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Oscillation analysis for nonlinear difference equation with non-monotone arguments
Advances in Difference Equations volume 2018, Article number: 166 (2018)
Abstract
The aim of this paper is to obtain some new oscillatory conditions for all solutions of nonlinear difference equation with non-monotone or non-decreasing argument
where \(( p(n) ) \) is a sequence of nonnegative real numbers and \(( \tau (n) ) \) is a non-monotone or non-decreasing sequence, \(f\in C(\mathbb{R},\mathbb{R})\) and \(xf(x)>0\) for \(x\neq 0\).
1 Introduction
Oscillation theory of difference equations has attracted many researchers. In recent years there has been much research activity concerning the oscillation and nonoscillation of solutions of delay difference equations. For these oscillatory and nonoscillatory results, we refer, for instance, to [1–23]. As far as we can see, there is not yet a study in the literature about the solutions of Eq. (1) to be oscillatory under the \(( \tau (n) ) \) is a non-monotone or non-decreasing sequence. So, in the present paper, our aim is to obtain new oscillatory conditions for all solutions of Eq. (1). Consider the nonlinear difference equation with general argument
where \(( p(n) ) _{n\geq 0}\) is a sequence of nonnegative real numbers and \(( \tau (n) ) _{n\geq 0}\) is a sequence of integers such that
and
Δ denotes the forward difference operator \(\Delta x(n)=x(n+1)-x(n)\).
Define
Clearly, r is a positive integer.
By a solution of the difference equation (1), we mean a sequence of real numbers \((x(n))_{n\geq -r}\) which satisfies (1) for all \(n\geq 0\).
A solution \((x(n))_{n\geq -r}\) of the difference equation (1) is called oscillatory, if the terms \(x(n)\) of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory.
If \(f(x)=x\), then Eq. (1) takes the form
In particular, if we take \(\tau (n)=n-\ell\), where \(\ell>0\), then Eq. (4) reduces to
In 1989, Erbe and Zhang [8] proved that each one of the conditions
and
is sufficient for all solutions of (5) to be oscillatory.
In the same year, 1989, Ladas, Philos and Sficas [12] established that all solutions of (5) are oscillatory if
Clearly, condition (7) improves to (5).
In 1991, Philos [15] extended the oscillation criterion (8) to the general case of Eq. (4), by establishing that, if the sequence \(( \tau (n) ) _{n\geq 0}\) is increasing, then the condition
suffices for the oscillation of all solutions of Eq. (4).
In 1998, Zhang and Tian [20] found that if \(( \tau (n) ) \) is non-decreasing,
and
then all solutions of Eq. (4) are oscillatory.
Later, in 1998, Zhang and Tian [21] found that if \(( \tau(n) ) \) is non-decreasing or non-monotone,
and (10) holds, then all solutions of Eq. (4) are oscillatory.
In 2008, Chatzarakis, Koplatadze and Stavroulakis [3] proved that if \(( \tau (n) ) \) is non-decreasing or non-monotone \(h(n)=\max_{0\leq s\leq n}\tau (s)\),
then all solutions of Eq. (3) are oscillatory.
In 2008, Chatzarakis, Koplatadze and Stavroulakis [4] proved that if \(( \tau (n) ) \) is non-decreasing or non-monotone, \(h(n)=\max_{0\leq s\leq n}\tau (s)\),
and (10) holds, then all solutions of Eq. (4) are oscillatory.
In 2006, Yan, Meng and Yan [18] found that if \(( \tau (n) ) \) is non-decreasing,
and
then all solutions of Eq. (4) are oscillatory.
In 2016, Öcalan [16] proved that if \(( \tau (n) ) \) is non-decreasing or non-monotone, \(h(n)=\max_{0\leq s\leq n}\tau (s)\) and (16) holds, then all solutions of Eq. (4) are oscillatory.
Set
Clearly
Observe that it is easy to see that
and therefore condition (16) is better than condition (11).
When the case \(\tau (n)=n-\ell\), where \(\ell>0\), then Eq. (1) reduces to
For Eq. (19), we can suggest references [11] and [17] for the reader.
2 Main results
In this section we investigated the oscillatory behavior of all solutions of Eq. (1). We present new sufficient conditions for the oscillation of all solutions of Eq. (1) under the assumption that the argument \(( \tau (n) ) \) is non-monotone or non-decreasing sequence. Set
Clearly, \((h(n))\) is non-decreasing, and \(\tau (n)\leq h(n)\) for all \(n\geq0\). We note that if \((\tau (n))\) is non-decreasing, then we have \(\tau(n)=h(n)\) for all \(n\geq 0\).
Assume that the f in Eq. (1) satisfies the following condition:
Theorem 1
Assume that (2), (3) and (21) hold. If \(( \tau (n) ) \) is non-monotone or non-decreasing, and
then all solutions of Eq. (1) oscillate.
