Second-Order Neutral Differential Equations: Improved Criteria for Testing the Oscillation

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Department of Mathematics, Faculty of Education–Al-Nadirah, Ibb University, Ibb, Yemen Department of Mathematics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt

In real-world life problems, the NDDEs have interesting applications.
e NDDEs appear in the modeling of the networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar, in the theory of automatic control, and others, see [1]. It is easy-in recent times-to observe the great development in the theory of oscillation for differential equations of different orders.
In the following, we review some of the works that contributed to the development of the oscillation theory of second-order NDDEs were the motivation for this work.
At studying the oscillatory behavior of NDDEs with canonical Case (2), the relationship between the solution and the corresponding function has been commonly used in the literature. For canonical case (2), by using the Riccati technique, Xu and Meng [2] presented some oscillation criteria for (1) when c � β. In the case, where 0 < p(t) ≤ p 0 < ∞, Baculikova and Dzurina [3] established the oscillation criteria for (1).
Using relationship (3), Grace et al. [4] studied the oscillatory behavior of solutions of (1) when c � β and p < 1. Moreover, they improved previous results in the literature.
e similar results as those above have been extended for even-order NDDEs in [7][8][9][10][11]. For the works that dealt with the noncanonical case, that is, see, for example, [12][13][14]. e objective of this paper is to establish new oscillation criteria for the NDDE (1) by improving (3). e new relationship enables us to (i) Create more effective criteria for studying neutral equations in both cases p < 1 and p > 1 (ii) Essentially take into account the influence of the delay argument ϱ(l) that has been careless in all related results (iii) Exclude some restrictions that are usually imposed on the coefficients of the neutral equations in the case where p > 1 Moreover, we use an iterative technique to establish new oscillation criteria for the NDDE (1) when β � c and p < 1. One purpose of this paper is to further improve eorems 2 and 1. e results reported in this paper generalize, complement, and improve those in [3][4][5][6]. To show the importance of our results, we provide an example.

Main Results I: New Relationship between x and u
For simplicity, we just write the functions without the independent variable, such as f(l) ≔ f and f(g(l)) � f(g). Moreover, assuming where c 1 and c 2 are positive constants, the set of all eventually positive solutions of (1) is denoted by X + .