Second-order general Emden-Fowler differential equations of neutral type: Improved Kamenev-type oscillation criteria

1.   Introduction

Studying the properties of solutions to differential equations (DEs) is the main aim of qualitative theory. This theory is concerned with investigating features such as stability, periodicity, bifurcation, oscillation, synchronization, etc. This theory emerged from researchers' attempts to obtain sufficient information about the nonlinear models that appear when modeling biological, physical, and other phenomena; see ^[1,2,3,4]. The qualitative theory was also extended to include functional, fractional, and partial differential equations.

A functional differential equation (FDE) is a DE with a deviating argument. That is, the FDE contains a dependent variable and some of its derivatives for various argument values. These equations are distinguished by the fact that they take into account the previous and subsequent times when studying any phenomenon or system. In fact, the majority of studies on FDEs until the time of Volterra ^[5] focused only on some properties of some very special equations. A few books on FDEs were published. In the latter portion of the 1940s and the initial phase of the 1950s. Mishkis ^[6] laid the groundwork for a broad theory of linear systems in his book by introducing a general class of delayed-argument equations. In 1954, Bellman and Danskin ^[7] noted the many uses of equations containing past information in disciplines such as economics and biology. Bellman and Cook ^[8] provide a more comprehensive treatment of the theoretical framework concerning linear equations and the fundamentals of stability theory.

Oscillation theory, as part of qualitative theory, revolves around creating the criteria for the presence of oscillatory and non-oscillatory solutions to FDEs, investigating zero-order distribution laws, estimating the number of zeros in a certain period of time and the distance between neighboring zeros, and other basics. Oscillation theory has now become an important mathematical instrument for numerous advanced fields and technological applications. Obtaining oscillation conditions for specific FDEs has been a widely studied area in the last few decades; see ^{[9,10,11,12,13]}.

1.1.   FDEs with a neutral delay argument

Delay DEs that have the derivative of the solutions of the highest order with and without delay are known as neutral differential equations. These equations are apparent in the investigation of oscillatory masses and the modeling of electrical circuitry with ideal transmission lines (see ^[14]). Understanding the qualitative characteristics of delay differential equations is becoming more and more important as new models and developments in biology, economics, physics, and engineering keep appearing.

In the canonical scenario, we find new conditions to test the oscillation of nonlinear neutral FDEs of second order. Namely, we consider the FDE

where $t\in \mathbb{I}: = \left[t_{0}, \infty \right)$, $\alpha \in \mathbb{Q} ^{+}$ is a ratio of odd numbers, and

In our study, we consider the following hypotheses:

${\rm{A1}}:$ $a\in {\mathbf C}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$, $q\in {\mathbf C}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$, and $p\in {\mathbf C}\left(\mathbb{I}, \left[0, p_{0}\right] \right),$ where $p_{0} < 1$.

${\rm{A2}}:$ $h,$ $g\in {\mathbf C}\left(\mathbb{I}, \mathbb{R} \right)$, $h \leq t$, $g \leq t$, $g^{\prime } \geq 0$, $\lim_{t\rightarrow \infty }h\left(t\right) = \infty$, and $\lim_{t\rightarrow \infty }g\left(t\right) = \infty$.

${\rm{A3}}:$ $\psi \in {\mathbf C}^{1}\left(\mathbb{R}, \left[m, M\right] \right)$, where $0 < m \leq M$, and $\kappa = m^{-1/\alpha }M^{1/\alpha }$.

${\rm{A4}}:$ $F\in {\mathbf C}^{1}\left(\mathbb{R}, \mathbb{R} \right)$, $\varkappa\, F\left(\varkappa\right) > 0$ for $\varkappa\neq 0$, $F^{\prime }\left(\varkappa\right) \geq 0$, and $-F\left(-\varkappa w\right) \geq F\left(\varkappa w\right) \geq F\left(\varkappa\right) F\left(w\right)$ for $\varkappa w > 0$.

