On oscillatory second order impulsive neutral difference equations

1.   Introduction

The problem of oscillation of solution by imposing proper impulse controls, arises in a wide variety of real world phenomena observed in Sciences and Engineering. Indeed, impulsive differential equations arise in circuit theory, bifurcation analysis, population dynamics, loss less transmission in computer network, biotechnology, mathematical economic, chemical technology, mechanical system with impact, merging of solution, noncontinuity of solution, etc. ^[4,10].

With the development of computer techniques, it is essential to formulate discrete dynamical systems while implementing the continuous dynamical systems for computer simulation, for experimental or computational purpose. These discrete time systems, which are described by the difference equations, inherit the similar dynamical characteristics. Because of that, many researchers pay their attentions to dynamical behaviours of difference equations with impulse.

In ^[9], M. Peng has investigated the oscillation criteria for second order impulsive delay difference equations of the form:

In another work ^[8], Peng has extended the work of ^[9] to the second order impulsive neutral delay difference equations of the form:

and obtained the sufficient conditions for oscillation of the system $(E'')$ when $p(n) = -1$.

From the above works ^[8] and ^[9], we have a common question:

(Q) Can we find some oscillation criteria for $(E')$ and $(E'')$ when the neutral coefficient $p(n)\in\mathbb{R}$ viz. $-\infty < p(n) < -1$, $-1 < p(n)\leq 0$ and $0\leq p(n) < \infty$?

The aim of this paper is to give a positive answar to this question by using the techniques developed in ^[16], to obtain some oscillation and nonoscillation criteria for a class of second order nonlinear neutral impulsive difference systems of the form:

where $\tau$, $\sigma > 0$ are integers, $p$, $q$, $r$, $a$ are real valued functions with discrete arguments such that $q(n) > 0$, $r(n) > 0$, $a(n) > 0$, $|p(n)| < \infty$ for $n\in\mathbb{N}(n_0) = \{n_0, n_0+1, \cdots\}$, $G\in C(\mathbb{R}, \mathbb{R})$ with the property $xG(x) > 0$ for $x\neq 0$, and $\Delta$ is the forward difference operator defined by $\Delta u(n) = u(n+1)-u(n).$ Let $m_1, m_2, m_3, \cdots$ be the moments of impulsive effect with the properties $0 < m_1 < m_2 < \cdots, \lim_{j\rightarrow\infty} m_j = +\infty$. And $\underline{\Delta}$ is the difference operator defined by $\underline{\Delta} u(m_j-1) = u(m_{j})-u(m_{j}-1)$.

We refer the reader to some of the related works ^{[2,3,5,6,7,11,12,13,15,17,20]} and the references cited there in.

Definition 1.1. By a solution of $(E)$ we mean a real valued function $x(n)$ defined on $\mathbb{N}(n_0-\rho)$ which satisfy $(E)$ for $n\geq n_0$ with the initial conditions $x(i) = \phi(i), \, i = n_0-\rho, \cdots, n_0,$ where $\phi(i), \, i = n_0-\rho, \cdots, n_0$ are given and $\rho = \max\{\tau, \sigma\}.$

Definition 1.2. A nontrivial solution $x(n)$ of $(E)$ is said to be nonoscillatory, if it is either eventually positive or eventually negative. Otherwise, the solution is said to be oscillatory. The system $(E)$ is said to be oscillatory, if all its solutions are oscillatory.

Theorem 1.3. ^[1] (Krasnoselskii's Fixed Point Theorem)

Let $X$ be a Banach space and $S$ be a bounded closed subset of $X$. Consider two maps $T_{1}$ and $T_{2}$ of $S$ into $X$ such that $T_{1}x+T_{2}y\in S$ for every pair $x, y\in S$. If $T_{1}$ is a contraction and $T_{2}$ is completely continuous, then the equation $T_{1}x+T_{2}x = x$ has a solution in $S$.

