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 Open Access
Oscillation theorems for second order nonlinear neutral difference equations
 Srinivasan Selvarangam^{1},
 Ethiraju Thandapani^{2} and
 Sandra Pinelas^{3}Email author
https://doi.org/10.1186/1029242X2014417
© Selvarangam et al.; licensee Springer. 2014
 Received: 27 May 2014
 Accepted: 17 September 2014
 Published: 21 October 2014
Abstract
This paper deals with the oscillation of second order neutral difference equations of the form
The oscillation of all solutions of this equation is established via comparison theorems. Examples are provided to illustrate the main results.
MSC:39A10.
Keywords
 neutral difference equation
 second order
 comparison theorems
 oscillation
1 Introduction

(H_{1}) $\{{p}_{n}\}$ and $\{{q}_{n}\}$ are nonnegative real sequences with $\{{q}_{n}\}$ not identically zero for infinitely many values of n;

(H_{2}) $\{{a}_{n}\}$ is a positive sequence such that ${R}_{n}={?}_{s={n}_{0}}^{n1}\frac{1}{{a}_{s}^{\frac{1}{a}}}?\mathrm{8}$ as $n?\mathrm{8}$;

(H_{3}) a and ß are ratio of odd positive integers;

(H_{4}) $\{t(n)\}$ and $\{s(n)\}$ are nondecreasing sequences of integers such that ${lim}_{n?\mathrm{8}}t(n)={lim}_{n?\mathrm{8}}s(n)=\mathrm{8}$ and $t\xb0s=s\xb0t$;

