Some Properties of Third-Order Differential Equations with Mixed Arguments

There is a permanent interest in studying the properties of third-order differential equations. Various techniques have been presented for such equations (see [1–21]). Comparison theorems, especially, are a very strong and effective tool in the oscillation theory. Mahfoud [18] has presented a very useful comparison technique for studying the properties of delay differential equations through those of differential equations without delay. However, the corresponding result for advanced differential equation is still missing. In this paper, we fill up this gap in the oscillation theory; and moreover, we present such a comparison result that works also for the differential equation with mixed arguments.

In this paper, we study the canonical case of (), that is, There is a permanent interest in studying the properties of third-order differential equations.Various techniques have been presented for such equations (see ).Comparison theorems, especially, are a very strong and effective tool in the oscillation theory.Mahfoud [18] has presented a very useful comparison technique for studying the properties of delay differential equations through those of differential equations without delay.However, the corresponding result for advanced differential equation is still missing.In this paper, we fill up this gap in the oscillation theory; and moreover, we present such a comparison result that works also for the differential equation with mixed arguments.

The Main Results
The following result is crucial for our intended comparison theorem.
Proof.It follows from the monotonicity of that is, On the other hand, since () → ∞ as  → ∞, then for any  ∈ (0, 1) there exists a  1 large enough, such that or equivalently Using (6) We introduce the structure of nonoscillatory solutions.
This structure follows from canonical case of (), and the proof can be omitted.It is well known that for a particular case of (), namely, for the equation there always exists a solution satisfying ( 0 ), and this fact leads to the following definition, which is due to Kiguradze and Chanturia [15].
We are prepared to present the main result.
Proof.Assume the contrary, that is, we admit that () possesses a positive solution () satisfying ( 2 ).Thus, it follows from Lemma 1 that, for every  ∈ (0, 1), Integrating from  to ∞, we obtain that Integrating twice from  1 to , we are led to Let us denote the right-hand side by ().Then, () ≥ (), () satisfies ( 2 ), and which contradicts with the assumptions of the theorem, and we conclude that () has property (A).
Theorem 4 reduces the examination of properties of differential equations with mixed arguments to that of simpler equations without deviating arguments and permits to extend immediately the criteria known for property (A) of equations without deviating arguments to more general equations with both advanced and delay arguments.We provide some applications of our main result.
For our further references, we set that where  ∈ (0, 1).
Proof.By Theorem 4, it is sufficient to show that the differential inequality has property (A) for some  ∈ (0, 1).Assuming the contrary, we admit that ( 2 ) does not property (A), that is, it possesses an eventually positive solution () satisfying ( 2 ).We define have the following: Differentiating (), one gets in view of ( 2 ) that On the other hand, using the monotonicity of [()[  ()]  ]  , we have eventually; let us say that for  ≥  1 or equivalently where ℓ ∈ (0, 1) will be specified latter.Setting the last inequality into (17), we obtain Integrating the last inequality from  to ∞, we have or eventually; let us say that  ≥  1 .Since () > (), then inf Thus, From ( 15), we see that there exist some ℓ ∈ (0, 1) and some positive , such that Combining ( 24) with (25), we have ≥ 1 +  1+1/  > 1 +  1+1/ ( + 1) or equivalently This contradicts with the fact that the function is positive for all  > 0, and we conclude that () cannot satisfy ( 2 ); consequently, ( 1 ) has property (A).
In the following criteria, we utilize existing results known for differential equations without deviating arguments to provide new criteria for differential equations with mixed arguments.Proof.By Theorem 4, ( 4 ) enjoys property (A), provided that, for some  ∈ (0, 1), differential inequality has property (A).But by Corollary 1 in [16], this is equivalent to property (A) of the corresponding differential equation ( 3 ).

Corollary 7.
Assume that at least one of the following conditions: holds, then ( 3 ) has property (A).
Proof.It is known (see, e.g., [12,15]) that lim inf guarantees property (A) of The rest follows from Corollary 6, where constant  ∈ (0, 1) is eliminated due to the sharp inequality of the used criterion.
We support our results by several illustrative examples.
Our results take into account the value of advanced argument, which can be seen in the following example.where ,  > 0, 0 <  < 1.Both conditions of Corollary 7 reduce to which yields property (A) of ( 3 ).

Conclusion and Discussion
In the paper, we presented new comparison theorems for studying the properties of third-order differential equations with mixed arguments through those of the corresponding differential equations without deviating arguments.The presented technique permits to extend immediately the results known for an equation without deviating arguments to a more general equation with advanced and delay arguments.The results obtained have been supported by several illustrative examples.