2.   Oscillation properties

In this section, we discuss the oscillation criteria for neutral impulsive difference equations $(E)$. We assume that $a(n)$ satisfies

$(A_{0})\; \; A(n) = \sum_{s = n_{0}}^{n}\frac{1}{a(s)}\; and\; \lim_{n\to\infty}A(n) = \infty.$

Theorem 2.1. Let $-1\leq p(n)\leq 0$ and $\tau < \sigma$. In addition to $(A_{0})$, assume that

$(A_1)\; \; G(-u) = -G(u), \; u\in\mathbb{R},$

$(A_2)\; G\; {satisfies}\; \int_{0}^{\pm\alpha}\frac{du}{G(u)} < \infty, \alpha>0,$

$(A_{3})\; \; \sum_{n = 1}^{\infty}q(n)+\sum_{j = 1}^{\infty} r(m_j-1) = \infty$

and

$(A_{4})\; \; \sum_{n = 1}^{\infty}q'(n)+\sum_{j = 1}^{\infty}r'(m_{j}-1) = \infty$

hold, where $q'(n) = \min\left\{\frac{q(n)}{a(n)}, \frac{q(n)}{a(n+1)}\right\}$ and $r'(n) = \min\left\{\frac{r(m_j-1)}{a(m_j-1)}, \frac{r(m_j-1)}{a(m_j)}\right\}$. Then every solution of $(E)$ oscillates.

Proof. Suppose on the contrary that $x(n)$ is a nonoscillatory solution of $(E)$ for $n\geq n_0$. Without loss of generality and due to $(A_1)$, we may assume that $x(n) > 0$, $x(n-\tau) > 0$ and $x(n-\sigma) > 0$ for $n\geq n_0 > \rho$. Setting

in $(E)$, we have

for $n\geq n_1 > n_0+\sigma$. Therefore, $a(n)\Delta y(n)$ and $y(n)$ are monotonic for $n\geq n_1$. Here, we arise four possible cases, viz.,

Case 1. We can choose $n_2 > n_1 +1$ and a constant $\beta > 0$ such that $y(n)\geq \beta$ for $n\geq n_2$. Indeed, $y(n) > 0$ and $-1\leq p(n)\leq 0$ implies that $y(n)\leq x(n)$ and hence $y(m_{j}-1)\leq x(m_{j}-1)$. Now, the impulsive system $(E)$ reduces to

Summing $(E_1)$ from $n_2$ to $n-1$, we get

that is,

a contradiction to $(A_3)$ due to $\lim_{n\rightarrow\infty}a(n)y(n) < \infty$.

Case 2. Since $y(n) < 0$ for $n\geq n_2$, then we can find $n_3 > n_2$ such that

Therefore, from $(E_1)$ we get

that is,

implies that

due to the nondecreasing nature of $y$ and $\tau < \sigma$. Clearly,

that is,

If $y(n)\leq u\leq y(n+1)$ and $y(m_{j}-1)\leq v\leq y(m_{j+1}-1)$, then $\frac{1}{G(y(n))}\geq \frac{1}{G(u)}$ and $\frac{1}{G(y(m_{j}-1))}\geq \frac{1}{G(v)}$. Therefore, the preceding inequalities reduce to

As a result,

Since for nonimpulsive points $m_j -1$ and $n$ we have $\lim_{n\rightarrow\infty}y(n) < \infty$ and $\lim_{j\rightarrow\infty}y(m_j -1) < \infty$, then

a contradiction to $(A_4)$ due to $(A_2)$.

Case 3. As $a(n)\Delta y(n)$ is nonincreasing for $n\geq n_1$, we can find a constant $\gamma > 0$ and $n_2 > n_1 +1$ such that $a(n)\Delta y(n) < -\gamma$ for $n\geq n_2$ and hence $a(m_{j}-1)\Delta y(m_j-1) < -\gamma$ for $n\geq n_2$. Summing $\Delta y(n) < -\frac{\gamma}{a(n)}$ from $n_2$ to $n-1$, we get

that is,

a contradiction to the fact that $y(n) > 0$ for $n\geq n_2$.

Case 4. Here, $\lim_{n\to\infty}y(n) = -\infty$ and so also $\lim_{j\to\infty}y(m_{j}-1) = -\infty$. By Sandwich theorem, it follows that $\lim_{j\rightarrow\infty}y(m_{j}) = -\infty$. Clearly, $y(n) < 0$ for $n\geq n_1$ implies that

Analogously,

due to the nonimpulsive points $m_j-1, m_j-\tau-1, m_j-2\tau-1, \cdots$. Therefore, $x(n)$ is bounded for all nonimpulsive points. We assert that $x(m_j)$ is bounded. If not, let it be $\lim_{j\rightarrow\infty}x(m_j) = +\infty$. Ultimately,

implies that $y(m_j) > 0$ as $j\rightarrow\infty$, a contradiction, where $x(m_j -\tau)\leq B_1$. So, our assertation holds and $y(n)$ is bounded for every $n$. Again this leads to a contradiction to the fact that $y(n)$ is unbounded. This complete the proof of the theorem.

