Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument

The chapter is devoted to study the oscillation of all solutions to second-order nonlinear neutral damped differential equations with delay argument. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and integral averaging techniques. 2010 Mathematics Subject Classification: 34C10, 34K11.

By a solution of Eq. (1), we mean a nontrivial real-valued function xðtÞ, which has the property zðtÞ∈C 1 ð½T x , ∞ÞÞ, rðtÞ z ′ ðtÞ α ∈C 1 ð½T x , ∞ÞÞ, T x ≥t 0 , and satisfies Eq. (1) on ½T x , ∞Þ. In the sequel, we will restrict our attention to those solutions xðtÞ of Eq. (1) that satisfy the condition sup {jxðtÞj : T≤t < ∞} > 0 for T≥T x : We make the standing hypothesis that Eq. (1) admits such a solution. As is customary, a solution of Eq. (1) is said to be oscillatory if it is neither eventually positive nor eventually negative on ½T x , ∞Þ and otherwise, it is termed nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.
Remark 1. All the functional inequalities considered in the sequel are assumed to hold eventually, that is, they are satisfied for all t large enough.
Oscillation theory was created in 1836 with a paper of Jacques Charles François Sturm published in Journal des Mathematiqués Pures et Appliqueés. His long and detailed memoir [1] was one of the first contributions in Liouville's newly founded journal and initiated a whole new research into the qualitative analysis of differential equations. Heretofore, the theory of differential equations was primarily about finding solutions of a given equation and so was very limited. Contrarily, the main idea of Sturm was to obtain geometric properties of solutions (such as sign changes, zeros, boundaries, and oscillation) directly from the differential equation, without benefit of solutions themselves.
Henceforth, the oscillation theory for ordinary differential equations has undergone a significant development. Nowadays, it is considered as coherent, self-contained domain in the qualitative theory of differential equations that is turning mainly toward the study of solution properties of functional differential equations (FDEs).
The problem of obtaining sufficient conditions for asymptotic and oscillatory properties of different classes of FDEs has experienced long-term interest of many researchers. This is caused by the fact that differential equations, especially those with deviating argument, are deemed to be adequate in modeling of the countless processes in all areas of science. For a summary of the most significant efforts and recent findings in the oscillation theory of FDEs and vast bibliography therein, we refer the reader to the excellent monographs [2][3][4][5][6].
In a neutral delay differential equation the highest-order derivative of the unknown function appears both with and without delay. The study of qualitative properties of solutions of such equations has, besides its theoretical interest, significant practical importance. This is due to the fact that neutral differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays. We refer the reader to the monograph [7] for further applications in science and technology.
So far, most of the results obtained in the literature has centered around the special undamped form of Eq. (1), i.e., when pðtÞ ¼ 0 (for example, see Refs. [8][9][10][11][12][13][14][15][16][17][18]). For instance, in one of the pioneering works on the subject, Grammatikopoulos et al. [8] studied the second-order neutral differential equation with constant delay of the form and proved that Eq. (4) is oscillatory if ð ∞ t0 qðsÞ 1−aðs−τÞ ds ¼ ∞: Later on, Grace and Lalli [9] extended the results from [8] to the more general equation and showed that Eq. (6) In Ref. [10], Dong has involved to study the oscillation problem for a half-linear case of Eq. (1) and by defining a sequence of continuous functions has obtained various kinds of better results. Afterward, his approach has been further developed by several authors, see, e.g., [11][12][13][14]. However, it appears that very little is known regarding the oscillation of Eq. (1) with pðtÞ≠0 and α≠β. Motivated by the results of Ref. [10], this chapter presents some new oscillation criteria, which are applicable on Eq. (1).
On the other hand, Eq. (1) can be considered as a natural generalization of the second-order delay differential equation of the form rðtÞ x ′ ðtÞ α þ pðtÞ x ′ ðtÞ α þ qðtÞf xðσðtÞÞ ¼ 0: Very recently, the authors of [19] studied the oscillation problem of Eq. (9) with pðtÞ ¼ 0 and α ¼ β. Their ideas, which are based on careful investigation of classical techniques covering Riccati transformations and integral averages, will be extended to the more general equation (1).

