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In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. The effectiveness of the obtained criteria is illustrated via examples.
In this work, we study the oscillation of a fourth-order delay differential equation
where is a quotient of odd positive integers and we assume that is positive is a nondecreasing function in , and satisfies the following conditions:
By a solution of Equation (1), we mean a function y, which has the property , and satisfies Equation (1) on ,. We consider only those solutions y of Equation (1) which satisfy for all We assume that (1) possesses such a solution. A solution of (1) is called oscillatory if it has arbitrarily large zeros on , and otherwise it is called to be nonoscillatory. The Equation (1) is said to be oscillatory if all its solutions are oscillatory.
The problem of the oscillation of higher and fourth order differential equations have been widely studied by many authors, who have provided many techniques for obtaining oscillatory criteria for higher and fourth order differential equations. We refer the reader to the related books (see [1,2,3,4]) and to the papers (see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] ). In what follows, we present some relevant results that have provided the background and the motivation, for the present work.
In the following, we present some related results that served as a motivation for the contents of this paper.
Tunc and Bazighifan  studied the oscillatory behavior of the fourth-order nonlinear differential equation with a continuously distributed delay
Grace et al.  studied the oscillation behavior of the fourth-order nonlinear differential equation
Bazighifan et al.  and Zhang et al.  consider the oscillatory properties of the higher-order differential equation
under the conditions
Our aim in the present paper is to employ the Riccati technique and comparison technique with first order delay differential equation to establish some new conditions for the oscillation of all solutions of (1). Some examples are presented to illustrate our main results.
We begin with the following lemmas.
(See , Lemma 2.1) Let be a ratio of two numbers. Then
(See , Lemma 1.1) If the function z satisfies and then
(See , Lemma 1.1) Let and assume that is of fixed sign and not identically zero on a subray of If moreover, and then, for every , there exists such that
(See , Lemma 2.1) Assume that (2) holds and y is an eventually positive solution of (1). Then there are the following two possible cases eventually:
2. Main Results
In this section, we shall establish some oscillation criteria for Equation (1). For convenience, we denote
Let (2) holds and assume that for some constant , the differential equation
is oscillatory. If there exists a positive function such that
for some then every solution of (1) is oscillatory.
Assume that (1) has a nonoscillatory solution Without loss of generality, we can assume that . From Lemma 4, we get that there exist y two possible cases for where is sufficiently large.
Assume that we have Case 1 and hold. From Lemma 3, we have
for every Using (5) in Equation (1), we see that
is a positive solution of the differential inequality
By Theorem 1 in , we conclude that the corresponding Equation (1) also has a positive solution. This is a contradiction.
Assume that we have Case 2 and hold. From Lemma 2, we get that , by integrating this inequality from to z, we get
Hence, we have
Integrating (1) from z to u and using , we obtain
Letting , we see that
Integrating again from z to ∞, we get
Now, we define
Then for . By differentiating and using (6), we find
Thus, we obtain
Using Lemma 1 with and we get
From (1), (7) and (8), we obtain
This implies that
Thus, inequality (9) yields
Applying the Lemma 1 with , and , we get
Integrating from to z, we get
this contradicts (4). □
Let (1.2) holds. If
and (4) hold for some constant λ , then every solution of (1) is oscillatory.
In this section, we give the following examples to illustrate our main results.
Consider the fourth-order differential equation
We note that and . If we choose , then it easy to see that the condition (4) holds. Hence, by Theorem 1, every solution of Equation (13) is oscillatory.
Consider the following differential equation of fourth order
where is a constant. Let
Then, we get
If we now set , thus, by Corollary 1, every solution of Equation (14) is oscillatory, provided .
Our results supplement and improve the results obtained in .
The results of this paper are presented in a form which is essentially new and of a high degree of generality. In this paper, using the generalized Riccati transformations and comparison technique, we offer some new sufficient conditions which ensure that any solution of Equation (1) oscillates under the condition (2). Further, we can consider the case of in the future work.
The authors claim to have contributed equally and significantly in this paper. All authors read and approved the final manuscript.
The authors received no direct funding for this work.
The authors thank the reviewers for for their useful comments, which led to the improvement of the content of the paper.
Conflicts of Interest
The authors declare no conflicts of interest.
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