Theorem 2.2. Let $-\infty < b\leq p(n)\leq c < -1$ and $\tau-\sigma\geq 1$. If $(A_{0})$, $(A_{1})$, $(A_{3})$, $(A_{4})$ and

$(A_{5})\; \; G \; satisfies \; \int_{\pm \alpha}^{\pm\infty}\frac{dv}{G(v)} < \infty, \alpha>0$

hold, then every solution of $(E)$ either oscillates or satisfies $\lim_{n\to\infty}x(n) = 0$.

Proof. Suppose on the contrary that $x(n)$ is a nonoscillatory solution of $(E)$ for $n\geq n_{0} > \rho$. Proceeding as in the proof of Theorem 2.1, we have that $a(n)\Delta{y(n)}$ and $y(n)$ are of one sign for $n\geq n_1 > n_0$. So, we have following four cases:

The proofs for Case 1 and Case 3 are similar to that of Theorem 2.1.

Case 2. Let $\lim_{n\to\infty}y(n) = l, \; -\infty < l\leq 0$. We claim that $l = 0$. Otherwise, there exists $n_{2} > n_{1}+1$ such that $y(n+\tau-\sigma)\leq l$ and so also, $y(m_{j}+\tau-\sigma-1)\leq l$. Indeed, $y(n) < 0$ implies that $y(n) > p(n)x(n-\tau)\geq bx(n-\tau)$ and analogously, $y(m_j-1)\geq bx(m_j-\tau-1)$ due to nonimpulsive points $m_j-1, m_j-\tau-1, \cdots$. Hence, there exists $n_3 > n_2$ such that $(E)$ takes the form

for $n\geq n_3$. Summing the above impulsive system from $n_3$ to $n-1$, it follows that

that is,

a contradiction to $(A_{3})$. So, our claim holds and thus $\lim_{n\to\infty}y(n) = 0$, $\lim_{j\to\infty}y(m_{j}-1) = 0$. Now,

implies that $\limsup_{n\to\infty}x(n) = 0$ due to $(1+c) < 0$ and hence $\lim_{n\to\infty}x(n) = 0$. We encounter that $\lim_{j\to\infty}x(m_{j}-1) = 0$ because of nonimpulsive points $m_j-1, j\in\mathbb{N}$. Since $m_{j}-1 < m_{j} < n$, then an application of the Sandwich theorem implies that $\lim_{j\to\infty}x(m_{j}) = 0$. Therefore, $\lim_{n\to\infty}x(n) = 0$ for all $n$ and $m_{j}, j\in\mathbb{N}$.

Case 4. For $y(n) < 0$,

Analogously,

due to the nonimpulsive point $m_{j}-1, m_{j}-\tau-1, \cdots$ and so on. Therefore,

for $n\geq n_2 > n_1 +1$. Ultimately, $(E)$ becomes

that is,

Using the fact that $y$ is nonincreasing for $n\geq n_{2}$ and $\tau-\sigma\geq 1$, we get

Consequently,

If $b^{-1}y(n+1)\leq u \leq b^{-1}y(n+2)$ and $b^{-1}y(m_{j})\leq v \leq b^{-1} y(m_{j+1})$, then the last two inequalities can be written as

that is,

Since for nonimpulsive points $m_j -1$ and $n$ we have $\lim_{n\rightarrow\infty}y(n) = \infty$ and $\lim_{j\rightarrow\infty}y(m_j -1) = \infty$, then an application of Sandwich theorem shows that $\lim_{j\to\infty}y(m_{j}) = \infty$. Therefore,

a contradiction to $(A_{4})$ due to $(A_{5})$. This completes the proof of the theorem.

Theorem 2.3. Let $0\leq p(n)\leq d < \infty$ and $\tau\leq \sigma$. In addition to $(A_0)$ and $(A_1)$, assume that

$(A_{6})\; G(u)G(v)\geq G(uv) \; \text{for}\; u, v\in \mathbb{R}_{+},$

$(A_7)\; \; there\; exists\; \lambda>0\; such\; that\; G(u)+G(v)\geq \lambda G(u+v)\; for\; u, v\in \mathbb{R}_{+},$

$(A_8)\; \; \sum_{n = \tau}^{\infty}Q(n)+\sum_{j = 1}^{\infty}R(m_j-1) = \infty$

$and$

$(A_9)\; \; \sum_{n = \tau}^{\infty}Q'(n)+\sum_{j = 1}^{\infty}R'(m_j-1) = \infty$

hold, where $Q(n) = \min\{q(n), q(n-\tau)\}$, $R(m_j-1) = \min\{r(m_j-1), r(m_j-\tau-1)\}$, $Q'(n) = \min\left\{\frac{q(n)}{a(n+1)}, \frac{q(n-\tau)}{a(n+1-\tau)}\right\}$ for $n\geq\tau$ and $R'(m_j-1) = \min\left\{\frac{r(m_j-1)}{a(m_{j})}, \frac{r(m_j-\tau-1)}{a(m_{j}-\tau)}\right\}$ for $m_{j}\geq\tau+1$. Then every solution of $(E)$ oscillates.

