Abstract
The oscillation of even-order nonlinear differential equations (NLDiffEqs) with mixed nonlinear neutral terms (MNLNTs) is investigated in this work. New oscillation criteria are obtained which improve, extend, and simplify the existing ones in other previous works. Some examples are also given to illustrate the validity and potentiality of our results.
1. 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 [2–4]), rd-order neutral delay dynamic equations on time scales [5], nd-order nonlinear neutral delay differential equation solutions’ asymptotic behavior [6] (see also [7]), and nd-order superlinear Emden-Fowler neutral DEqsOs [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 [14–16], 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 is an even integer, and
From Equations (1) and (2), the following are assumed: (i), , , and are the ratios of two positive odd integers with (ii) are continuous functions(iii) are continuous functions; and as for (iv), and as
Let us suppose that for which
A continuous function satisfying Equation (1) on , , is said to be a solution of Equation (1) on where is defined in (2). We only consider those solutions of (1) which satisfy
A solution of (1) is said to be oscillatory if there exists a sequence such that 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 super-linear 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.
2. Main Results
Some oscillation criteria for Equation (1) are studied when and .
To obtain our results, the following lemma is needed:
Lemma 1 ([5]). Let and be two nonnegative real numbers. Then, the following inequality is obtained:
where equality holds if and only if .
In what follows, we let
for for some , where is a continuous function.
Theorem 2. Let and , conditions (i)-(iv), and (3) hold, and let such that and the equation is oscillatory for all constant . Let us assume that there exist constants , , and such that and the equations are oscillatory, and Then, every solution of Equation (1) is oscillatory, or
Proof. Without loss of generality, the solution of Equation (1) is assumed to be positive and for for some (i.e., a nonoscillatory solution). From Equation (1), we have the following: and
Hence, is nonincreasing with a constant sign. Namely, or for for some , so the following four cases are examined separately:
(a) and (b) and (c) and (d) and Let us first consider the case (a). Since for , we obtain the following:
for some positive constant , i.e.,
for . Integrating the last inequality -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 , i.e., we obtain the following:
If we apply the first inequality in (7) with , , and
then we have
In a similar manner, by applying the second inequality in (7) with , , and
we obtain the following:
By using (21) and (23), (25) turns out that
Since in nondecreasing, we have the following: for some . Hence, (26) turns that
Now, we see
from (9) and (27) for some . (28) implies that Equation (1) turns to be
There exists a constant such that
for (see [16, 18, 19]). By setting , we obtain the following:
By using (31), (29) turns that
where
From Corollary 1 in [20], it can easily be concluded that there exists a positive solution of Equation (10) with , which contradicts the fact that Equation (10) is oscillatory.
Now, let us consider the cases when for . Suppose that
which implies
or
On the other hand, we obtain the following:
Now, let us consider the case (c). Clearly, we see that and either or for . First, we assume that for . It is easy to see that
(refer to [18]). Now, we may express
for . By taking and for in inequality (39), we see that
By using (40), (38) turns out to be
By setting for , (41) turns that
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 for . Clearly, we have the following:
There exists a constant such that
for . Now, we see that
The rest of the proof is similar to that of the above case and hence is omitted.
Finally, let us consider the case (d). Clearly, we have and so
or that
for some . Thus, we obtain the following:
for . By using (50), (37) turns out
The rest of the proof is trivial and hence is omitted. This completes the proof.☐
Corollary 3. Let and , conditions (i)-(iv), and (3) hold, and let such that (9) holds. Assume that there exist real numbers , such that (11) is satisfied. If we have condition (14), then Then, Equation (1) is oscillatory.
3. Illustrative Examples
Two illustrative examples are presented in this section as follows:
Example 1. Consider the following second-order equation: Clearly, , , and , and hence, there exists a such that for . The verification of all the conditions of Theorem 2 gives that every solution of Equation (53) is oscillatory; otherwise, . It is worth mentioning that is such a solution of Equation (53).
Example 2. Consider the following even-order equation: By noting that , , , and and letting , , and , it can be easily seen that all the conditions of Corollary 3 hold, and hence, Equation (55) is oscillatory.
4. Conclusion
New results concerning the oscillation of NLDiffEq with MNLNTs have been successfully established in this paper. We have used novel technique which is based on a basic inequality and some comparison results to prove the main theorem. Demonstrating the validity and applicability of our results, two examples have been presented in this regard. It is worth mentioning that the oscillation of Equations (53) and (55) cannot be commented by previous works.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors have taken equal part in this research, and they read and approve the final manuscript.
Acknowledgments
The authors thank the referees for valuable comments that improve the presentation of the results in this paper. J. Alzabut expresses his sincere thanks to Prince Sultan University and OSTİM Technical University for supporting this research.