For a solution of Eq (1.1), we define a function $x\in {\mathbf C} ^{1}\left(\left[t_{x}, \infty \right), \mathbb{R} \right),$ $t_{x}\in \mathbb{I}$, which has the properties: $a\cdot \psi \cdot \left(z^{\prime }\right) ^{\alpha }\in {\mathbf C}^{1}\left(\left[t_{x}, \infty \right), \mathbb{R} \right)$, $\sup \left\{ \left\vert x\left(t\right) \right\vert :t\geq t_{\ast }\right\} > 0$ for all $t_{\ast }\geq t_{x}$, and satisfies Eq (1.1) on $\left[t_{x}, \infty \right)$. If a solution $x$ of FDE (1.1) has arbitrarily large zeros, it is referred to as oscillatory; if not, it is referred to as non-oscillatory. The FDE (1.1) is said to be in the canonical case if

1.2.   Literature review

We can divide previous related works in the literature into two main parts. The first section consists of studies that are concerned with using different techniques to study oscillation, and the second section consists of studies that focus on improving the relationships and inequalities used in studying oscillation.

Oscillation criteria are often only sufficient conditions to test the oscillation and are not necessary. Therefore, finding different techniques and methods for studying oscillation is inevitable and necessary for the purpose of application to the largest area of special cases and also to get rid of some of the restrictions that may be imposed by some techniques.

In 2000, Džurina and Mihalíková ^[15] used Riccati substitution to provide some oscillatory conditions for the solutions of the equation

with $h(t) = t-k$, $k > 0$ and $p(t) = -p_{0} < 1$. Han et al. ^[16] acquired some Kamenev-type oscillation conditions for (1.3) when $h^{\prime }(t) = h_{0} > 0$. By comparison method, Džurina ^[17] studied the oscillation of (1.3) with an advanced neutral term, i.e., when $h\left(t\right) \geq t$.

Şahiner ^[18] presented Philos-type conditions for oscillation of (1.1) when $\alpha = 1$, $F^{\prime }\left(\upsilon\right) \geq K > 0$, and $\pm F\left(\pm \upsilon w\right) \geq L F\left(\upsilon\right) F\left(w\right)$, for all $\upsilon w > 0$ and for some $L > 0$.

Theorem 1. (^[18], Theorem 2.1) Assume that

By $P\in \Im$, we mean that the continuous real-valued function $P$ with domain $D$ belongs to class $\Im$ and satisfies

If there is a function $\rho \in {\mathbf C}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$ such that

then all solutions of (1.1) are oscillatory, where $Q : = q \left[1-p\left(g \right) \right]$, and

Putting $P\left(\varkappa, \varsigma \right) = \left(\varkappa-\varsigma \right) ^{n},$ $n > 1$, in the results of Theorem 1.1, we obtain a Kamenev-type criterion of Eq (1.1).

On the other hand, in the most recent period, there was a significant surge in research activity concentrated on improving the inequalities that are used in the study of oscillation. One of the influential relationships in investigating the oscillation of neutral equations involves examining the interplay between $x$ and its $z$, as well as the relationship between the corresponding function and its derivatives.

The classical substitution $x > \left(1-p\right) z$ is often used in studying the oscillation of neutral equations. Moaaz et al. ^[19] devised improved conditions for the oscillation of

They provided the following relationship as an improvement on the classical relationship:

for $p_{0} > 1$ and $\nu$ is an even natural number, and

for $p < 1,$ and $\nu$ is an odd natural number, where $h^{\left[\pm l\right] } = h^{\pm 1}\left(h^{\left[\pm \left(l-1\right) \right] } \right)$, for $l = 1, 2, ...$, and

For the non-canonical situation, Hassan et al. ^[20] used the improved relationship

when $z^{\prime } < 0$.

A research study for the fourth-order DE

Moaaz et al. ^[21] devised improved substitution for $x$ by $z$ in all cases of positive solutions.

Lemma 1. (^[21], Lemma 1) Let $x$ be a solution of (1.1) and $x > 0$ eventually. Then,

for any integer $m\geq 0$, where

See ^[22,23,24] for other intriguing findings on the oscillatory nature of solutions to third- and fourth-order DEs that have been published more recently.

The oscillation criteria of neutral DEs depend on several factors:

$–$ The substitution $x$ by $z$;

$–$ The monotonic and asymptotic features of non-oscillatory solutions;

$–$ The technique used to obtain the oscillation criteria.

Improving any of these factors directly affects the oscillation criteria. In our paper, we test the extent to which the oscillation criteria are affected by improving the relationship between $x$ and $z$. We used a well-known technique that produces criteria known as Kamenev criteria; but the improvement lies in the new relationships used in the study.

1.3.   The function class $\mathbb{Y}$

Here, we define a class of functions to obtain the oscillation condition of the Kamenev-type criteria. Suppose that

A $\varphi \in {\mathbf C}\left(\mathbb{K}, \mathbb{R} \right)$ belongs to the class $\mathbb{Y}$ ($\varphi \in \mathbb{Y}$), if

$({\rm{i)}}$ $\varphi \left(u, u, w\right) = 0$, $\varphi \left(u, w, w\right) = 0$, and $\varphi \left(u, s, w\right) \neq 0$ for $w < s < u$.

