In recent years, scholars have shown a growing interest in studying the oscillatory behavior of solutions to second-order differential and dynamic equations. This attention is motivated by the significance of such behavior in real-life applications, such as steam turbine regulation, neural networks, and governing equations which describe the temporal variation of hormones; see [1–9]. Moreover, there exist numerous sophisticated equations that serve as applications or direct representations of problems reliant on both present and future rates of change for further applications in science and technology; see [10]. An advanced argument is characterized by its ability to characterize the impact of a hypothetical future acts. It is commonly observed in phenomena such as population dynamics and economic difficulties.

In this study, we discuss the oscillatory behavior of second-order non-linear neutral differential equations with mixed non-linear neutral terms of the form

and

where Φ*(s)=|s|*-1for*>0 and ε = ±1. Throughout this article, we assume that the following hypotheses hold

By a solution of Equation (1), we mean a non-trivial function x ∈ C([T_x, ∞), ℝ), T_x ≥ ζ₀, which has the properties z∈C1([Tx,∞),ℝ), a(z′)α∈C([Tx,∞),ℝ) and satisfies Equation (1) on [T_x, ∞). Our attention is restricted to those solutions x(ζ) of Equation (1) satisfying sup {|x(ζ)| : ζ ≥ T} > 0 for all T ≥ T_x. We assume that Equation (1) possesses such a solution. A solution of Equation (1) is called oscillatory if it has arbitrarily large zeros on [T_x, ∞); otherwise, it is termed non-oscillatory. Equation (1) is said to be oscillatory if all solutions are oscillatory.

The literature extensively addresses the oscillation and asymptotic behavior of solutions across various classes of delay and advanced differential equations. Recently, Tunç and Özdemir [11] introduced novel sufficient conditions for solutions to second-order half-linear differential equations.

where g₁(ζ) < ζ, g₂(ζ) > ζ, p₁(ζ) ≥ 0 and p₂(ζ) ≥ 1, p₂(ζ) ≢ 1 for large ζ. In the study by Agwa et al. [12], developed new oscillation criteria for the second-order non-linear neutral dynamic equation with mixed arguments

in the canonical and non-canonical cases. In the study by Moaaz et al. [9], investigated the oscillatory and asymptotic properties of a specific class of delay differential equations of mixed neutral type

with the non-canonical operator.

Very recently, in the study by Grace et al. [13], introduced novel criteria for the oscillation of second-order non-linear differential equations featuring mixed non-linear neutral terms and mixed deviating arguments

Upon reviewing the literature, it becomes evident that numerous results pertain to the oscillation of second-order differential equations with linear neutral terms. In contrast, there is a paucity of articles dedicated to exploring differential equations featuring sublinear or superlinear neutral terms, as evidenced in the studies by Agarwal et al. [14], Bohner et al. [15], Džurina et al. [16], Grace et al. [17], Lin and Tang [18], and Muhib et al. [19]. Furthermore, there is a notable scarcity of results concerning equations incorporating both sublinear and superlinear neutral terms. Motivated by this gap, this study aims to establish oscillation criteria for a specific class of second-order mixed functional differential equations (Equation 1) characterized by sublinear and superlinear neutral terms under the conditions where either ε = −1 or ε = +1, η differs from 1. To the best of our knowledge, there does not appear to be any oscillation results for Equation (1) when η ≠ 1. Additionally, it is worth noting that the findings presented in this study are novel even in the linear case.

In discussing oscillation results for Equation (1), we assume that any functional inequality holds for all large ζ. To lay the groundwork for our key results, we first define a few lemmas. To keep things simple, we will use these symbols

Lemma 1. Baculíková [20] and Philos [21] Let q₁:[ζ₀, ∞) → (0, ∞), g₁:[ζ₀, ∞) → ℝ and f : ℝ → ℝ are continuous functions, f is non-decreasing with xf(x) > 0 for x ≠ 0 and g₁(ζ) → ∞ as ζ → ∞. If

Lemma 2. Hardy et al. [22] If X, Y ≥ 0, then

and

where equalities hold if and only if X = Y.