Main results
For the simplicity and without further mention, we use the following notations: where φðtÞ∈C 1 ðℐ, ℝÞ is a given function and will be specified later.
The organization of this chapter is as follows. Before stating our main results, we present two lemmas that ensure that any solution xðtÞ of Eq. (1) satisfies the condition for t sufficiently large. Next, we get our main oscillation results for Eq. (1) by employing the generalized Riccati transformations and integral averaging techniques. We base our arguments on the assumption that the function PðtÞ is positive or negative.
Integrating the above inequality from t 2 to t, we find that for t≥t 2 : By condition (14), zðtÞ approaches to −∞ as t ! ∞, which contradicts the fact that zðtÞ is eventually positive. Therefore, z ′ ðtÞ > 0 and from Eq. (1), we have that rðtÞ z ′ ðtÞ α ′ < 0. The proof is complete.
for s≥t≥t 2 . Integrating the above inequality from t to t ′ , t ′ ≥t≥t 2 , we get On the other hand, from Eq. (23), we have that is, whereγ : ¼ kγ β . Now, if we define the function It follows from this last inequality that AðuÞ rðuÞ for t≥t 2 : As t ! ∞, then by condition Eq. (21), zðtÞ approaches to −∞, which contradicts the fact that zðtÞ is eventually positive. Therefore, z ′ ðtÞ > 0 and from Eq. (1), we have rðtÞ z ′ ðtÞ α ′ < 0. The proof is complete.
AðuÞ rðuÞ holds and Eq. (1) has a positive solution xðtÞ on ℐ. Then there exists T∈ℐ, sufficiently large, such that either Proof. As in the proof of Lemma 1, we assume that there exists t 2 ∈ℐ such that z ′ ðtÞ < 0 on ½t 2 , ∞Þ. So, zðtÞ is decreasing and exists. Therefore, there exists t 3 ∈½t 2 , ∞Þ such that zðσðtÞÞ > zðtÞ≥b > 0: As in the proof of Lemma 2, we obtain Eq. (27) Integrating the above inequality twice from t 3 to t, one gets AðuÞ rðuÞ for t≥t 3 : As t ! ∞, then by condition (39), zðtÞ approaches to −∞, which contradicts the fact that zðtÞ is eventually positive. Thus, b ¼ 0 and from 0≤xðtÞ≤zðtÞ, we see that lim Using results of Lemmas 1 and 2, we can obtain the following oscillation criteria for Eq. (1).
Theorem 1. Let conditions ðiÞ-ðvÞ and one of the conditions (14) or (21) where ψðtÞ ¼ Then, Eq. (1) As in the proof of Lemma 1, we get Eq. (18), which in view of the assumption ðvÞ yields Now, we consider the following three cases: Case I: β > α.
In this case, since z ′ ðtÞ > 0 for t≥T, then there exists T 1 ≥T such that zðσðtÞÞ≥c for t≥T 1 . This implies that Case II: β ¼ α.
In this case, we see that z β−α α ðσðtÞÞ ¼ 1: Case III: β < α. Since Hence, there exists T 1 ≥T and a constant d 1 depending on d such that and thus for some positive constant d 2 .
Using these three cases and the definition of ψðtÞ, we get for t≥T 1 ≥T. Setting and using the inequality Taking the lim sup on both sides of the above inequality as t ! ∞, we obtain a contradiction to the condition (48). This completes the proof.
Next, we present some complementary oscillation results for Eq. (1) by using an integral averaging technique due to Philos. We need the class of functions F. Let D 0 ¼ fðt, sÞ : t > s≥t 0 g and D ¼ fðt, sÞ : t > s≥t 0 g (75) The function Hðt, sÞ∈CðD, ℝÞ is said to belong to a class F if (a) Hðt, tÞ ¼ 0 for t≥T, Hðt, sÞ > 0 for ðt, sÞ∈D 0 and using the inequality which is a contradiction to Eq. (77). The proof is complete. [15,20] studied a partial case of Eq. (1) by employing the generalized Riccati substitution (80). Note that the function ρðtÞ used in the generalized Riccati substitution (80) finally becomes unimportant. Thus, we can put ρðtÞ ¼ 0 and obtain similar results to those from [15,20].

Remark 3. Authors in
In the next part, we provide several oscillation results for Eq. (1) under the assumption that the function PðtÞ is nonpositive. These results generalize those from [10] for Eq. (1) in such sense that α≠β and pðtÞ≠0.
that is, Integrating the above inequality from T 1 to t, we have Taking the lim sup on both sides of the above inequality as t ! ∞, we obtain a contradiction to condition Eq. (91). This completes the proof.
for T≥t 0 sufficiently large.
We assert that that is, Moreover, by induction, we have that wðtÞ≥y n ðtÞ, for t≥T 1 , n ¼ 1; 2; 3;…: Thus, since the sequence {y n ðtÞ} ∞ n ¼ 0 is monotone increasing and bounded above, it converges to yðtÞ. Letting n ! ∞ and using Lebesgue monotone convergence theorem in Eq. (100), we get Eq. (103). The proof is complete.
where ψðtÞ is as in Theorem 1, then Eq. (1) is oscillatory. Then it is easy to see that λ≥1 and which contradicts the admissible value of λ and γ, and thus completes the proof.
And therefore, which contradicts Eq. (120). The proof is complete.
for T sufficiently large and T 1 ≥T, where ψðtÞ is as in Theorem 1, then Eq. (1) is oscillatory.
Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the conclusions of Lemma 1 or 2 on ½T, ∞Þ with xðtÞ > 0; xðτðtÞÞ > 0; xðσðtÞÞ > 0 (128)  Taking the derivative of uðtÞ, one gets