Proof. Proceeding as in the proof of Theorem 2.1, we have following two possible cases:

Case 1. In this case, $y(n)$ is nondecreasing for $n\geq n_1$. So, there exist $n_2 > n_1 +1$ and a constant $\beta > 0$ such that $y(n)\geq \beta$ for $n\geq n_2$. From (1.1), we have

and

Combining (2.2) and (2.3), we have

which on applying $(A_6)$, we obtain

that is,

due to $(A_7)$. Since $y(n -\sigma)\leq x(n -\sigma)+ax(n-\tau -\sigma)$, then the preceding inequality can be written as

By using a similar argument in (1.2), we get

Summing (2.4) from $n_2$ to $n-1$ and then using (2.5), we get

that is,

Therefore,

a contradiction to $(A_8)$.

Case 2. From (2.2) and (2.3), we have

Consequently, (2.4) reduces to

By a similar argument to (2.5), we get

Hence, the impulsive system $(E)$ reduces to

Using the fact that $y$ is nonincreasing and $\tau\leq \sigma$, we can find $n_{3} > n_2 +1$ such that the above inequality can be written as

for $n\geq n_3$. If

then form $(E_2)$ it is easy to verify that

that is,

As a result,

implies that

a contradiction to $(A_{9})$ due to $(A_{2})$. This completes the proof of the theorem.

Next, we establish the criteria for existence of positive solution of the impulsive system $(E)$.

Theorem 2.4. Let $-1 < p_{1}\leq p(n)\leq p_{2}\leq 0$. Assume that

$(A_{10})\; \; \sum_{s = n}^{\infty}\frac{1}{a(s)}\Big[\sum_{t = n^{*}}^{s-1}q(s)+\sum_{n^{*}\leq m_{j}-1\leq s-1} r(m_j-1)\Big] < \infty$

holds. Then $(E)$ has a bounded non-oscillatory solution.

Proof. Let $X = l_{\infty}^{n_{1}}$ be the Banach space of all real valued bounded sequence $x(n)$ for $n\geq n_{1}$ with the norm defined by

Consider a closed subset $S$ of $X$, where

where $\beta_{1} > 0$ and $\beta_{2} > 0$ are so chosen that $\beta_{1}-p_{1}\beta_{2} < \beta_{2}$. Due to $(A_{10})$, we can find $n_{2} > n_{1}$ and $\beta_{1} < \gamma < (1+p_{1})\beta_{2}$ such that

where $M = \max\{G(x):\beta_{1}\leq x \leq \beta_{2}\}$. For $x\in S$ and $n\geq n_2$, define two maps

and

Indeed, for $x_{1}, x_{2}\in S$ and using (2.6) for $n\geq n_{2}$, we have

and

Therefore, $\beta_{1}\leq T_{1}x_{1}+T_{2}x_{2}\leq \beta_{2}$ for $n\geq n_{2}$. Also, for $x_{1}, x_{2}\in S$ and $n\geq n_{2}$, we have

that is,

and hence $T_{1}$ is a contraction mapping with the contraction constant $-p_{1} < 1$.

Next, we show that $T_{2}$ is completely continuous. For this, we need to show that $T_{2}x$ is continuous and relatively compact. Let $x_{k}\in S$ be such that $x_{k}(n)\to x(n)$ as $k\to\infty$. Since $S$ is closed, then $x = x(n)\in S$. Now, for $n\geq n_{2}$

Since $|G(x_{k}(n-\sigma))-G(x(n-\sigma))|\to 0$ as $k\to\infty$, by applying the Lebesgue's dominated convergence theorem ^[1], we have that $\lim_{k\to\infty}|(T_{2}x_k)(n)-(T_{2}x)(n)|\to 0$. Therefore, $T_{2}x$ is continuous. To show that $T_{2}x$ is relatively compact, we show that the family of functions $\{T_{2}x:x\in S\}$ is uniformly bounded and equicontinuous on $[n_{2}, \infty)$. It is easy to see that $T_{2}x$ is uniformly bounded.