(H_{5}) there is a positive constant p such that $0={p}_{n}=p<\mathrm{8}$.
By a solution of equation (1.1) we mean a real sequence $\{{x}_{n}\}$ defined, and satisfying equation (1.1), for all $n\ge {n}_{0}$. We consider only those solutions $\{{x}_{n}\}$ of equation (1.1) which satisfy $sup\{{x}_{n}:n\ge N\}>0$ for all $N\ge {n}_{0}$. We assume that equation (1.1) possesses such a solution. A solution $\{{x}_{n}\}$ of equation (1.1) is oscillatory if it is neither eventually positive nor eventually negative, and nonoscillatory otherwise.
Since the second order equations have applications in various problems in physics, biology, and economics, there is a permanent interest in obtaining new sufficient conditions for the oscillation or nonoscillation of solutions of various types of second order equations; see for example [1–12], and the references cited therein.
and presented oscillation criteria which improved the existing results for the equation.
In [17], the authors established some sufficient conditions for the oscillation of all solutions of equation (1.3) via comparison theorems. In [18], Sun and Saker considered equation (1.5) and obtained some new oscillation criteria, which improve and complement that given in [16].
Recently in [19, 20], the authors studied the oscillatory behavior of all solutions of equation (1.1) under the conditions $0\le {p}_{n}\le p<1$ and $\alpha >\beta $ or $\alpha <\beta $ or $\alpha =\beta $.
Motivated by the above observation in this paper we shall investigate the oscillatory properties of equation (1.1) without assuming any usual restrictions on $\{{p}_{n}\}$, $\{\sigma (n)\}$, and $\{\tau (n)\}$. We shall provide new comparison theorems in which we compare the second order equation (1.1) with the first order difference equations in the sense that the oscillation nature of these first order difference equations yields the oscillation of equation (1.1). In Section 2, we present some basic lemmas, and in Section 3 we establish oscillation results. In Section 4, we provide several examples to illustrate the main results and we present our conclusion in Section 5.
2 Some basic lemmas
In this section we present some basic lemmas which will be used to prove the main results.
for $x>0$. Therefore, $f(x)$ is convex and $f(\frac{A+B}{2})\le \frac{f(A)+f(B)}{2}$, which implies the result (2.1). □
Proof If $A=0$ or $B=0$, then the result is true. For $A\ne 0$, let $x=\frac{B}{A}$. Then the result (2.2) takes the form ${(1+x)}^{\alpha}\le 1+{x}^{\alpha}$, which is clearly true for all $x>0$. □
Next, we present the structure of positive solutions of equation (1.1) since the opposite case is similar.
eventually.
This contradiction proves (2.3). □
The proof of the following lemma is found in [21].
are oscillatory.
Next we state a result given in [22].
has no eventually positive solution.
is oscillatory.
But this contradicts Lemma 2.5 according to which condition (2.5) ensures that (2.8) has no positive solution. The proof is now complete. □
We conclude this section with the following lemma.
also has an eventually positive solution.
Clearly, $\{{u}_{n}\}$ is an eventually positive solution of (2.10) for all $n\ge N$. Since ${u}_{n}>0$ for all ${n}_{0}\le n\le N$, it follows that ${u}_{n}>0$ for all $n\ge {n}_{0}$. Hence equation (2.10) has an eventually positive solution for all $n\ge {n}_{0}$. This completes the proof. □
3 Oscillation results
where ${n}_{1}$ is sufficiently large. We first study the case $\alpha =\beta =1$.
has no positive solution, then every solution of equation (1.1) is oscillatory.
Therefore, (3.6) together with (3.5) ensures that $\{{y}_{n}\}$ is a positive solution of inequality (3.1). This contradicts our assumption and therefore the proof is complete. □
Remark 3.1 The condition $\tau \circ \sigma =\sigma \circ \tau $ of the hypothesis (H_{4}) is satisfied, for example $\tau (n)=n\tau $ and $\sigma (n)=n\sigma $.
Remark 3.2 When studying the oscillatory properties of neutral type equations one usually assumes $\sigma (n)\le \tau (n)$, $\sigma (n)\le n$, $\tau (n)\le n$, $0\le {p}_{n}<1$, etc. In Theorem 3.1 no such constraint is involved and what is more we do not impose is $\tau (n)$ is delayed or advanced and accordingly $\sigma (n)$ can be delayed or advanced. Hence, our result is of high generality and extends and complements the known ones.
has no positive solution, then every solution of equation (1.1) is oscillatory.
Proof We assume that $\{{x}_{n}\}$ is a positive solution of equation (1.1). It follows from Lemma 2.3 and the proof of Theorem 3.1 that ${y}_{n}={a}_{n}\mathrm{\u25b3}{z}_{n}>0$ is decreasing and it satisfies (3.1). Let us denote ${w}_{n}={y}_{n}+p{y}_{\tau (n)}$.
Substituting this term into (3.1), we obtain $\{{w}_{n}\}$ is a positive solution of (3.8), a contradiction. This completes the proof. □
Adding the restriction that $\sigma (n)=n\sigma $, and using suitable criterion for the absence of positive solutions of equation (3.8) (see e.g. Lemma 2.5), we obtain an easily verifiable oscillation result for equation (1.1).
then every solution of equation (1.1) is oscillatory.
