Oscillation criterion for half-linear sublinear functional noncanonical dynamic equations

1.   Introduction

Oscillation has drawn significant interest from researchers in engineering and science due to its essential applications in mechanical vibrations. Models may include advanced terms or delays to account for the effects of temporal contexts on solutions. Numerous studies have been conducted regarding oscillation in delay differential equations, such as ^[1,2,3,4,5], advanced differential equations, such as^[6,7,8,9], and dynamic equations, such as ^[10,11], Also, various models are used to study oscillation phenomena in practical applications. In biology, mathematical models have been enhanced with cross-diffusion factors to better account for delay and oscillation effects; see ^{[12,13,14,15]}. Current research focuses on differential equations essential for analyzing real-world phenomena. This paper explores their application in the turbulent flow of a polytropic gas through porous materials and non-Newtonian fluid theory. A solid understanding of the underlying mathematics is crucial for these fields; for more details, refer to ^{[16,17,18,19,20]}. Therefore, this work aims to investigate the oscillatory behavior of a particular class of second-order noncanonical half-linear sublinear functional dynamic equations

on an arbitrary unbounded above time scale $\mathbb{T}$, where $s\in \lbrack s_{0}, \infty)_{\mathbb{T}}$, $s_{0}\geq 0$, $s_{0}\in \mathbb{T}$; $\Phi (u): = \left\vert u\right\vert ^{\kappa }\mathrm{sgn}u$, $0 < \kappa \leq 1;$ $r_{1}, r_{2}:\mathbb{T}\rightarrow \left(0, \infty \right)$ and $\mathbb{ \zeta }:\mathbb{T\rightarrow T}$ are rd-continuous functions such that $\lim_{s\rightarrow \infty }\mathbb{\zeta }(s) = \infty$.

By a solution of equation (1.1), we mean a nontrivial real-valued function $\varkappa \in \mathrm{C}_{\mathrm{rd}}^{1}[T_{\varkappa }, \infty)_{\mathbb{T}}$, $T_{\varkappa }\in \lbrack s_{0}, \infty)_{\mathbb{T}}$ such that $r_{1}\Phi \left(\varkappa ^{\Delta }\right) \in \mathrm{C}_{ \mathrm{rd}}^{1}[T_{\varkappa }, \infty)_{\mathbb{T}}$ and $\varkappa$ satisfies (1.1) on $[T_{\varkappa }, \infty)_{\mathbb{T}}$, where $\mathrm{C}_{\mathrm{rd}}$ represents rd-continuous functions. We propose ^{[21,22,23,24]} as a very helpful introduction to time scale calculus. According to Trench ^[25], Equation (1.1) is said to be in noncanonical form when

and canonical form when

A solution $\varkappa$ of (1.1) is oscillatory if it is not positive or negative; otherwise, it is nonoscillatory. Solutions that vanish at infinity are excluded. We call that Eq (1.1) is oscillatory if all its solutions oscillate.

The subsequent presents oscillation results for differential equations associated with the oscillation results for (1.1). It also provides a comprehensive summary of the significant contributions of this paper, with our results demonstrating that they can be applied to consolidate specific outcomes regarding oscillation in differential and difference equations and expanded to ascertain oscillatory behavior in additional cases. When $\mathbb{T} = \mathbb{R},$ then (1.1) transforms into the half-linear sublinear differential equation.

Fite ^[26] proved that the differential equation

is oscillatory if

Hille ^[27] improved criterion (1.6) and showed that if

then Eq (1.5) is oscillatory. Erbe ^[28] extended (1.7) to the delay equation

where $\mathbb{\zeta }\left(s\right) \leq s$ and proved that if

then Eq (1.8) is oscillatory. Ohriska ^[29] showed that equation (1.8) is oscillatory if

When $\mathbb{T} = {\mathbb{Z}}$, then (1.1) becomes the half-linear sublinear difference equation

Thandapani et al. ^[30] considered the equation

and it was proved that Eq (1.11) is oscillatory if

If $\mathbb{T} = \{s:s = q^{n}, \; n\in { \mathbb{N}}_{0}, \; q > 1\}$, then (1.1) converts the half-linear sublinear $q$-difference equation

