## Numerical Mathematical Techniques for Partial Differential Equations

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Omar Bazighifan, "Some New Oscillation Results for Fourth-Order Neutral Differential Equations with a Canonical Operator", *Mathematical Problems in Engineering*, vol. 2020, Article ID 8825413, 7 pages, 2020. https://doi.org/10.1155/2020/8825413

# Some New Oscillation Results for Fourth-Order Neutral Differential Equations with a Canonical Operator

**Academic Editor:**Hijaz Ahmad

#### Abstract

By this work, our aim is to study oscillatory behaviour of solutions to 4th-order differential equation of neutral type where . By using the comparison method with first-order differential inequality, we find new oscillation conditions for this equation.

#### 1. Introduction

Neutral differential equations are getting an increasing importance among the practitioners in the recent past, since they are used in numerous applications in technology and natural science [1]. During the last one and half decade, a multidimensional application of differential equation with delay has been observed in the field of statistics, science, and engineering [2–6].

In this paper, we are concerned with the oscillation of solutions of the even-order nonlinear differential equation:where and and are quotient of odd positive integers and

The operator is said to be in canonical form if ; otherwise, it is called noncanonical. The main results are obtained under the following conditions: F1. , , , F2. , , , , F3. For some the differential inequality is oscillatory, where . F4. For some the differential inequality is oscillatory. F5. For some the equationis oscillatory.

By a solution of (1), we mean a function which has the property and satisfies (1) on .

Over the years, several studies have been carried out on oscillation of neutral differential equations [7–13].

Li et al. [14] have studied oscillation fourth-order differential equations with neutral type.

In [15], Bazighifan has considered differential equation with positive and negative coefficients and studied oscillatory behaviour of equation:under the condition

In [16], Liu et al. have considered an even-order differential equation of the formwhere and obtained some new oscillation results for this equation.

Zafer [17], Zhang and Yan [18] proved that the equationis oscillatory ifwhere and is even.

Xing et al. [19] and Moaaz et al. [20] proved that the equationis oscillatory ifwhere is an even and .

Now, we consider the equation

By applying conditions (10), (11), (13), and (14) to equation (15), we obtain Table 1.

From above, we get that [20] enriched the results in [17–19].

Our aim of this paper is complement results in [17–20].

By the comparison method with first-order differential inequality, we obtain new conditions of oscillation for this equation. Some examples are provided to show effectiveness of the obtained results.

#### 2. Main Results

Lemma 1. *Reference [21] If satisfies , , and eventually. Then, for every eventually.*

Lemma 2 (see [22], Lemma 2.2.3). *Let and . If , thenfor every .*

Lemma 3 (see [23], Lemma 1 and 2). *Let , thenwhere is a positive real number.*

Theorem 1. *Assume that*

If holds, then (1) is oscillatory. Proof. Assume to the contrary that (1) has a nonoscillatory solution in . Without loss of generality, we let be an eventually positive solution of (1). Then, there exists a such that , and for . Since , we havefor . From (1), we get

It follows from definition of and Lemma 3 that

From (20) and (21), we obtainwhich with (18) gives

Since , we get that . Thus, from Lemma 2, we get

Combining (23) and (24), we see that

If we setthen it is easy to see that

Thus, from (25), we get that is a positive solution ofwhich is a contradiction. The proof is complete.

Theorem 2. *Assume that (18) holds. If holds, then (1) is oscillatory.*

*Proof*. Proceeding as in the proof of Theorem 1, we get (25). If we set , then is a positive solution of (4), which is a contradiction. The proof is complete.

Corollary 1. *Let and (18) holds. If andwhere or , then (1) is oscillatory.*

*Proof*. It is well-known (see, e.g., [24], Theorem 2.1.1]) that condition (29) implies the oscillation of (3) and (5).

