Oscillation theorems for higher order dynamic equations with superlinear neutral term

1.   Introduction

In this paper, we establish new oscillation criteria for a class of higher order dynamic equations with superlinear neutral term of the form

where $y(\mu): = x(\mu)+p(\mu)x^{\alpha}(\delta(\mu))$ on an arbitrary time scale $\mathbb{T} \subseteq \mathbb{R}$ and under the assumptions:

(i) $a, p$ and $q\in C_{rd}([\mu_0, \infty)_{\mathbb{T}}, (0, \infty))$ and $a^{\Delta}(\mu) \ge 0$ for $\mu \in [t_0, \infty)_{\mathbb{T}}$,

(ii) $\alpha, \beta$ are the ratios of positive odd integers with $\beta \le \alpha$ and $\alpha \ge 1$,

(iii) $\tau, \delta \in C_{rd}(\mathbb{T}_0, \mathbb{T})$ such that $x(\mu) = \delta^{-1}(\tau(\mu)) \le \mu$, $\delta(\mu) \ge \mu$, $\delta$ is nondecreasing and invertible and $x$ is nondecreasing with $\lim_{\mu \to \infty}x(\mu) = \infty$.

Assume that

and we also assume that

Define the time scale interval to be $[\mu_0, \infty)_{\mathbb{T}}: = [\mu_0, \infty)\bigcap \mathbb{T}$. Since we are interested in the oscillatory behavior of solutions of (1.1), we assume that $\sup \mathbb{T} = \infty$. Recall that a solution of (1.1) is a nontrivial real-valued function $x$ satisfying Eq (1.1) for $\mu \ge \mu_0$. We exclude from our consideration all solutions vanishing in some neighborhood of infinity. A solution $x$ of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.

The theory of dynamic equations has been introduced to unify difference and differential equations; see, for instance ^[1]. Meanwhile, different theoretical aspects of this theory have been discussed in the last years. Particularly, the oscillation of dynamic equations has been the target of many researchers who succeeded in reporting relevant results. Exploring the literature, however, one can observe that most of the obtained results for the oscillation of dynamic equations have been carried out using the comparison, integral averaging, and Riccati transformation techniques ^{[2,3,4,5,6,7,8,9,10,11,12]}. Based on authors' observation, never the less, there are no known results regarding the oscillation of higher order dynamic equations with nonlinear (superlinear/sublinear) neutral term and advanced argument. The reader can consult relevant results for difference and differential equations in ^[13,14], and the references cited therein. More precisely, the existing literature does not provide any criteria for the oscillation of Eq (1.1). Motivated by this inspiration, we consider new sufficient conditions that ensure that all solutions of (1.1) are either oscillatory or converge to zero. Eq (1.1) is commonly used in a variety of applied problems. We state, in particular, the use of Eq (1.1) in the study of non-Newtonian fluid theory and the turbulent flow of a polytrophic gas in a porous medium; see the papers ^[15,16] for further details.

2.   Main results

As our results will be based on Taylor monomials, we give the following definition.

Definition 2.1. ^[1] Taylor monomials are the functions $h_{n}:\mathbb{T} \times \mathbb{T} \to \mathbb{R}$, $n \in \mathbb{N}_{0} = \{0, 1, 2, \ldots\}$ which are recursively defined as

and for $n \in \mathbb{N}_{0}$

It follows that $h_{1}(\mu, s) = \mu-s$ on any time scale.

One should observe that finding $h_n$ for $n \ge 2$ is not an easy task in general. For a particular time scale such as $\mathbb{T} = \mathbb{R}$ or $\mathbb{T} = \mathbb{Z}$, we can easily find the functions $h_n$. Indeed, we have

where $\mu^{\overline{n}}: = \mu(\mu+1)\ldots(\mu+n-1)$.

Lemma 2.2. (Kneser's Theorem) [1,Theorem 5] Let $\sup \mathbb{T} = \infty$, $n \in \mathbb{N}$ and $x \in C_{rd}^{n}(\mathbb{T}, \mathbb{R}^{+})$. Suppose that $x^{\triangle^{n}}(\mu)\ne 0$ is either nonnegative or nonpositive on $\mathbb{T}$. Then there exists $m \in [0, n)_{\mathbb{Z}}$ such that $(-1)^{n-m}x^{\triangle^{n}}(\mu) \ge 0$ holds for all sufficiently large $\mu$. Moreover, both of the following conditions hold:

(i) $0 \le k < m$ implies $x^{\triangle^{k}}(\mu) > 0$, for all sufficiently large $\mu$.

