Abstract
Our aim in the present paper is to employ the Riccatti transformation which differs from those reported in some literature and comparison principles with the second-order differential equations, to establish some new conditions for the oscillation of all solutions of fourth-order differential equations. Moreover, we establish some new criterion for oscillation by using an integral averages condition of Philos-type, also Hille and Nehari-type. Some examples are provided to illustrate the main results.
1. Introduction
In this work, we study the fourth-order nonlinear differential equation
where is a quotient of odd positive integers, such that
and under the condition
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 .A solution of Equation (1) is called oscillatory if it has arbitrarily large zeros on otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory.
One of the main reasons for this lies in the fact that differential and functional differential equations arise in many applied problems in natural sciences and engineering, see [].
Fourth-order differential equations are quite often encountered in mathematical models of various physical, biological, and chemical phenomena. Applications include, for instance, problems of elasticity, deformation of structures, or soil settlement; see []. In mechanical and engineering problems, questions related to the existence of oscillatory and nonoscillatory solutions play an important role.
During the last years, significant efforts have been devoted to investigate the oscillatory behavior of fourth-order differential equations. For treatments on this subject, we refer the reader to the texts [,,,,,,,,,,,,,,,,,,,,,,,,,,]. In what follows, we review some results that have provided the background and the motivation, for the present work.
In our paper, by careful observation and employing some inequalities of a different type, we provide a new criterion for the oscillation of differential Equation (1). Here, we offer different criteria for oscillation which can cover a larger area of different models of fourth-order differential equations. We introduce a generalized Riccati substitution to obtain a new Philos-type criteria and Hille- and Nehari-type. In the last section, we apply the main results to two different examples.
In the following, we show some previous results in the literature which relate to this paper: Many researchers in [,,,] have studied the oscillatory behavior of equation
under the condition in Equation (3), where is a positive real number. In [], the authors studied the oscillation of the equation
under the condition
By comparison theory, Baculikova et al. [] proved that the equation
is oscillatory if the delay equations
is oscillatory. Moaaz et al. [] established the oscillation criterion for solutions of the equation
under the condition in Equation (4).
Next, we begin with the following lemmas.
Lemma 1.
[] Letbea ratio of two odd numbers. Then
and
Lemma 2.
[] If the function u satisfiesfor allandthen
Lemma 3.
[] The equation
whereandis nonoscillatory if and only if there exist aand a functionsuch that
for.
Lemma 4.
[] Suppose thatis of a fixed sign onnot identically zero and there exists asuch that
for all. If we have, then there existssuch that
for everyand.
2. Main Results
In this section, we shall establish some oscillation criteria for Equation (1). For convenience, we denote
and
where are constants and .
Also, we define the generalized Riccati substitutions
and
After studying the asymptotic behavior of the positive solutions of Equation (1), there are only two cases:
Moreover, from Equation (1) and the condition in Equation (2), we have that . In the following, we will first study each case separately.
Lemma 5.
Proof.
Let y is an eventually positive solution of (1) and for all . Thus, from Lemma 4, we get
for every and for all large ℓ. From Equation (6), we see that for and
Using Lemma 1 with and , we get
From Lemma 2, we have that and hence,
This implies that
Thus,
The proof is complete. □
Lemma 6.
Proof.
Let y is an eventually positive solution of Equation (1), for and . From Lemma 2, we get that . By integrating this inequality from to ℓ, we get
Hence, from Equation (2), we have
Letting , we see that
and so
Integrating again from ℓ to ∞, we get
From the definition of , we see that for . By differentiating, we find
Using Lemma 1 with and we get
This implies that
Thus,
The proof is complete. □
Lemma 7.
Assume that y will eventually be a positive solution of Equation (1). If there exists a positive functionsuch that
for some, then y does not fulfill Case.
Proof.
Lemma 8.
Assume that y will eventually be a positive solution of Equation (1),forand. If there exists a positive functionsuch that
then y does not fulfill Case.
Proof.
Theorem 1.
In the next theorem, we establish new oscillation results for Equation (1) by using the integral averaging technique.
Definition 1.
Let
A kernel functionis said to belong to the function class ℑ, written by, if, for,
- (i)
- for
- (ii)
- has a continuous and nonpositive partial derivativeonand there exist functionsandsuch that
Lemma 9.
Proof.
Assume that is eventually positive solution of Equation (1). From Lemma 5, we get that Equation (8) holds. Multiplying Equation (8) by and integrating the resulting inequality from to ℓ; we find that
From Equation (22), we get
Lemma 10.
Proof.
Theorem 2.
In the next theorem, we establish new oscillation results for (1) by using the theory of comparison with the second order differential equation:
Theorem 3.
Proof.
Suppose to the contrary that Equation (1) has a eventually positive solution x and by virtue of Lemma 3. If we set in Equation (8), then we get
Thus, we can see that Equation (27) is nonoscillatory.Which is a contradiction. If we now set in Equation (13), then we obtain
Hence, Equation (28) is nonoscillatory, which is a contradiction.
Theorem 3 is proved. □
It is well known (see []) that if
then Equation (5) with is oscillatory.
From the previous results that we have concluded and Theorem 3, we can easily obtain Hille and Nehari type oscillation criteria for Equation (1), in the next theorem:
3. Discussion and Application
Theorems 1 and 2 can be used in a wide range of applications for oscillation of Equation (1) depending on the appropriate choice of functions and . To applying the conditions of theorems, we search on for suitable restitution for functions and such that .
In the following, by using our results, we study the oscillation behavior of some differential equations with a fourth-order.
Example 1.
Consider a differential equation
where. Note thatand. Hence, we have
Remark 1.
In Example 1, by using Theorem 1, the new criterion for oscillation of Equation (29) is
Example 2.
Consider a differential equation
whereandis a constant. Note thatandHence,
If we setandthen
and
for some constant. Hence, by Theorem 1, every solution of Equation (30) is oscillatory if
Author Contributions
The authors claim to have contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.
Funding
The authors received no direct funding for this work.
Acknowledgments
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 conflict of interest.
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