Theorem 3. *Assume that and . If holds, then (1) is oscillatory.*

*Proof*. Proceeding as in the proof of Theorem 1, we get (19). From definition of , we getwhich with (1) gives

From Lemma 2, we obtain

Combining (31) and (32), we get

Hence, if we set , then we get that is a positive solution of the inequality

In view of ([12], Corollary 1), the associated delay differential equation (5) also has a positive solution, which is a contradiction. The proof is complete.

Corollary 2. *Let , and . Ifthen (1) is oscillatory.*

*Proof*. It is well-known (see, e.h., [24], Theorem 2.1.1) that condition (35) implies the oscillation of (5).

Theorem 4. *Assume that and . If there exists a positive functions such thatfor some and every , wherethen (1) is oscillatory.*

*Proof*. Proceeding as in the proof of Theorem 3, we find (19) and (31). From (19), we have is of one sign.

In the case where , we define a generalized Riccati substitution by

By differentiating and using (31), we obtain

From Lemma 1, we have that , and hence,

Using Lemma 2, we get for all . Thus, by (40)–(42), we obtain

Since , there exist and a constant such thatfor all . Using the inequalitywith and , we find

This implies thatwhich contradicts (36).

For , integrating (31) from to , we obtain

From Lemma 1, we see that , and hence,

For (48), letting and using (49), we get

Integrating (50) from to , we getfor all . Now, we define

Then for . By using (44) and (51), we obtain

Thus, we findand so

Then, we obtainwhich contradicts (37). This completes the proof.

*Example 1. *Consider the differential equationwhere . We note that , , , , , and . Hence, it is easy to see thatUsing Corollary 1, the equation (57) is oscillatory ifFrom Corollary 2, the equation (57) is oscillatory if

*Example 2. *Consider the differential equationwhere and . We note that , , , , and . Hence, it is easy to see thatUsing Corollary 1, the equation (61) is oscillatory ifFrom Corollary 2, ifthen (61) is ocillatory.

Finally, if we set and , then we haveThus, from Theorem 4, equation (61) is oscillatory if

#### Data Availability

No data were used in the study.

#### Conflicts of Interest

The authors declare that they have no competing interests.

#### Authors’ Contributions

The authors declare that it has been read and approved the final manuscript.

