Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior

: In this paper, we establish new sufﬁcient conditions for the oscillation of solutions of a class of second-order delay differential equations with a mixed neutral term, which are under the non-canonical condition. The results obtained complement and simplify some known results in the relevant literature. Example illustrating the results is included.

Let y be a real-valued function defined for all l in a real interval [l y , ∞), l y ≥ l 0 , and having a second derivative for all l ∈ [l y , ∞). The function y is called a solution of the differential Equation (1) on [l y , ∞) if y satisfies (1) on [l y , ∞). A nontrivial solution y of any differential equation is said to be oscillatory if it has arbitrary large zeros; otherwise, it is said to be nonoscillatory. We will consider only those solutions of (1) which exist on some half-line [l b , ∞) for l b ≥ l 0 and satisfy the condition sup{|y(l)| : l c ≤ l < ∞} > 0 for any l c ≥ l b .
A delay differential equation of neutral type is an equation in which the highest order derivative of the unknown function appears both with and without delay. During the last decades, there is a great interest in studying the oscillation of solutions of neutral differential equations. This is due to the fact that such equations arise from a variety of applications including population dynamics, automatic control, mixing liquids, and vibrating masses attached to an elastic bar, biology in explaining self-balancing of the human body, and in robotics in constructing biped robots, it is easy to notice the emergence of models of the neutral delay differential equations, see [1,2].
In the following, we review some of the related works that dealt with the oscillation of the neutral differential equations of mixed-type.
Li [7] and Li et al. [8] studied the oscillation of solutions of the second-order equation with constant mixed arguments: Arul and Shobha [5] established some sufficient conditions for the oscillation of all solutions of Equation (3) in the canonical case, that is, Thandapani et al. [9] studied the oscillation criteria for the differential equation of the form (z α (l)) + q(l)y β (l − τ 1 ) + p(l)y γ (l + τ 1 ) = 0.
In this paper, we study the oscillatory behavior of solutions of the second-order differential equation with a mixed neutral term (1) under condition (5). We follow a new approach based on deducing a new relationship between the solution and the corresponding function. Using this new relationship, we first obtain one condition that ensures oscillation of (1). Moreover, by introducing a generalized Riccati substitution, we get a new criterion for oscillation of (1). Often these types of equations (such as (1), (2), and (3)) are studied under condition (4). On the other hand, the works that studied these equations under the condition (5) obtained two conditions to ensure the oscillation. Therefore, our results are an extension and simplification as well as improvement of previous results in [3][4][5]8,11].
Let ψ be a increasing function on [l 1 , ∞). Then, we obtain and so ψ(l) κ(l 1 , l) Thus, the proof is complete.
then, all solutions of (1) are oscillatory.
then, all solutions of (1) are oscillatory.