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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
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.
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 . 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.
Li et al.  have studied oscillation fourth-order differential equations with neutral type.
In , Bazighifan has considered differential equation with positive and negative coefficients and studied oscillatory behaviour of equation:under the condition
In , Liu et al. have considered an even-order differential equation of the formwhere and obtained some new oscillation results for this equation.
Now, we consider the equation
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  If satisfies , , and eventually. Then, for every eventually.
Lemma 2 (see , Lemma 2.2.3). Let and . If , thenfor every .
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
Since , we get that . Thus, from Lemma 2, we get
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 3. Assume that and . If holds, then (1) is oscillatory.
From Lemma 2, we obtain
Hence, if we set , then we get that is a positive solution of the inequality
Corollary 2. Let , and . Ifthen (1) is oscillatory.
Theorem 4. Assume that and . If there exists a positive functions such thatfor some and every , wherethen (1) is oscillatory.
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,
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,
Integrating (50) from to , we getfor all . Now, we define
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
No data were used in the study.
Conflicts of Interest
The authors declare that they have no competing interests.
The authors declare that it has been read and approved the final manuscript.
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