Lemma 3. Hardy et al. [22] If B, L be non-negative numbers and if m, n > 1 are real numbers such that 1n+1m=1, then

Equality holds if and only if Bⁿ = L^m.

Lemma 4. Bohner et al. [23] Let G(U)=AU-B(U-R)μ+1μ, where B > 0, A and B are constants, μ is a ratio of odd positive integers. Then, G attains its maximum value at U*=R+(μA/((μ+1)B)μ) and

In this section, we investigate the oscillatory characteristics of solutions to Equation (1) under the conditions where ε = −1, τ₁(ζ) = τ₂(ζ) = τ(ζ), and either of the following conditions is satisfied either λ < 1 and ν > 1, or λ < ν ≤ 1.

Theorem 1. Suppose τ₁(ζ) = τ₂(ζ) = τ(ζ), λ < 1, ν > 1, and conditions (H₁)-(H₃) are satisfied. Additionally, assume that

and there exists a non-decreasing function ϱ(ζ) ∈ C([ζ₀, ∞), ℝ) such that

for ζ ≥ ζ₀. If there exist constant c₂ ∈ (0, 1) such that the first-order differential equations

are oscillatory for sufficiently large ζ₂ > ζ₁ ≥ ζ₀, then every solution of Equation (1) is oscillatory.

Proof. Consider a non-oscillatory solution x(ζ) of Equation (1). Without loss of generality, let us assume that x(ζ) is eventually positive with limζ→∞x(ζ)≠0 for ζ ≥ ζ₀. Hence, x(ζ) > 0, x(τ(ζ)) > 0, x(σ(ζ)) > 0, and x(φ(ζ)) > 0 for ζ ≥ ζ₁ ≥ ζ₀. It is worth noting that the proof for the case where x(ζ) is eventually negative follows a similar path and is therefore omitted. Now, from Equation (1), it can be inferred that

Thus, (a(ζ)(z′(ζ))^α) is decreasing and eventually of one sign. In other words, there exists ζ₂ ≥ ζ₁ such that z′(ζ) > 0 or z′(ζ) < 0 for ζ ≥ ζ₂. Therefore, we will distinguish the following four cases:

Initially, let us examine Case (I). Depending on the fact that z′(ζ) < 0 and (a(ζ)(z′(ζ))^α)^′ < 0, in accordance with Equation (2), we conclude that limζ→∞z(ζ)=-∞, contradicting the established condition z(ζ) > 0. Therefore, Case (I) is deemed impossible. For the second case, based on the definition of z(ζ), we have

Utilizing inequality (5) with U = x^η(ζ), A = 1, B = p₂(ζ), R = x^η(ζ)−x(τ(ζ)), and μ=1ν-1, we deduce

This with Equation (11) leads to

Applying the inequality (4) to [p1(ζ)xλ(τ(ζ))-x(τ(ζ))] with ξ = λ, X = x(τ(ζ)) and Y=1(λp1(ζ))1/(λ-1), we obtain

It is deduced from Equation (12) that

As z(ζ) is positive and increasing, there exists a constant c > 0, ensuring z(ζ) ≥ c for ζ ≥ ζ₂. This implies

Therefore, based on Equations (6, 13), a constant c₁ ∈ (0, 1) exists, such that

Combining Equation (14) with Equation (1), we get

Consequently, since z(ζ) is increasing,

This with Equation (15) leads to

Letting ϝ(ζ): = a(ζ)(z′(ζ))^α, we have

By applying Lemma 1 (i), the differential Equation (8) associated with the inequality (17) reveals the existence of a positive solution, leading to a contradiction.

