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.