Next, we show that $T_{2}x$ is equicontinuous. For $n_{4} > n_{3}\geq n_{2}$ and $x\in S$ such that

Therefore, there exists $\epsilon > 0$ and $\delta > 0$ such that for $\epsilon < \frac{(1+p_{1})\beta_{2}-\gamma}{M}$

and this relation continue to hold for every $n_3, n_4 \in [n_2, \infty)$. Therefore, $\{T_{2}x:x\in S\}$ is uniformly bounded and equicontinuous on $[n_{2}, \infty)$ and hence $T_{2}x$ is relatively compact. By Theorem 1.3, $T_1 + T_2$ has a unique fixed point $x\in S$ such that $T_{1}x+T_{2}x = x$ for which

Indeed, $x(n)$ is a positive solution of the impulsive system $(E)$. This completes the proof of the theorem.

3.   Conclusion and example

We present some examples to illustrate our main results.

Example 3.1. Consider the impulsive difference equation

where $\tau = 1$, $\sigma = 3$, $a(n) = n$, $p(n) = -1/2$, $q(n) = 6n+3$, $r(m_j-1) = 6m_{j}-3$, $G(u) = u^{1/3}$, $m_j = 3j$ for $j\in\mathbb{N}$. Clearly,

and

Therefore, $(A_2)-(A_4)$ hold. It is easy to see that all conditions of Theorem 2.2 are satisfied. Hence, $(E_{3})$ is oscillatory. In particular, $x(n) = (-1)^{n}$ is an oscillatory solution of the first equation of $(E_{3})$ and $(-1)^{m_j}$ is an oscillatory solution of the second equation of $(E_{3})$.

Example 3.2. Consider the impulsive difference equation

where $\tau = 3$, $\sigma = 1$, $a(n) = 1$, $p(n) = -2$, $q(n) = (n-1)^{3}\left(\frac{1}{n+2}+\frac{2}{n+1}+\frac{1}{n}+\frac{2}{n-3}+\frac{4}{n-2}+\frac{2}{n-1}\right)$, $r(m_j-1) = (m_{j}-2)^{3}\left(\frac{1}{m_{j}+5}+\frac{1}{m_{j}+4}+\frac{2}{m_{j}+2}+\frac{2}{m_{j}+1}+\frac{1}{m_{j}}+\frac{2}{m_{j}-1}+\frac{2}{m_{j}-3}+\frac{2}{m_{j}-4}\right)$, $m_j = 5j$ for $j\in\mathbb{N}$ and $G(u) = u^{3}$. Clearly,

Therefore, $(A_3)$ and $(A_4)$ hold. It is easy to see that all conditions of Theorem 2.1 are satisfied. In particular, $x(n) = \frac{(-1)^{n+1}}{n}$ is a solution of the first equation of $(E_{4})$ and $\frac{(-1)^{m_{j}}}{m_j-1}$ is a solution of the second equation of $(E_{4})$.

Remark 3.3. In Theorem 2.4, we have obtained the necessary condition for the existence of bounded positive solution of the impulsive system $(E)$ by using the Krasnoselskii's fixed point theorem in the range $-1 < p(n)\leq 0$. It would be interesting to prove the results in the other ranges of $p(n)$ by means of Krasnoselskii's fixed point theorem.

Remark 3.4. We may note that, Theorem 2.2 guarantees that every solution of $(E)$ either oscillates or converges to zero. Unfortunately, we can not establish sufficient condition that ensure that all solutions of $(E)$ are just oscillatory.

Remark 3.5. Based on Remark 3.4, we can raise following problems for future research:

(1) Is it possible to establish sufficient condition that ensure that all solutions of $(E)$ are oscillatory when $-\infty < p(n)\leq -1$?

(2) Is it possible to suggest a different method to study $(E)$ and find some sufficient conditions which ensure that all solutions of $(E)$ are oscillatory when $|p(n)| < \infty$?

(3) Is it possible to find the necessary and sufficient conditions which ensure that all solutions of $(E)$ are oscillatory?

Acknowledgments

The author thanks the anonymous editor and two referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper. This work is supported by Rajiv Gandhi National fellowship(UGC), New Delhi, India, through the Letter No. F1-17.1/2017-18/RGNF-2017-18-SC-ORI-35849, dated: july 11th, 2017.

Conflict of interest

The author declares no conflict of interest.