Now, suppose the case (III) holds, we have z(ζ) < 0 and z′(ζ) > 0. Let

it follows that

It can be inferred from Equation (1) that

Since z̄′(ζ)<0, we have

Letting m = τ⁻¹(σ(ζ)) and n = ϱ(ζ), we conclude

Combining Equations (19, 20) yields

That is as follows:

where Ω′(ζ):=(a(ζ)(z̄′(ζ))α). By applying Lemma 1 (i), we conclude that the Equation (9) corresponding to the inequality (21) also possesses a positive solution, leading to a contradiction.

Ultimately, examine the case (IV) wherein z(ζ) < 0 and z′(ζ) < 0. As in the preceding case, let us take z̄(ζ)=-z(ζ). Utilizing Equation (18) in conjunction with Equation (1), we obtain

Integrating from ϱ(ζ) to ζ, we get

which has a positive solution z̄(ζ). It follows from Lemma 1 (ii) that the Equation 10 corresponds the inequality (22) also has a positive solution. This completes the proof.

Corollary 1. Assume that τ₁(ζ) = τ₂(ζ) = τ(ζ), λ < 1, ν > 1, and (H₁)-(H₃) hold. Furthermore, assume that Equation (6) holds and there exists a non-decreasing function ϱ(ζ) ∈ C([ζ₀, ∞), ℝ) satisfies Equation (7). If

and

hold for sufficiently large ζ₂ > ζ₁ ≥ ζ₀, then every solution of Equation (1) is oscillatory.

Theorem 2. Assume that τ₁(ζ) = τ₂(ζ) = τ(ζ), λ < 1, ν > 1, β > αη, and (H₁)-(H₃) hold. Furthermore, assume that Equation (6) holds and there exists a non-decreasing function ϱ(ζ) ∈ C([ζ₀, ∞), ℝ) satisfies Equation (7). If there exist function ρ(ζ)∈C1([ζ0,∞),(0,∞)) and c_* > 0 such that

and the differential Equations (9) and (10) are oscillatory for sufficiently large ζ₁ ≥ ζ₀, then every solution of Equation (1) is oscillatory.

Proof. Let x(ζ) be a non-oscillatory solution of Equation (1). Following the same proof of Theorem 1, we arrive at Equation (16). Define the Riccati transformation by

Then, ω(ζ) > 0 and

Using the monotonicity of z(ζ) with Equations (15, 16), we get

Given that z(ζ) > 0 and z′(ζ) > 0, it follows that there exists a positive constant c_* > 0 such that z(ζ) ≥ c_* for ζ ≥ ζ₂. Consequently, we have

Applying the inequality (5), we get

Integrating from (ζ₃ > ζ₂) to ζ, we obtain

This contradiction contradicts Equation (23). By completing the proof for the two cases (III) and (IV) in a manner similar to the proof of Theorem 1, we arrive at the conclusion of the theorem.

Theorem 3. Assume that τ₁(ζ) = τ₂(ζ) = τ(ζ), λ < ν ≤ 1, and (H₁)-(H₃) are satisfied. Additionally, assume that all other conditions of Theorem 1 are met, replacing Equation (6) with

then the conclusion of Theorem 1 remains valid.

Proof. Let x(ζ) be a non-oscillatory solution of Equation (1). For convenience, assume, without loss of generality, that x(ζ) becomes eventually positive. Consequently, there exists ζ₁ ≥ ζ₀ such that x(ζ) > 0, x(τ(ζ)) > 0, x(σ(ζ)) > 0, and x(φ(ζ)) > 0 for ζ ≥ ζ₁. Employing a similar approach as in the proof of Theorem 1, we analyze the four cases (I)–(IV) for z(ζ).

Firstly, consider case (I). As demonstrated in the proof of case (I) for Theorem 1, it is evident that this case is not feasible.

Next, let us examine case (II). Expressing [p1(ζ)xλ(τ(ζ))-p2(ζ)xν(τ(ζ))] in the form

Applying inequality (3) with n=νλ, B = x^λ(τ(ζ)), L=λνp1(ζ)p2(ζ) and m=νν-λ, we get