(ii) $m \le k < n$ implies $(-1)^{m+k}x^{\triangle^{k}}(\mu) > 0$, for all sufficiently large $\mu$.

In what follows, we provide the following lemma which plays a crucial role in the sequel.

Lemma 2.3. ^[1] Let $\sup \mathbb{T} = \infty$ and $x \in C_{rd}^{n}(\mathbb{T}, \mathbb{R}^{+}), \; n \ge 2$. Moreover, suppose that Kneser's Theorem holds with $m \in [0, n)_{\mathbb{Z}}$ and $x^{\triangle^{n}}(\mu)\le 0$ on $\mathbb{T}$. Then, there exists a sufficiently large $\mu_1 \in \mathbb{T}$ such that

If $m = n-1$ in the above inequality, then upon integration it becomes

For convenience, we let

for any constant $c > 0$ and $\mu \ge \mu_1$, $\mu_{1} \in [\mu_0, \infty)_{\mathbb{T}}$. We introduce the notations

and

for $\mu \ge \mu_1, \; \; \mu_1 \in [\mu_0, \infty)_{\mathbb{T}}$.

Theorem 2.4. Assume that conditions (i)–(iii) and (1.2) hold. If

and there exists $\mu_1 \in [\mu_0, \infty)_{\mathbb{T}}$ such that

then every solution of Eq (1.1) is either oscillatory or converges to zero.

Proof. Let $x$ be a nonoscillatory solution of Eq (1.1), say $x(\mu) > 0$ with $\lim_{\mu \to \infty}x(\mu) \ne 0,$ $x(\tau(\mu)) > 0$ and $x(\delta(\mu)) > 0$ for $\mu \ge \mu_1$ and $\mu_1 \in [\mu_0, \infty)_{\mathbb{T}}$. Then Eq (1.1) implies that

Hence $a(\mu)y^{\triangle^{n-1}}(\mu)$ is nonincreasing and of one sign. That is, there exists $\mu_2 \ge \mu_1$ such that

(I) $y^{\triangle^{n-1}}(\mu) > 0,$ and $\big(a(\mu)y^{\triangle^{n-1}}(\mu) \big)^{\triangle} < 0$, for $\mu \ge \mu_2$, $

or

(II) $y^{\triangle^{n-1}}(\mu) < 0,$ and $\big(a(\mu)y^{\triangle^{n-1}}(\mu) \big)^{\triangle} < 0$, for $\mu \ge \mu_2$.

We observe that the above cases hold for all $n$, while the subcases including $y^{\triangle}(\mu) \le 0$ on $[\mu_1, \infty)$ are excluded due to the condition $\lim_{\mu \to \infty}x(\mu) > 0$.

For case I: Here, we have $y^{\triangle}(\mu) > 0$ for $\mu \ge \mu_2$. There exist positive constants $c_1$ and $\mu_3 \ge \mu_2$ such that $y(\delta^{-1}(\mu)) \ge c_1$ and $0 < y^{\frac{1}{\alpha}-1}(\delta^{-1}(\mu)) \le c_{1}^{\frac{1}{\alpha}-1}: = c$.

Clearly, we get $y(\mu) \ge x(\mu)$ and

It is easy to see that

and

Using the fact that $y$ is a nondecreasing, $\delta(\mu) \ge \mu$ and the fact that $\delta^{-1}(\delta^{-1}(\mu)) \le \delta^{-1}(\mu)$, we have $y^{\frac{1}{\alpha}}(\delta^{-1}(\delta^{-1}(\mu))) \le y^{\frac{1}{\alpha}}(\delta^{-1}(\mu))$. Therefore, we get

or

Using (2.7) in (2.6), we get

By Lemma 2.3, we find

Using (2.9) in (2.8), we get

or

where $Y(\mu) = a(\mu)y^{\triangle^{n-1}}(\mu)$ and $Q_1$ is defined in (2.2). Integrating (2.11) from $x(\mu)$ to $\mu$, and using that $x(\mu)$, $a(\mu)$ and $y^{\triangle^{n-1}}(\mu)$ are nondecreasing, we have

or

Taking limsup of both sides as $\mu \to \infty$, we arrive at the desired contradiction.

For case II: Here, we see that

Proceeding as in the above case, we obtain (2.8) and by Lemma 2.3 with $m = n-2$, we see that

Using (2.12) in (2.8), we have

Integrating this inequality from $\mu_3$ to $\mu$, we have

or

where $Q_2$ is defined in (2.3). It follows that

where $Z(\mu) = y^{\triangle^{n-2}}(\mu)$. The remaining part of the proof is similar to that of the above case and hence is omitted.

