \(C=C_{11}\). Then there exists a positive integer \(n_{6}\) such that
Consider the Banach space B of all real sequences \(y=(y_{n})\) such that
where \(n_{7} = n_{6} + \tau1\). We have
It is easy to see that S is a bounded, convex and closed subset of B.
Now we define an operator \(T \colon S \to B\) in the following way:
First we show that \(T(S)\subset S\). Indeed, if \(y\in S\) it is clear from (32) that \((Ty)_{n} \geq\frac{C_{10}}{2}Q_{n}\) for \(n\geq1\). Furthermore, by (30), we have
Thus T maps S into itself.
Next we prove that T is continuous. Let \((y^{(m)})\) be a sequence in S such that \(y^{(m)} \to y\) as \(m \to \infty\). Because S is closed, \(y\in S\). Now, we get
and therefore
By (10), (28), and (30), it implies that
We see that T is a continuous mapping.
Finally, we need to show that \(T(S)\) is uniformly Cauchy. To see this, we have to show that, given any \(\varepsilon>0\), there exists an integer \(n_{8}\) such that for \(m>n>n_{8}\); we have
for any \(y \in S\). Indeed, we have
Therefore, by Theorem 1, there exists \(y\in S\) such that \(y_{n} = (Ty)_{n}\) for \(n\geq n_{7}\). It is easy to see that \((y_{n})\) is a solution to (6).
Furthermore, by Stolz’s theorem (see Theorem 2) and (8), we have
Thus
This completes the proof. □
Remark 3
Note that if the sequences \((\frac{1}{a_{n}})\) and \((\frac{1}{b_{n}})\) are both polynomial sequences, then \((Q_{n})\) is a polynomial sequence, too.
For example, let \(\frac{1}{a_{n}}=n\) and \(\frac{1}{b_{n}}=n\). Hence \((Q_{n})\), defined by (8), takes the following form:
So, \((Q_{n})\) is a quartic polynomial.
Now, let \(\frac{1}{a_{n}}\equiv1\) and \(\frac{1}{b_{n}}\equiv1\). This means that \(a_{n} \equiv1\) and \(b_{n}\equiv1\). Hence \(Q_{n} = \frac{1}{2} n^{2}  \frac{3}{2}n +1\) is a quadratic polynomial. Obviously, by virtue of (9), this case holds only if \(p_{n} \equiv1\).
Theorem 7
Let conditions (H1)(H4) be satisfied and
Then a necessary and sufficient condition for (1) to have a solution \((x_{n})\) which is asymptotically equivalent to the sequence \(( Q_{n+1} \prod_{i=1}^{n}p_{i} )\) is the convergence of the series
where C is some nonzero constant.
Proof
Using the notation of (2), (5), and (7) in condition (28) the conclusion of this theorem follows directly from Theorem 6. □
Note that for particular cases of (1), if \((\frac{1}{a_{n}})\) is a polynomial sequence and \(p_{n} \equiv1\), from Theorem 7 we get the existence of asymptotically polynomial solutions.
Example 4
In Example 3 (25) is considered. In this equation \(a_{n} \equiv1\) and \(p_{n} \equiv1\). All assumptions of Theorem 7 are satisfied. Hence (25) has an asymptotically \((Q_{n})\) solution, where \(Q_{n} = \frac{1}{2} n^{2}  \frac{3}{2}n +1\). It means that (25) has an asymptotically polynomial solution.
Some results concerning asymptotically polynomial solutions to difference equations can be found, for example, in [30–34].
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This work was partially supported by the Ministry of Science and Higher Education of Poland (PB43081/14DS).
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Bezubik, A., Migda, M., NockowskaRosiak, M. et al. Trichotomy of nonoscillatory solutions to secondorder neutral difference equation with quasidifference. Adv Differ Equ 2015, 192 (2015). https://doi.org/10.1186/s1366201505316
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DOI: https://doi.org/10.1186/s1366201505316