$({\rm{ii}})$ $\varphi$ has the partial derivative $\partial \varphi /\partial v$ on $\mathbb{K}$ and $\partial \varphi /\partial v$ is locally integrable with respect to $v$ in $\mathbb{K}.$

For any function $f\in {\mathbf C}^{1}\left(\mathbb{I}, \mathbb{R} \right)$, we define the following operator:

for $w\leq v\leq u$. Moreover, we define the function $\mu \left(u, v, w\right)$ by

for all $w < v < u$. We notice that $\mathcal{T}\left[\cdot \,; u, w\right]$ is linear. Using integration by parts, we find that

2.   Main results

Here, we investigate the improved monotonic features of non-oscillatory solutions and then establish new oscillation conditions for (1.1). For simplicity, the class of all positive non-oscillatory solutions of (1.1) is denoted by the symbol $\mathbb{S}$. Also, we assume that $\gamma = \frac{1}{\alpha }\left(\alpha /\left(\alpha +1\right) \right) ^{\alpha +1}$,

for $t\geq t_{\ast }$, and

2.1.   Monotonic properties and preliminary results

First, we infer the monotonic behavior of positive solutions in the next lemma.

Lemma 2. If $x \in \mathbb{S}$, then the corresponding function $z$ of $x$ conforms to $z > 0$, $z^{\prime } > 0$, and $\left(a \psi \left(x \right) \left[z^{\prime } \right] ^{\alpha }\right) ^{\prime } < 0$, eventually.

Proof. Suppose that $x \in \mathbb{S}$. From (A2), there exists a $t_{1}\in \mathbb{I}$ whereby $\left(x\circ h\right) \left(t\right)$ and $\left(x\circ g\right) \left(t\right)$ are positive for $t\geq t_{1}$. Therefore, $z\left(t\right) > 0$. Using (A4), we have that $F\left(x\left(g\left(t\right) \right) \right) > 0$. Hence, (1.1) becomes

Thus, we find that $a \psi \left(x \right) \left[z^{\prime } \right] ^{\alpha }$ has a constant sign. This is the same as stating that $z^{\prime } > 0$ or $z^{\prime } < 0$ for $t\geq t_{2}$, where $t_{2}$ is large enough. But, when $z^{\prime }\left(t\right) < 0$, this case contradicts (1.2), as displayed next:

Let $z^{\prime }\left(t\right) < 0$ for $t\geq t_{2}$. Then

Hence,

which with (A3) gives

Thus,

But condition (1.2) results in $z\left(t\right) \rightarrow -\infty$ as $t\rightarrow \infty$, a contradiction.

Consequently, the proof ends. □

Lemma 3. Assume that $x \in \mathbb{S}$. Then

Moreover,

is nonincreasing, for $t\geq t_{1}$.

Proof. Let $x \in \mathbb{S}$. From (A2), there is a $t_{1}\in \mathbb{I}$ such that $\left(x\circ h\right) \left(t\right) > 0$ and $\left(x\circ g\right) \left(t\right) > 0$ for $t\geq t_{1}$. Using Lemma 2, we obtain

Hence, for any $t\geq t_{1}$,

Since $A_{t_{1}}^{\prime }\left(t\right) \geq 0$, $A_{t_{1}}\left(t_{1}\right) = 0$ and $A_{t_{1}}\left(\infty \right) = \infty$, there is a $t_{2}\geq t_{1}$ such that $A_{t_{1}}\left(t_{2}\right) = 1$. Now, from (2.1), we obtain

Consequently, this proof ends. □

Next, we improve the oscillation results by deriving a new substitution for $x$ by $z$.

Lemma 4. Assume that $x \in \mathbb{S}$. Then (1.1) can be articulated as

for $t > t_{1}$ and any integer $m\geq 0$.

Proof. Let $x \in \mathbb{S}$. From (A2), there is a $t_{1}\in \mathbb{I}$ such that $\left(x\circ h\right) \left(t\right) > 0$ and $\left(x\circ g\right) \left(t\right) > 0$ for $t\geq t_{1}$. Using Lemma 1, we obtain

From Lemmas 2–3, we find that

Then,

and

Using previous relationships in (2.3), we arrive at

which with (1.1) gives

where assumption (A4) was used.

