The main purpose of this manuscript is to show asymptotic properties of a class of differential equations with variable coefficients , where and . By using integral averaging technique, we get conditions to ensure oscillation of solutions of this equation. The obtained results improve and generalize the earlier ones; finally an example is given to illustrate the criteria.
In this paper, we study the oscillatory properties of solutions of the following fourth-order neutral differential equation
From now on we make the following assumptions:
and are quotient of odd positive integers;
, , and under the condition
, , ;
, , .
Here we present some preliminary definitions that will clarify the terms used throughout the paper.
The function y, is said to be a solution of (1), if , and satisfies (1) on .
A solution of (1) is said to be oscillatory if it has arbitrarily large zeros on . Otherwise, a solution that is not oscillatory is said to be nonoscillatory.
The equation (1) is said to be oscillatory if every solution of it is oscillatory.
A differential equation is said to be neutral if the highest-order derivative of the unknown function appears both with and without delay.
A kernel function is said to belong to the function class ℑ if, for
exists on and it is continuous and non-positive. Moreover, there exist three functions and such that
The historical background of neutral differential equations is extremely varied. In fact, they find numerous applications in natural science but also in technology: in the study of distributed networks containing lossless transmission lines, in high-speed computers, in the theory of automatic control and in aeromechanical systems (see ). In last years, the asymptotic properties of solutions of differential equations has been the subject of intensive study (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]). The model of human balancing is considered in [21,22,23] where results on stability are presented.
Furthermore, many researchers investigate regularity and existence properties of solutions to partial differential equations. See for instance [11,24,25,26,27] and the references therein. Also, we mention the study of the exact solutions to partial differential equations performed with a Lie symmetry analysis. A recent result in this direction is represented by  where the authors study a modified Schrödinger equation.
Interesting applications of neutral differential equations can be found in the study of the effects of vibrating systems fixed to an elastic bar, for example the Euler equations of the fluid dynamics (see the recent paper ).
In , the author obtained the necessary and sufficient conditions under which a general fourth-order ordinary differential equation admits a unique Lagrangian. Nevertheless, there exist examples of fourth-order ordinary differential equations which do not have a second-order Lagrangian.
Many papers have been concerned to the solution of the inverse problem of calculus of variations, namely finding a Lagrangian of differential equations. Also, the use of the Jacobi last multiplier and its connection with Lie theory, in order to find the Lagrangian for ordinary differential equations, can be found in .
Now we state some preliminary and interesting results related to the contents of this paper. Zafer , Zhang and Yan  studied the equation
where n is even and established some new sufficient conditions for oscillation.
From above, we see that  enriched the results in .
Thus, the motivation in studying this paper is to extend the already interesting and pioneer results in [33,35]. By using integral averaging technique, new oscillatory criteria for (1) are established. Furthermore, in order to illustrate the criteria presented here, an example is given.
The following lemmas will be very useful:
(, Lemma 2.2.3). Let . Assume that is of fixed sign and not identically zero on and that there exists a such that for all . If , then for every there exists such that
(, Lemma 1.2). If the function y satisfies , , and , then
(, Lemma 1.1). Let bea ratio of two odd numbers, and U are constants. Then
(, Lemma 1.2). Assume that is an eventually positive solution of (1). Then, there exist two possible cases:
for where is sufficiently large.
2. Oscillation Criteria
For convenience, we denote
Let is an eventually positive solution of (1). Then
Let y be an eventually positive solution of (1) on . Since and from the definition of w, we get
Thus, by using Corollary 1, Equation (31) is oscillatory if
This paper deals with a class of fourth-order neutral differential equations with variable coefficients. Using the famous Riccati’s transformation, we establish a new asymptotic criterion that improves and complements the findings contained in [33,35]. Moreover, we get Philos type oscillation criteria to ensure oscillation of solutions of the Equation (1). Furthermore, in a future work we will get some oscillation criteria for (1) under the condition
Conceptualization, O.B., M.R. and A.S.; formal analysis, O.B., M.R. and A.S.; investigation, O.B., M.R. and A.S.; writing–original draft preparation, O.B., M.R. and A.S.; writing–review & editing, O.B., M.R. and A.S.; supervision, O.B., M.R. and A.S. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
The authors wish to thank the anonymous reviewers for their remarks and constructive criticism which helped them to improve the presentation.
Conflicts of Interest
The authors declare no conflict of interest.
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