Define

By virtue of the above theorem, we conclude the following immediate consequence.

Corollary 2.5. Assume that conditions (i)–(iii) and (1.2) are satisfied. If

then every solution of Eq (1.1) is either oscillatory or converges to zero.

Remark 2.6. For $\mathbb{T} = \mathbb{R}$, conditions (2.4) and (2.5) become:

and

For the case when $\mathbb{T} = \mathbb{Z}$, on the other hand, Conditions (2.4) and (2.5) reduce to

and

for $\mu \ge \mu_1, \; \; \mu_1 \in [\mu_0, \infty)_{\mathbb{T}}.$.

The following example is illustrative.

Example 2.7. Consider the following differential equations

and

where $y(\mu): = x(\mu)+\mu x^{3}(2\mu)$. We consider the cases $\tau(\mu) = \frac{3}{2}\mu$, $\tau(\mu) = \mu$ or $\tau(\mu) = \frac{2}{3}\mu$. Since $\delta(\mu) = 2\mu$, we have $x(\mu) = \frac{3}{4}\mu$, $x(\mu) = \frac{1}{2}\mu$ or $x(\mu) = \frac{1}{3}\mu$ respectively. Clearly, $B(\mu) = \frac{2(\mu^{\frac{1}{3}}-4^{\frac{1}{3}})}{\mu^{\frac{4}{3}}} \ge 0$ for $\mu \ge 4$. One can easily prove that conditions (2.15) and (2.16) are satisfied and hence we conclude that every solution of Eq (2.19) or (2.20) is either oscillatory or tends to zero.

Remark 2.8. Results reported in the literature cannot comment on the oscillatory behavior of Eqs (2.19) and (2.20).

Remark 2.9. One may observe that the selection of different values of the argument $\tau(\mu)$ implies that the current results work for all types of equations, that is, for equations with delay, ordinary or advanced arguments.

For the case

we can derive the following immediate result.

Theorem 2.10. Let $n \ge 2$ and condition (2.21) be satisfied. Then, the following statements hold true:

(i) If condition (2.4) holds, then every solution of even order Eq (1.1) is oscillatory.

(ii) If condition (2.4) holds, then every solution of odd order Eq (1.1) is either oscillatory or converges to zero.

Theorem 2.11. Let $n \ge 2$ and condition (2.21) hold. If $x(\mu) \ge \mu$ and

for $\mu_1 \in [\mu_0, \infty)_{\mathbb{T}}$, then the conclusions of Theorem 2.10 hold.

Proof. Let $x$ be a nonoscillatory solution of Eq (1.1), say $x(\mu) > 0$ with $\lim_{\mu \to \infty}x(\mu) \ne 0,$ $x(\tau(\mu)) > 0$ and $x(\delta(\mu)) > 0$ for $\mu \ge \mu_1$ for $\mu_1 \in [\mu_0, \infty)_{\mathbb{T}}$. Proceeding as in the proof of Theorem 2.4 until we obtain (2.8). Integrating (2.8) from $\mu$ to $u$ and letting $u \to \infty$, we have

By virtue of Lemma 2.3, we find

Using (2.24) in (2.23), we get

Taking limsup of both sides of this inequality as $\mu \to \infty$, we obtain a contradiction to (2.22).

Example 2.12. Consider the dynamic equation

where $y(\mu): = x(\mu)+ x^{3}(2\mu)$. It is easy to see that conditions of Theorem 2.11 are satisfied and hence we conclude that every solution of Eq (2.25) is either oscillatory or tends to zero.

Remark 2.13. The oscillatory behavior of Eq (2.25) cannot be addressed via results existing in the literature.

3.   Conclusions

Unlike previous techniques, a new approach is employed to establish easily verifiable sufficient conditions for the oscillation of higher order dynamic equations with superlinear neutral term and advanced argument. The obtained results improve and complement some previous theorems in the literature. Examples are provided to support theoretical findings. The results in this paper are presented in an essentially new form and of a high degree of generality. For future consideration, it will be of great importance to study the oscillation of Eq (1.1) when $\beta > \alpha$.

Acknowledgments

J. Alzabut and K. Abodayeh would like to thank Prince Sultan University for supporting and funding this work.

Conflict of interest

The authors declare that there is no competing interest concerning this work.