Therefore, the proof ends. □

The following theorem transforms the studied equation into the form of a Riccati inequality, or what is known as the Riccati technique.

Theorem 2.1. Assume that $x \in \mathbb{S}$, and there is a constant $L > 0$ such that

for $u\neq 0$. If we define the function $\omega$ as

then $\omega$ satisfies

where $\upsilon \in {\mathbf C}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$.

Proof. Assume that $x \in \mathbb{S}$. By differentiating $\omega$, we find

which with (2.2) yields

Since $\left(a\left(t\right) \psi \left(x\left(t\right) \right) \left[z^{\prime }\left(t\right) \right] ^{\alpha }\right) ^{\prime } < 0$ and $g\left(t\right) \leq t$, we have

and then

Combining (2.7) and (2.8), we arrive at

Using (2.4), we find

Therefore, the proof ends. □

2.2.   Kamenev-type oscillation criteria

Here, we employ the previous results to derive a novel criterion that tests whether all solutions are oscillatory.

Theorem 3. Assume that $\varphi \in \mathbb{Y}$, $\upsilon \in {\mathbf C} ^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$, and there is a constant $L > 0$ such that (2.4) holds. If

then (1.1) oscillates.

Proof. Assuming the opposite of what is required means that there is a non-oscillatory solution to the studied equation, and this necessarily leads to guaranteeing the existence of a solution that eventually becomes positive for this equation. Let it be $x$. From Theorem 2.1, if we define the function $\omega$ as in (2.5), then $\omega$ satisfies (2.6).

Next, applying the operator $\mathcal{T}\left[\cdot \,; t, l\right]$ to (2.6), we obtain

Using the property (1.5), we obtain

By simple calculation, we find that the function $H\left(\omega \right) = c_{1}\omega -c_{2}\omega ^{1+1/\alpha }$, where $c_{1},$ $c_{2} > 0$, has the maximum

at $\omega _{_{\max }} = \left(\alpha c_{1}/\left(\left(\alpha +1\right) c_{2}\right) \right) ^{\alpha }$. Using (2.12), (2.11) becomes

Taking the super limit for (2.13), we have

This contradicts assumption (2.10).

Therefore, the proof ends. □

Corollary 1. Assume that there is a constant $L > 0$ such that (2.4) holds. If there are $\upsilon,$ $\eta \in {\mathbf C}^{1}\left(\mathbb{I}, \mathbb{R} ^{+}\right)$ and $\varrho,$ $k > \max \left\{ 1/2, \alpha \right\}$ such that

for $l\geq t_{0}$, then all solutions of (1.1) are oscillatory, where

Proof. By choosing

we find

Using Theorem 3, condition (2.10) reduces to (2.14).

Therefore, the proof ends. □

Corollary 2. Assume that there is a constant $L > 0$ such that (2.4) holds. If $g\left(t\right) = \lambda t,$ $\lambda \in \left(0, 1\right]$, $\alpha \in \mathbb{Z} ^{+}$, $a\left(t\right) \geq 1$, and there is $\varrho,$ $k > \alpha$ such that

for $l\geq t_{0}$, then all solutions of (1.1) are oscillatory, where $\beta \left(\cdot, \cdot \right)$ is the beta function.

Proof. Assume that

Then, we find that

Since $a^{\prime }\left(t\right) \geq 0$, we have that $a\left(\lambda s\right) /a\left(s\right) \leq 1$. Hence,

Let $w: = A_{t_{1}}\left(s\right) -A_{t_{1}}\left(l\right)$. Thus,

Now, as in the proof of Theorem 3, if we assume the contrary, then we arrive at (2.13). Under assumptions in (2.17), inequality (2.13) reduces to

which with (2.18) gives

Thus, we obtain

This contradicts assumption (2.16).

Therefore, the proof ends. □

In the following corollaries, we present an oscillation criterion for a special case of the studied equation, which is

It is easy to notice that the function $F\left(u\right) = u^{\alpha }$ satisfies (A4) and (2.4) with $L = \alpha$.

Corollary 3. If there are $\upsilon,$ $\eta \in {\mathbf C}^{1}\left(\mathbb{I }, \mathbb{R} ^{+}\right)$ and $\varrho,$ $k > \max \left\{ 1/2, \alpha \right\}$ such that

for $l\geq t_{0}$, then all solutions of (1.1) are oscillatory, where $H\left(t, s, l\right)$ is defined as in (2.15).

