On the Oscillation of Even-Order Nonlinear Differential Equations with Mixed Neutral Terms

Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia Gofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, 26649 Addis Ababa, Ethiopia Jabalia Camp, United Nations Relief andWorks Agency (UNRWA) Palestinian Refugee Camp, Gaza Strip Jabalia, State of Palestine Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza 12221, Egypt Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Industrial Engineering, OSTİM Technical University, Ankara 06374, Turkey Department of Mathematics, Atılım University, 0683 Incek, Ankara, Turkey


Introduction
Recently, numerous research studies have been carried out concerning the oscillatory behavior of the differential equations with a linear neutral term. Some previous notable studies include the investigation of even-order quasilinear neutral functional differential equations' oscillation (DEqsOs) [1] (see also [2][3][4]), 3rd-order neutral delay dynamic equations on time scales [5], 2nd-order nonlinear neutral delay differential equation solutions' asymptotic behavior [6] (see also [7]), and 2nd-order superlinear Emden-Fowler neutral DEq-sOs [8]. On one hand, higher-order neutral delay DEqsOs was studied in [9]. On the other hand, even-order of DEqsOs and nonlinear neutral DEqsOs with variable coefficients were investigated in [10,11], respectively. A neutral functional delay differential equation was investigated in the sense of fractional calculus [12] (for more information about the applications of fractional calculus, refer to [13]). However, differential equations' oscillation with nonlinear neutral terms has been rarely studied in literature. For the case of differential equations with a sublinear neutral term [14][15][16], Grace et al. [17] proposed differential equations with both sublinear and super-linear neutral terms, where a second-order half-linear differential equation of the following form was investigated: where n > 0 is an even integer, and (i) α, β, γ, and δ are the ratios of two positive odd integers with α ≥ 1 (ii) p 1 , p 2 , q : ½t 0 , ∞Þ ⟶ ℝ + are continuous functions (iii) τ k : ½t 0 , ∞Þ ⟶ ℝ are continuous functions; τ k ðtÞ ≤ t and τ k ðtÞ ⟶ ∞ as t ⟶ ∞ for k = 1, 2 (iv) hðtÞ = τ −1 2 ðτ 1 ðtÞÞ ≤ t, and hðtÞ ⟶ ∞ as t ⟶ ∞ Let us suppose that for which A continuous function x satisfying Equation (1) on ½t * , ∞Þ, t * ≥ t 0 , is said to be a solution of Equation (1) on ½t * , ∞Þ where yðtÞ is defined in (2). We only consider those solutions x of (1) which satisfy A solution x of (1) is said to be oscillatory if there exists a sequence fξ n g such that xðξ n Þ = 0 and Otherwise, it is called nonoscillatory. Equation (1) is said to be an oscillatory (or nonoscillatory) equation if all its solutions are oscillatory (or nonoscillatory).
According to the best of our knowledge, the higher-order differential equations with nonlinear neutral terms have not been studied yet in any other research work. Inspired by the above studies, the oscillation of the proposed differential equations in (1) is investigated in this paper. New oscillation results for Equation (1) are obtained by comparing with the first-order delay differential equations whose oscillatory characters are well-known via an integral criterion. All results in this work are totally new, and more general oscillation results can be obtained by extending our obtained results to more general differential equations with both sublinear and super-linear neutral terms. As a result, a special research interest is hopefully stimulated from our work for possible general investigation of various neutral differential equations' classes, particularly the ones with sublinear and/or superlinear neutral terms.
This article consists of the following sections: our main results are investigated in Section 2. Two illustrative examples are given in Section 3. Then, a short conclusion of our work is provided in Section 4.
To obtain our results, the following lemma is needed: . Let X and Y be two nonnegative real numbers. Then, the following inequality is obtained: where equality holds if and only if X = Y.

Journal of Function Spaces
Proof. Without loss of generality, the solution xðtÞ of Equation (1) is assumed to be positive and xðτ 1 ðtÞÞ > 0 for t ≥ t 1 for some t 1 ≥ t 0 (i.e., a nonoscillatory solution). From Equation (1), we have the following: xðτ 2 ðtÞÞ > 0and Hence, rðtÞ½y ðn−1Þ ðtÞ α is nonincreasing with a constant sign. Namely, y ðn−1Þ ðtÞ > 0 or y ðn−1Þ ðtÞ < 0 for t ≥ t 2 for some t 2 ≥ t 1 , so the following four cases are examined separately: Let us first consider the case (a). Since y ðn−1Þ ðtÞ < 0 for t ≥ t 2 , we obtain the following: for some positive constant c, i.e., for t ≥ t 2 . Integrating the last inequality ðn − 1Þ-times and by condition (3), we conclude that which is a contradiction. Next, let us consider the case (b). It is obvious that From Equation ((2)) of yðtÞ, i.e., we obtain the following: If we apply the first inequality in (7) with λ = δ > 1, X = p 1/δ 2 ðtÞxðτ 2 ðtÞÞ, and then we have In a similar manner, by applying the second inequality in (7) with λ = β < 1, X = p 1/β 1 ðtÞxðτ 2 ðtÞÞ, and we obtain the following: By using (21) and (23), (25) turns out that Since yðtÞ in nondecreasing, we have the following: yðtÞ ≥ c 0 for some c 0 > 0. Hence, (26) turns that Now, we see from (9) and (27) for some c 1 ∈ ð0, 1Þ. (28) implies that Equation (1) turns to be There exists a constant θ 0 ∈ ð0, 1Þ such that for t ≥ t 1 (see [16,18,19]). By setting wðtÞ = rðtÞ ½y ðn−1Þ ðtÞ α , we obtain the following: By using (31), (29) turns that 3 Journal of Function Spaces From Corollary 1 in [20], it can easily be concluded that there exists a positive solution wðtÞ of Equation (10) with lim t⟶∞ wðtÞ = 0, which contradicts the fact that Equation (10) is oscillatory. Now, let us consider the cases when yðtÞ < 0 for which implies On the other hand, we obtain the following: Now, let us consider the case (c). Clearly, we see that v ðn−1Þ ðtÞ ≤ 0 and either v ′ ðtÞ < 0 or v ′ ðtÞ > 0 for t ≥ t 1 . First, we assume that v ′ ðtÞ < 0 for t ≥ t 1 . It is easy to see that (refer to [18]). Now, we may express for t 1 ≤ u 1 ≤ u 2 . By taking u 1 = μ 2 hðtÞ and u 2 = μ 3 hðtÞ for t ≥ t 1 in inequality (39), we see that By using (40), (38) turns out to be By setting VðtÞ ≔ −rðtÞ½v ðn−1Þ ðtÞ From (42) and (31), we obtain the following: which implies The proof can be easily completed by following the same steps as we did for the case (a) and hence is omitted.
Next, we assume that v′ðtÞ > 0 for t ≥ t 1 . Clearly, we have the following: There exists a constant θ 1 ∈ ð0, 1Þ such that Journal of Function Spaces The rest of the proof is similar to that of the above case and hence is omitted.