Regarding canonical dynamic equations on time scales, Karpuz ^[31] studied the canonical dynamic equation

and obtained that if

and

then Eq (1.13) is oscillatory. Erbe et al. ^[32] created the Hille-type and Ohriska-type criteria for the canonical dynamic equation

where $\mathbb{\zeta }(s)\leq s$, $0 < \kappa \leq 1$ is a quotient of odd positive integers,

and obtained that if

and one of the following criteria holds:

where $\mathbb{\ell }: = \liminf_{s\rightarrow \infty }\dfrac{s}{{\sigma (s)}}{ > 0}$, then Eq (1.14) is oscillatory. Hassan et al. ^[33] studied (1.14) and showed that if (1.15) holds and

Then Eq (1.14) is oscillatory. By using (1.19), it is clear that the second-order Euler dynamic equations

and

are oscillatory if $\lambda > \dfrac{1}{4}$. It is well known that this is the best possible case for the second-order Euler differential equation

Also, we note that criterion (1.19) improves (1.17) since

and

For more Hille-type and Ohriska-type criteria, see ^{[34,35,36,37,38]}.

Concerning the noncanonical form, Hassan et al. ^[39] established some interesting oscillation criteria for the delay noncanonical linear dynamic equation

where $\mathbb{\zeta }\left(s\right) \leq s$ and $\int_{s_{0}}^{\infty } \dfrac{\Delta \mathbb{\varsigma }}{r_{1}(\mathbb{\varsigma })} < \infty$, which are as follows:

Theorem 1.1 (see ^[39]). Equation (1.22) is oscillatory if one of the following criteria holds:

for sufficiently large $T\in \lbrack s_{0}, \infty)_{\mathbb{T}}$.

Also, Hassan et al. ^[40] established, in particular, Hille-type and Ohriska-type oscillation criteria for the advanced noncanonical linear dynamic equation (1.22) where $\mathbb{\zeta }\left(s\right) \geq s$, as shown in the following result:

Theorem 1.2 (see ^[40]). Equation (1.22) is oscillatory if one of the following conditions is satisfied

for sufficiently large $T\in \lbrack s_{0}, \infty)_{\mathbb{T}}$.

It is crucial to emphasize that previous research, such as ^{[31,33,34,35,38]}, primarily focuses on the canonical form, indicating that condition (1.3) holds. This study aims to expand on the findings of ^[39,40] by determining the oscillatory Hille-type and Ohriska-type criteria for the noncanonical half-linear sublinear dynamic equation (1.1) in both cases $\mathbb{\zeta }(s)\leq s$ and $\mathbb{\zeta }(s)\geq s$. The results given in this study have successfully solved a previously unsolved problem that was discussed in several of the author's articles, such as ^{[16,33,35,39,40]}.

This paper is organized as follows: After this introduction, we present the main results in Section 2 for $\mathbb{\zeta }(s)\leq s$, the main results in Section 3 for $\mathbb{\zeta }(s)\geq s$, and the discussion and conclusion in Section 4.

2.   Oscillation criteria for (1.1) when $\mathbb{\zeta }\left(s\right) \leq s$

In the following results, we will present Hille-type and Ohriska-type oscillation criteria for the noncanonical case of Eq (1.1) when $\mathbb{\zeta }(s)\leq s$ on $[s_{0}, \infty)_{{\mathbb{T}}}$.

Theorem 2.1. If for sufficiently large $T\in \lbrack s_{0}, \infty)_{{\mathbb{ T}}},$

where

then Eq (1.1) is oscillatory.

Proof. Suppose that $\varkappa$ is a nonoscillatory solution of (1.1) on $[s_{0}, \infty)_{{\mathbb{T}}}$. Without loss of generality, let $\varkappa (\mathbb{\zeta }\left(s\right)) > 0$ on $[s_{0}, \infty)_{{\mathbb{ T}}}$. By applying the same method as in the proof of Case (a) of [39, Theorem 1], we obtain

eventually. Then there exists an $s_{1}\in \lbrack s_{0}, \infty)_{{\mathbb{T }}}$ such that for $s\in$ $[s_{1}, \infty)_{{\mathbb{T}}},$