But this condition, according to Lemma 2.5, guarantees that (3.8) has no positive solution and the assertion now follows from Theorem 3.2. □
Now, we turn our attention to the case when $\tau (n)$ is delayed. We use the notation ${\tau}^{1}(n)$ for its inverse function.
has no positive solution, then every solution of equation (1.1) is oscillatory.
Proof Assume that $\{{x}_{n}\}$ is a positive solution of equation (1.1). Then ${y}_{n}={a}_{n}\mathrm{\u25b3}{z}_{n}>0$ is decreasing solution of (3.1). We denote ${w}_{n}={y}_{n}+p{y}_{\tau (n)}$. Then by (3.11), we have ${w}_{n}\le {y}_{\tau (n)}(1+p)$. Substituting this into (3.1), we see that $\{{w}_{n}\}$ is a positive solution of inequality (3.12), a contradiction. This completes the proof. □
then every solution of equation (1.1) is oscillatory.
Proof The proof is very similar to that of Corollary 3.1 and hence it is omitted. □
Next, we consider the case $0<\beta \le 1$, and $\sigma (n)\le n$ in equation (1.1).
has no positive solution, then every solution of equation (1.1) is oscillatory.
which is a contradiction. This completes the proof. □
Next, we shall deduce new sufficient conditions for inequality (3.15) to have no positive solutions, to obtain new oscillation criteria for equation (1.1). We shall discuss both the cases when $\tau (n)$ is delayed and advanced.
is oscillatory, then so is equation (1.1).
It follows from Lemma 2.7 that the associated difference equation (3.21) also has a positive solution, which contradicts the oscillatory nature of equation (3.21). □
is oscillatory, then so is equation (1.1).
By Lemma 2.7, the associated difference equation (3.23) also has a positive solution, which is a contradiction. This completes the proof. □
Now, we derive a criterion for equations (3.21) and (3.23) to be oscillatory. Employing this criterion, one can easily verify sufficient conditions for the oscillation of all solutions of equation (1.1).
Applying condition (2.6) to equations (3.21) and (3.23) in view of Theorem 3.5 and Theorem 3.6, immediately we obtain the following oscillatory criteria for equation (1.1).
then every solution of equation (1.1) is oscillatory.
then every solution of (1.1) is oscillatory.
In view of Lemma 2.4, and Theorems 3.5 and 3.6, we have the following oscillation criteria for equation (1.1).
then every solution of equation (1.1) is oscillatory.
then every solution of equation (1.1) is oscillatory.
Next, we turn our attention to the case $\beta \ge 1$ and we rewrite our previous results to cover this case.
has no positive solution, then every solution of equation (1.1) is oscillatory.
which follows from Lemma 2.1. □
The following results are equivalent to Theorems 3.5 and 3.6 and the proofs are omitted.
is oscillatory, then every solution of equation (1.1) is oscillatory.
is oscillatory, then solution of equation (1.1) is oscillatory.
Combining Lemma 2.6 with Theorems 3.8 and 3.9, we have the following oscillation criteria for equation (1.1).
then every solution of equation (1.1) is oscillatory.
then every solution of equation (1.1) is oscillatory.
Combining Lemma 2.4 with Theorems 3.8 and 3.9, we have following results.
then every solution of equation (1.1) is oscillatory.
then every solution of equation (1.1) is oscillatory.
4 Examples
In this section, we present some examples to illustrate the main results.
If the condition (4.2) is satisfied, then by Corollary 3.1, every solution of equation (4.1) is oscillatory.
If the condition (4.3) holds, then by Corollary 3.2, every solution of equation (4.1) is oscillatory. Hence, we have covered the oscillation of equation (4.1) when $\tau (n)$ is delayed or advanced.
Therefore, by Corollary 3.3, every solution of equation (4.4) is oscillatory if condition (4.5) holds.
Therefore, by Corollary 3.4, every solution of equation (4.4) is oscillatory if condition (4.6) holds.
where $b>0$, $\alpha =\frac{1}{5}$, $\beta =\frac{1}{3}$, $\tau =1$, $\sigma =1$. Choose $\lambda =2$, then condition (3.26) reduces to $\frac{b}{{2}^{\frac{1}{3}}}>0$. Therefore, by Corollary 3.5 every solution of equation (4.7) is oscillatory.
Therefore, by Corollary 3.7, every solution of equation (4.8) is oscillatory if condition (4.9) holds.
Therefore, by Corollary 3.8, every solution of equation (4.8) is oscillatory if condition (4.10) holds.
5 Conclusions
In this paper, we have introduced new comparison theorems for investigation of the oscillation of equation (1.1). The established comparison principles reduce the study of the oscillation of the second order neutral difference equations to a study of the oscillation properties of various types of first order difference inequalities, which clearly simplifies the investigation of the oscillation of equation (1.1). Further, the method used here permits us to relax the restrictions usually imposed on the coefficients of equation (1.1). So the results obtained here are of high generality and easily may be applicable, as illustrated with suitable examples.
Declarations
Acknowledgements
This paper was supported by CINAMIL  Centro de Investigação, Desenvolvimento e Inovação da Academia Militar.
Authors’ Affiliations
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