Oscillation Test for Second-Order Differential Equations with Several Delays

: In this paper, the oscillatory properties of certain second-order differential equations of neutral type are investigated. We obtain new oscillation criteria, which guarantee that every solution of these equations oscillates. Further, we get conditions of an iterative nature. These results complement and extend some beforehand results obtained in the literature. In order to illustrate the results we present an example

A solution is said to be oscillatory if it is distinguished that it is neither positive nor negative eventually.A differential equation whose solutions all oscillate is called an oscillatory equation.
NDDEs appear in a variety of situations, including issues with electric networks using lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching circuits).Additional applications in population dynamics, automatic control, mixing liquids, and vibrating masses attached to an elastic bar are available, see Hale [1,2].Dynamical systems with several delays have been intensively studied in control theory; see [3,4].To encourage special interest in the oscillatory behavior of solutions to second-order NDDEs through their applications in the natural sciences and engineering, There is a continuing need to discover new necessary conditions for the oscillation or nonoscillation of solutions varietal type equations; see, e.g., papers [5][6][7][8][9][10][11].
Bohner et al. [16] and Agarwal et al. [6] studied the oscillation of the equation where Recently, Moaaz [17] created conditions for the oscillation of NDDEs under condition (2).
In this paper we will use some important lemmas and notation where c 1 and c 2 are positive constants.We will denote by the symbol κ + the class of all eventually positive solutions of (1).Lemma 1. [18] Let κ ∈ κ + .Then, y > 0, y > 0 and r y for ≥ 1 , where 1 is sufficiently large.
There is no doubt that the concept of symmetry is of great importance as it appears in many natural phenomena and has many applications.The approach adopted in our paper is based on exploiting the symmetry between positive and negative solutions in studying only positive solutions.
The aim of this work is to find new NDDE oscillation criteria (1).We establish more effective criteria by considering the equation in two cases: p < 1 and p > 1.To create more efficient criteria, we take into account the influence of the delay argument τ( ), and we abandon some of the constraints that are usually imposed on the coefficients of the equation in the case p > 1.When γ = β and p < 1, we also utilize an iterative method to obtain the oscillation criterion of (1).

Main Results
For convenience, we write the functions without the independent variable, such as f ( ) = f and f (q( )) = f (q).In addition, we suppose that 2, ....We use some notations in this paper: and First, we will establish new criteria for the oscillation of solution (1) using the Riccati technique. ( Then κ( ) ≥ p( )y( ).

Using the inequality
Integrating the above inequality from 1 to , we obtain which contradicts (25).The proof is complete.
Theorem 3. Assume that (5) holds.If there exists a function is oscillatory, where Proof.It is enough to use (6) instead of (10) in the proof of Theorem 2 to prove this theorem.
To prove this theorem, we use Now, we will establish new criteria for oscillation of solution (1) by using an iterative technique.
Proof.Supposing that the result we want to achieve is incorrect.We suppose the opposite that κ is a non-oscillatory solution of (1).Without losing generalization, we assume that κ ∈ κ + .Thus, κ( ), κ(τ( )) and κ(ϑ i ( )) are positive for all ≥ 1 , where 1 large enough.Using Lemma 5, we arrive at (31) holds.As in the proof of Theorem 2, we get (26).Using ( 33) and (31), we obtain If we set w = (r(y )) γ , we have that w is a positive solution of the delay differential inequality Using Theorem 1 in [20] the associated delay differential equation also has a positive solution.But, the equation (36) with condition (35) is oscillatory; this is a contradiction.The proof is complete.
Theorem 5. Suppose that γ = β and p < 1.Then (1) is oscillatory if there is a function for some integers k ≥ 0, where dξ , p and φ k are defined as in ( 11) and (32), respectively.
Proof.Supposing that the result we want to achieve is incorrect.We suppose the opposite that κ is a non-oscillatory solution of (1).Without losing generalization, we assume that κ ∈ κ + .Thus, there exist 1 > 0 such that κ( ), κ(τ( )) and κ(ϑ i ( )) are positive for all ≥ 1 and 1 ≤ i ≤ n.Now, we define ψ = ρr(y /y) γ .Thus, ψ( ) > 0 and From Lemma 5, we have that (31) holds.By replacing (29) with (31) in the proof of Theorem 2, this part of the proof is similar to that of Theorem 2 and so the proof is obtained.
Using Theorem 1, we see that (38) is oscillatory if Remark 1.By comparing the conditions (39) and (41) for different values p, λ 1 and η, we obtain the following table when γ = 1.