Define

Hence,

due to Pötzsche chain rule (see [23, Theorem 1.90]),

Also, by using the fact that $\varkappa ^{\Delta }(s) < 0,$ we obtain

Integrating (1.1) and using the fact that $\varkappa ^{\Delta }(s) < 0,$ we obtain

that is,

In this case, we have

Substituting (2.6) into (2.4), we obtain

Integrating (2.7) from $s$ to $v$, we obtain

Due to $\varpi > 0$ and $\varpi ^{\Delta } < 0$ and letting $v\rightarrow \infty$, we obtain

By multiplying each side of (2.8) by $\tilde{P}\left(s\right)$, we obtain

For any $\varepsilon \in \left(0, 1\right)$, there exists an $s_{2}\in \lbrack s_{1}, \infty)_{\mathbb{T}}$ such that, for $s\in \lbrack s_{2}, \infty)_{\mathbb{T}}$,

where

in view of (2.2) and (2.5). It follows from (2.9) and (2.10) that

due to $\tilde{P}\left(s\right) \rightarrow \infty$ as $s\rightarrow \infty$. Taking the $\liminf$ of each side of the last inequality as $s\rightarrow \infty$, we obtain

Since $\varepsilon > 0$ is arbitrary, we achieve

which is a contradiction to (2.1)

Example 2.1. Consider the second-order half-linear sublinear delay dynamic equation

where $\tilde{\beta} > 0$ is a constant. Here,

It is easy to see that

by [24, Example 5.60]. Also,

In view of Theorem 2.1, Equation (2.11) is oscillatory if $\tilde{\beta} > \dfrac{1}{2}$.

Theorem 2.2. If for sufficiently large $T\in \lbrack s_{0}, \infty)_{{\mathbb{ T}}},$

then Eq (1.1) is oscillatory.

Proof. Suppose that $\varkappa$ is a nonoscillatory solution of (1.1) on $[s_{0}, \infty)_{{\mathbb{T}}}$. Without loss of generality, let $\varkappa (\mathbb{\zeta }\left(s\right)) > 0$ on $[s_{0}, \infty)_{{\mathbb{ T}}}$. By applying the same method as in the proof of Case (a) of [39, Theorem 1], we obtain

eventually. Then there exists an $s_{1}\in \lbrack s_{0}, \infty)_{{\mathbb{T }}}$ such that for $s\in$ $[s_{1}, \infty)_{{\mathbb{T}}},$

In accordance with the proof of Theorem 2.1, Case (b), we conclude that

Since $\left[r_{1}(s)\Phi \left(\varkappa ^{\Delta }(s)\right) \right]^{\Delta } < 0,$ we obtain

Therefore,

Consequently, we have

which contradicts (2.12)

3.   Oscillation criteria for (1.1) when $\mathbb{\zeta }\left(s\right) \geq s$

In this section, we will introduce Hille-type and Ohriska-type oscillation criteria for the noncanonical case of Eq (1.1) when $\mathbb{\zeta }\left(s\right) \geq s$ on $[s_{0}, \infty)_{{\mathbb{T} }}$.

Theorem 3.1. If for sufficiently large $T\in \lbrack s_{0}, \infty)_{{\mathbb{T }}},$

where

with

then Eq (1.1) is oscillatory.

Proof. Suppose that $\varkappa$ is a nonoscillatory solution of (1.1) on $[s_{0}, \infty)_{{\mathbb{T}}}$. Without loss of generality, let $\varkappa (s) > 0$ on $[s_{0}, \infty)_{{\mathbb{T}}}$. As in the proof of Case (a) of [39, Theorem 1], we obtain

eventually. Then there exists an $s_{1}\in \lbrack s_{0}, \infty)_{{\mathbb{T }}}$ such that for $s\in$ $[s_{1}, \infty)_{{\mathbb{T}}},$

In accordance with the proof of Theorem 2.1, Case (b), we achieve that

Since $\left[r_{1}(s)\Phi \left(\varkappa ^{\Delta }(s)\right) \right] ^{\Delta } < 0,$ we obtain

Hence,

which implies

Therefore, (3.2) becomes

By integrating (1.1) and by the facts that

we obtain

that is,

In this case, we have

Substituting (3.7) into (3.5), we conclude that

Integrating (3.8) from $s$ to $v$, we obtain

As a result of $\varpi > 0$ and $\varpi ^{\Delta } < 0$ and assuming $v\rightarrow \infty$, we obtain

