New Aspects for Non-Existence of Kneser Solutions of Neutral Differential Equations with Odd-Order

: Some new oscillatory and asymptotic properties of solutions of neutral differential equations with odd-order are established. Through the new results, we give sufﬁcient conditions for the oscillation of all solutions of the studied equations, and this is an improvement of the relevant results. The efﬁciency of the obtained criteria is illustrated via example.

If there exists l u ≥ l 0 such that the real valued function u is continuous, r z (n−1) α is continuously differentiable and satisfies (1), for all l ∈ I u ; then, u is said to be a solution of (1). We restrict our discussion to those solutions u of (1) which satisfy sup {|u (l)| : l 1 ≤ l 0 } > 0 for every l 1 ∈ I u . (1) is called an N-Kneser solution if there exists a l * ∈ I 0 such that z (l) z (l) < 0 for all l ∈ I * . The set of all eventually positive N-Kneser solutions of Equation (1) is denoted by .

Definition 2.
A solution u of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. The equation itself is termed oscillatory if all its solutions oscillate.
For applications of odd-order equations in extrema, biology, and physics, we refer to the following examples. In 1701, James Bernoulli published the solution to the Isoperimetric Problem-a problem in which it is required to make one integral a maximum or minimum, while keeping constant the integral of a second given function, thus resulting in a differential equation of third-order (see [13]). In the early 1950s, Alan Lloyd Hodgkin and Andrew Huxley developed a mathematical model for the propagation of electrical pulses in the nerve of a squid. The Hodgkin-Huxley Model is a set of nonlinear ordinary differential equations. The model has played a seminal role in biophysics and neuronal modeling.
Recently, researchers have paid attention to neutral differential equations, as well as studying the oscillation behavior of their solutions. There is a practical side to study the problem of the oscillatory properties of solutions of neutral equations besides the theoretical side. For example, the neutral equations arise in applications to electric networks containing lossless transmission lines. Such networks appear in high-speed computers where lossless transmission lines are used to interconnect switching circuits. For more applications in science and technology, see [14][15][16].
Karpuz et al. [17] studied the higher-order neutral differential equations of the following type: where oscillatory and asymptotic behaviors of all solutions of higher-order neutral differential equations are compared with first-order delay differential equations, depending on two different ranges of the coefficient associated with the neutral part Xing et al. [18] established some oscillation criteria for certain higher-order quasi-linear neutral differential equation (r (l) (u(l) + p(l)u(θ(l))) (n−1) α ) + q(l)u α (η(l)) = 0, n ≥ 2 where α ≤ 1 is the quotient of odd positive integers. Li and Rogovchenko [19] concerned with the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations where p 0 ≥ 0, p 0 = 1 and 0 are constants, 0 ≥ 0 (delayed argument) or 0 ≤ 0 (advanced argument).
Some results that are closely related to our work are presented as follows: , Corollary 2, see [20], Theorem 3.1.1 and [21]). Assume that p satisfies the condition (3) is almost oscillatory.

Main Results
For the sake of convenience, we use the following notation: The following lemma is a direct conclusion from Lemmas 2.1 and 2.4 in [18], so its proof was neglected.

Lemma 3.
Assume that u is an eventually positive solution of (1). Then, there exists a sufficiently large l 1 ≥ l 0 such that, for all l ≥ l 1 , either Now, in the following theorem, we will provide a new criterion for non-existence of N-Kneser solutions of (1) by using the comparison theorem.
In the following theorem, we will provide another criterion for the non-existence of N-Kneser solutions of (1) using the comparison theorem.
By using (1) and (I), we see that and, similarly, Combining the above inequalities yields that Now, we set From (II) and the fact that r (l) (z (n−1) (l)) is non-increasing, it is easy to see that By using (12) with ς = θ (l) and = l and (20), we have From definition Ψ and using the above inequality in (18), we get In view of [24], Theorem 1, we have that (17) also has a positive solution, a contradiction. Thus, the proof is complete.

New Oscillation Criteria
In the following lemma, we present criteria that ensure that non-existence of solutions satisfies case (1).
Substituting these terms into (26), we get that φ is a positive solution of In view of [24], Theorem 1, we have that (22) also has a positive solution, which is a contradiction (22). Thus, the proof is complete.
The following theorems give the criteria for oscillation for all solutions of Equation (1). Theorem 6. If (5) and (21) are oscillatory, then (1) is oscillatory.
Proof. Assume on the contrary that u is an eventually positive solution of (1). Then, from Lemma 3, we conclude that there are two possible cases for the behavior of z and its derivatives. By using Theorem 3 and Lemma 4, conditions (5) and (21) ensure that there are no solutions for Equation (1) satisfy case (1) and case (2) respectively. Thus, the proof is complete. (17) and (21) are oscillatory, then (1) is oscillatory.
Proof. Assume on the contrary that u is an eventually positive solution of (1). Then, from Lemma 3, we conclude that there are two possible cases for the behavior of z and its derivatives. By using Theorem 5 and Lemma 4, conditions (17) and (21) ensure that there are no solutions for Equation (1) satisfying case (1) and case (2), respectively. Thus, the proof is complete.