#### References

- J. K. Hale,
*Theory of Functional Differential Equations*, Springer-Verlag, Berlin, Germany, 1977. - C. Vetro and F. Vetro, “On problems driven by the -Laplace operator,”
*Mediterranean Journal of Mathematics*, vol. 17, no. 24, pp. 1–11, 2020. View at: Publisher Site | Google Scholar - A. Yokus, H. Durur, and H. Ahmad, “Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system,”
*Facta Universitatis, Series: Mathematics and Informatics*, vol. 35, no. 2, pp. 523–531, 2020. View at: Google Scholar - O. Bazighifan, H. Ahmad, and S. W. Yao, “New oscillation criteria for advanced differential equations of fourth order,”
*Mathematics*, vol. 8, pp. 1–12, 2020. View at: Publisher Site | Google Scholar - O. Bazighifan and T. Abdeljawad, “Improved approach for studying oscillatory properties of fourth-order advanced differential equations with p-laplacian like operator,”
*Mathematics*, vol. 8, pp. 1–11, 2020. View at: Publisher Site | Google Scholar - H. Ahmad, A. R. Seadawy, and T. A. Khan, “Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm,”
*Mathematics and Computers in Simulation*, vol. 177, pp. 13–23, 2020. View at: Publisher Site | Google Scholar - R. P. Agarwal, S. R. Grace, and D. O'Regan, “Oscillation criteria for certain nth order differential equations with deviating arguments,”
*Journal of Mathematical Analysis and Applications*, vol. 262, no. 2, pp. 601–622, 2001. View at: Publisher Site | Google Scholar - O. Bazighifan, “Oscillatory applications of some fourth-order differential equations,”
*Mathematical Methods in Applied Sciences*, vol. 15, 2020. View at: Publisher Site | Google Scholar - O. Bazighifan and G. E. Chatzarakis, “Oscillatory and asymptotic behavior of advanced differential equations,”
*Advances in Difference Equations*, vol. 2020, p. 414, 2020. View at: Publisher Site | Google Scholar - G. E. Chatzarakis, E. M. Elabbasy, and O. Bazighifan, “An oscillation criterion in 4th-order neutral differential equations with a continuously distributed delay,”
*Advances in Difference Equations*, vol. 336, pp. 1–9, 2019. View at: Google Scholar - O. Moaaz, E. M. Elabbasy, and A. Muhib, “Oscillation criteria for even-order neutral differential equations with distributed deviating arguments,”
*Advances in Difference Equations*, vol. 2019, p. 297, 2019. View at: Google Scholar - C. G. Philos, “On the existence of nonoscillatory solutions tending to zero at ? for differential equations with positive delays,”
*Archiv der Mathematik*, vol. 36, no. 1, pp. 168–178, 1981. View at: Publisher Site | Google Scholar - C. Zhang, R. P. Agarwal, M. Bohner, and T. Li, “New results for oscillatory behavior of even-order half-linear delay differential equations,”
*Applied Mathematics Letters*, vol. 26, no. 2, pp. 179–183, 2013. View at: Publisher Site | Google Scholar - T. Li, B. Baculikova, J. Dzurina, and C. Zhang, “Oscillation of fourth order neutral differential equations with -Laplacian like operators, Bound,”
*Boundary Value Problems*, vol. 56, pp. 41–58, 2014. View at: Google Scholar - O. Bazighifan, “On the oscillation of certain fourth-order differential equations with p-Laplacian like operator,”
*Applied Mathematics and Computation*, vol. 386, Article ID 125475, 2020. View at: Publisher Site | Google Scholar - S. Liu, Q. Zhang, and Y. Yu, “Oscillation of even-order half-linear functional differential equations with damping,”
*Computers & Mathematics with Applications*, vol. 61, no. 8, pp. 2191–2196, 2011. View at: Publisher Site | Google Scholar - A. Zafer, “Oscillation criteria for even order neutral differential equations,”
*Applied Mathematics Letters*, vol. 11, no. 3, pp. 21–25, 1998. View at: Publisher Site | Google Scholar - Q. Zhang and J. Yan, “Oscillation behavior of even order neutral differential equations with variable coefficients,”
*Applied Mathematics Letters*, vol. 19, no. 11, pp. 1202–1206, 2006. View at: Publisher Site | Google Scholar - G. Xing, T. Li, and C. Zhang, “Oscillation of higher-order quasi linear neutral differential equations,”
*Advances in Difference Equations*, vol. 2011, pp. 1–10, 2011. View at: Publisher Site | Google Scholar - O. Moaaz, J. Awrejcewicz, and O. Bazighifan, “A new approach in the study of oscillation criteria of even-order neutral differential equations,”
*Mathematics*, vol. 8, no. 2, p. 197, 2020. View at: Publisher Site | Google Scholar - G. E. Chatzarakis, S. R. Grace, I. Jadlovska, T. Li, and E. Tunc, “Oscillation Criteria for Third-Order Emden-Fowler Differential Equations with Unbounded Neutral Coefficients,”
*Complexity*, vol. 2019, Article ID 5691758, 7 pages, 2019. View at: Publisher Site | Google Scholar - R. Agarwal, S. Grace, and D. O’Regan,
*Oscillation Theory for Difference and Functional Differential Equations*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. - B. Baculíková and J. Džurina, “Oscillation theorems for second-order nonlinear neutral differential equations,”
*Computers & Mathematics with Applications*, vol. 62, no. 12, pp. 4472–4478, 2011. View at: Publisher Site | Google Scholar - G. S. Ladde, V. Lakshmikantham, and B. G. Zhang,
*Oscillation Theory of Differential Equations with Deviating Arguments*, Marcel Dekker, New York, NY, USA, 1987.

#### Copyright

Copyright © 2020 Omar Bazighifan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.