Corollary 4. If $g\left(t\right) = \lambda t,$ $\lambda \in \left(0, 1\right]$, $\alpha = 1$, $a\left(t\right) \geq 1$, and there is $\varrho > 1/2$ such that

for $l\geq t_{0}$, then all solutions of (1.1) are oscillatory.

2.3.   Examples and discussion

Example 1. Consider the delay equation

where $h_{0},$ $g_{0}\in \left(0, 1\right]$, and $q_{0} > 0$. We note that $\alpha = 1$, $a\left(t\right) = 1$, $p\left(t\right) = 1/2$, $h\left(t\right) = h_{0}t$, $g\left(t\right) = g_{0}t$, $q\left(t\right) = q_{0}/t^{2}$, $F\left(u\right) = u$, and $\psi \left(u\right) = 1/\left(1+\sin ^{2}u\right)$. It is easy to verify that

$\kappa = 2$, $A_{t_{0}}\left(\infty \right) = \infty$, $h^{\left[r\right] }\left(t\right) = h_{0}^{r}t$ for $r = 0, 1, ...$,

Now, we have

Hence, condition (2.20) becomes

Then, by using Corollary 4, all solutions of (2.21) are oscillatory if

Remark 1. Using Theorem 1.1 with $H\left(t, s\right) = \left(t-s\right) ^{2}$ and $\rho \left(t\right) = t^{2}$, we obtain that all solutions of (2.21) are oscillatory if

Applying to the special case when $g_{0} = 0.5$ and $h_{0} = 0.9$, we notice that criteria (2.22) and (2.23) lead to $q_{0} > 1.672$ and $q_{0} > 4$, respectively. Therefore, our results improve results in ^[18]. For example, we find that our results guarantee that all solutions of equation

oscillate while the criteria of ^[18] do not apply ($2 = q_{0}\ngtr \frac{ 2}{g_{0}} = 4$).

Figure 1 shows one of the numerical solutions to Eq (2.21).

Example 2. Consider the delay equation

where $h_{0},$ $g_{0},$ and $q_{0}$ are positive. We note that $\alpha = 1$, $a\left(t\right) = {\mathbf e}^{-t}$, $p\left(t\right) = p_{0}$, $h\left(t\right) = t-h_{0}$, $g\left(t\right) = t-g_{0}$, $q\left(t\right) = q_{0} {\mathbf e}^{-t}$, $F\left(u\right) = u$, and $\psi \left(u\right) = 1+ {\mathbf e}^{-u^{2}}$. It is easy to verify that $m = 1$, $M = 2$, $\kappa = 2$,

By choosing $\eta \left(t\right) = \upsilon \left(t\right) = e^{t}$ and $\varrho = k = 2$, we obtain

Thus,

which is fulfilled if

Then, by using Corollary 3, all solutions of (2.24) are oscillatory if (2.25) holds.

Remark 2. Recent results in papers ^{[26,27,28,29,30]} provided many improved criteria that test the oscillatory characteristics of second-order neutral DEs. However, these results fail to apply to Eqs (2.21) and (2.24), because these results only apply in the case of $\psi \left(u\right) = u$. Figure 2 shows one of the numerical solutions to Eq (2.24).

3.   Conclusions

The investigation into the oscillatory behavior of FDEs is affected by the accuracy of the relationships and inequalities used. In this article, we studied the oscillations of solutions of the class of FDEs of the neutral type (1.1). As an extension of the results in ^[21], we have derived a novel relation between $x$ and $z$. We used the Riccati approach to couple the studied equation with an inequality of the Riccati type. Then, we presented Kamenev-type criteria that ensure the oscillation of (1.1).

Our results—as shown in Remark 2—have the advantage of being applied to a more general class of second-order FDEs of the neutral type compared to the results in ^[26,27,28]. Our results also presented more sharp conditions in the oscillation test than the results that dealt with the same equation (see Remark 1).

It would be of interest to formally extend our findings to the noncanonical case ($A_{t_{\ast }}\left(\infty \right) < \infty$). Also, an interesting point, as a future work, is to obtain an improved relation between $x$ and $z$ without the need for the constraint $\psi \left(u\right) \geq m > 0$, which excludes a large class of bounded functions such as $\sin ^{2}u$, ${\mathbf e}^{-u^{2}}$ and $\frac{1}{ 1+u^{2}}$.

Use of AI tools declaration

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

Acknowledgements

The authors would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare there is no conflicts of interest.