By multiplying each side of (3.9) by $\tilde{Q}\left(s\right)$, we obtain

For any $\varepsilon \in \left(0, 1\right)$, there is an $s_{2}\in \lbrack s_{1}, \infty)_{\mathbb{T}}$ such that, for $s\in \lbrack s_{2}, \infty)_{ \mathbb{T}}$,

where

in view of (2.2) and (3.6). From (3.10) and (3.11), we infer that

due to $\tilde{Q}\left(s\right) \rightarrow \infty$ as $s\rightarrow \infty$. Taking the $\liminf$ of (3.12) as $s\rightarrow \infty$, we obtain

Since $\varepsilon > 0$ is arbitrary, we see that

which is a contradiction to (3.1).

Theorem 3.2. If for sufficiently large $T\in \lbrack s_{0}, \infty)_{{\mathbb{T} }},$

where

then Eq (1.1) is oscillatory.

Proof. Suppose that $\varkappa$ is a nonoscillatory solution of (1.1) on $[s_{0}, \infty)_{{\mathbb{T}}}$. Without loss of generality, let $\varkappa (s) > 0$ on $[s_{0}, \infty)_{{\mathbb{T}}}$. By applying the same method as in the proof of Case (a) of [39, Theorem 1], we obtain

eventually. Then there exists an $s_{1}\in \lbrack s_{0}, \infty)_{{\mathbb{T }}}$ such that for $s\in$ $[s_{1}, \infty)_{{\mathbb{T}}},$

In accordance with the proof of Theorem 3.1, Case (b), we conclude that

and

Therefore,

Consequently,

which contradicts (3.13). This completes the proof.

Example 3.1. Consider the half-linear sublinear advanced dynamic equation

where $\tilde{\beta} > 0\ $is a constant. Here,

Thus,

By application of Theorem 3.2, if $\tilde{\beta} > 1$, then Eq (3.14) is oscillatory.

4.   Discussion and Conclusions

(1) The results in this paper presented are applicable across all time scales without any restrictive conditions, including ${\mathbb{T}} = \mathbb{R},$ ${\mathbb{T}} = \mathbb{N}$, and ${\mathbb{T}} = q^{\mathbb{N} _{0}}: = {\mathbb{\{}}q^{n}:$ $n\in {\mathbb{N}}_{0}$ for $q > 1\}$.

(2) These results, unlike previous findings ^{[2,26,27,28,29,31,33,34,35,36,37,38]}, do not require a condition (1.3) (the canonical case), thereby addressing an open problem noted in several papers ^{[16,33,35,39,40]}.

(3) Our results extend related contributions to the second-order dynamic equations for both cases $\mathbb{\zeta }(s)\leq s$ and $\mathbb{ \zeta }(s)\geq s$ on $[s_{0}, \infty)_{{\mathbb{T}}}$; see the following details:

(ⅰ) Criterion (2.1) reduces to (1.23) in the case where $\kappa = 1$ and $\mathbb{\zeta } (s)\leq s$;

(ⅱ) Criterion (2.12) becomes (1.24) in the case when $\kappa = 1$ and $\mathbb{\zeta }(s)\leq s$;

(ⅲ) Criterion (3.1) reduces to (1.25) assuming that $\kappa = 1$ and $\mathbb{\zeta }(s)\geq s$;

(ⅳ) Criterion (3.13) becomes (1.26) under the assumption that $\kappa = 1$ and $\mathbb{\zeta }(s)\geq s$.

(4) It would be of interest to establish Hille-type and Ohriska-type oscillation criteria for the second-order half-linear noncanonical dynamic equation (1.1) when $\kappa > 0$.

Author contributions

Taher S. Hassan: writing—original draft, Investigation, writing—review editing, and Supervision; Amir Abdel Menaem: Formal analysis, Resources; Mouataz Billah Mesmouli: Formal analysis, Resources; Wael W Mohammed: Formal analysis, Resources; Ismoil Odinaev: Formal analysis, Resources; Bassant M. ElMatary: writing—review editing, Validation, and Investigation. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This research has been funded by Scientific Research Deanship at University of Ha'il - Saudi Arabia through project number RG-23 097.

Conflict of interest

The authors declare there are no conflicts of interest.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.