Proof
Assume, for the sake of contradiction, that \((x(n))\) is an eventually positive solution of (1). Then there exists \(n_{1}\geq n_{0}\) such that \(x(n),x ( \tau (n) ),x ( h(n) ) >0\) for all \(n\geq n_{1}\). Thus, from Eq. (1) we have
Thus \((x(n))\) is non-increasing and has a limit \(k\geq 0\) as \(n\to \infty \). Now, we claim that \(k=0\). Otherwise, \(k>0\). By (3), \(f(x)>0\) and then \(\lim_{n\to \infty }f(x ( n) ) =f(k)>0\). So, summing up (1) from \(n_{1}\) to \(n-1\), we get
On the other hand, condition (22) implies that
In view of (23) and (24), we obtain for \(n\to \infty \)
This is a contradiction to the fact that \(k>0\). Therefore \(\lim_{n\to \infty }x ( n ) =0\). Now, suppose \(M>0\). Then, in view of (21) we can choose \(n_{2}\geq n_{1}\) so large that
On the other hand, we know from [4, Lemma 1.5] (also see [16, Lemma 1]) that
Since \(h(n)\geq \tau (n)\) and \((x(n))\) is non-increasing, by (1) and (25) we have
Also, from (22) and (26), it follows that there exists a constant \(c>0\) such that
So, from (28), there exists an integer \(n^{\ast }\in (h(n),n)\), for all \(n\geq n_{3}\) such that
Summing up (27) from \(h(n)\) to \(n^{\ast }\) and using \((x(n))\) is non-increasing, then we have
or
Thus, by (29), we have
Summing (27) from \(n^{\ast }\) to n and using the same facts, we get
Thus, by (29), we have
Combining the inequalities (30) and (31), we obtain
and hence we have
Let
and because of \(1\leq w\leq (4\mathrm{e})^{2}\), w is finite.
Now dividing (1) with \(x(n)\) and then summing up from \(h(n)\) to \(n-1\), we obtain
It is well known that
Since \(h(n)\geq \tau (n)\) and \((x(n))\) is non-increasing, we get
Taking lower limits on both of (35) and using (21), (22) and (32), we obtain \(\ln(w)>\) \(\frac{w}{\mathrm{e}}\). But this is impossible since \(\ln(x)\leq \frac{x}{\mathrm{e}}\) for all \(x>0\).
Now, we consider the case where \(M=0\). In this case, it is clear that by (21), we have
Since \(\frac{x}{f(x)}>0\), by (36), for sufficiently large integers, we get
and
where \(\varepsilon >0\) is an arbitrary real number. Thus, since \(\tau (n)\leq h(n)\) and \((h(n))\) is non-decreasing, by (1) and (37), we have
Summing up (38) from \(h(n)\) to n, we obtain
and so, we get
Thus, by (28) and (39), we can write
or
This contradicts \(\lim_{x\to 0}\frac{x}{f(x)}=0\). The proof of the theorem is completed. □
Theorem 2
Assume that (2), (3), (24) and (21) hold with \(0< M<\infty \). If \((\tau (n))\) is non-monotone, and
where \(h(n)\) is defined by (20) and \(\theta >1\) is a constant, then all solutions of Eq. (1) oscillate.
Proof
Assume, for the sake of contradiction, that there exists a nonoscillatory solution \((x(n))\) of (1). In view of (24), we know from the proof of Theorem 1 that \(\lim_{n\to \infty }x(n)=0\) for \(n\geq n_{1}\).
On the other hand, by (21) and for every \(\theta >1\), there exists a \(\delta >0\) such that
Since \(x(n)\to 0\) as \(n\to \infty\), we can find a \(n_{2}\) such that \(0< x(n)<\delta \) for \(n\geq n_{2}\), which yields
or equivalently
From Eqs. (1) and (41), we get
Since \(h(n)\geq \tau (n)\) and \((x(n))\) is non-increasing, we obtain
Summing up (42) from \(h(n)\) to n, and using the fact that \((h(n))\) is non-decreasing
or
This implies
and hence
Therefore, we obtain
This is a contradiction to (40). The proof is completed. □
Now, assume that f is non-decreasing function, then we have the following result.
Theorem 3
Assume that (2), (3), (24) and (21) hold with \(0< M<\infty\). If f is non-decreasing, \((\tau (n))\) is non-monotone and
where \(h(n)\) is defined by (20), then all solutions of Eq. (1) oscillate.
Proof
Assume, for the sake of contradiction, that there exists a nonoscillatory solution \((x(n))\) of (1). In view of (24), we know from the proof of Theorem 1 that \(\lim_{n\to \infty }x(n)=0\) for \(n\geq n_{1}\).
Since \(\tau (n)\leq h(n)\), \((x(n))\) is non-increasing and \((h(n))\), f are non-decreasing, for Eq. (1), we have
Summing up (44) from \(h(n)\) to n, we get
or
and so
Therefore
and hence, we have
This is a contradiction to (43). The proof is completed. □
Remark 1
We remark that if \((\tau (n))\) is non-decreasing, then we have \(\tau (n)=h(n)\) for all \(n\in \mathbb{N}\). Therefore, the condition (40) in Theorem 2 and the condition (43) in Theorem 3, respectively, reduce to
and
Now, we present an example to show the significance of our results.
Example 1
Consider the nonlinear delay difference equation
where
By (20), we see that
If we put \(p(n)=\frac{1}{\mathrm{e}} \) and \(f(x)=x\ln (10+ \vert x \vert )\). Then we have
and
that is, all conditions of Theorem 1 are satisfied and therefore all solutions of (47) oscillate.
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Öcalan, Ö., Özkan, U.M. & Yildiz, M.K. Oscillation analysis for nonlinear difference equation with non-monotone arguments. Adv Differ Equ 2018, 166 (2018). https://doi.org/10.1186/s13662-018-1630-y
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DOI: https://doi.org/10.1186/s13662-018-1630-y