Oscillations for Neutral Functional Differential Equations

We will consider a class of neutral functional differential equations. Some infinite integral conditions for the oscillation of all solutions are derived. Our results extend and improve some of the previous results in the literature.

There are numerous numbers of oscillation criteria obtained for oscillation of all solutions of (1). In particular, many various sufficient conditions for oscillation are established in [3-5, 9-15, 18, 19]. In reviewing the literature, (1) is much studied in the case when which has been considered as an essential condition for the oscillation.
Recently, Ahmed et al. [2,3] investigated the oscillation behaviour of (1) and obtained some new oscillation results. Additional results on the oscillation behaviour of (1) can also be found in the articles of Kulenović et al. [14], Kubiaczyk and Saker [13], and Greaf et al. [11].
Define the functions ( ) and ( ) as follows: If ( ) is an eventually positive solution of the equation 2 The Scientific World Journal then ( ) and ( ) are also solutions of (7). Furthermore, ( ) is a differentiable solution, while ( ) is twice differentiable. (see Győri and Ladas [12]). As usual, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is either eventually positive or eventually negative. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
In the sequel, unless otherwise specified, when we write a functional inequality, we assume that it holds for all sufficiently large .

Auxiliary Lemmas
To specify the proofs of our main results, we need the following essential lemmas.

(a) ( ) is a decreasing function and either
The following statements are equivalent: (c) The following statements are equivalent: Lemma 2 (see [16]). Assume that If ( ) is an eventually positive solution of the delay differential equation then, for the same , Lemma 3 (see [16]). If the equation has an eventually positive solution, then one has eventually that Lemma 4 (see [12]). Assume that Then the differential inequality has an eventually positive solution if and only if the equation has an eventually positive solution.

Main Results
In this section, we establish some infinite integral conditions for all solutions of (1) to oscillate. We assume that condition (3) holds. (2) and (3) hold with −1 ≤ ≤ 0,

Theorem 5. Let conditions
Then every solution of (1) is oscillatory.
Proof. Assume that (1) has a nonoscillatory solution on [ 0 , ∞). Then, without loss of generality, there is a 1 ∈ [ 0 , ∞), sufficiently large, so that ( ) > 0, ( − ) > 0 and ( − ) > 0 on [ 1 , ∞). Set ( ) to be defined as in (5). Then by Lemma 1, it follows that As ( ) > ( ), it follows from (1) that Dividing the last inequality by ( ) > 0, we obtain Let The Scientific World Journal 3 This implies that ( ) > 0. Substituting in (22) yields So by Lemma 4, we have that the delay differential equation has an eventually positive solution as well. Let Then ( ) is positive and continuous, and there exists 1 ≥ 0 such that ( 1 ) > 0, and Furthermore, ( ) satisfies the generalized characteristic equation where to (31), we have By interchanging the order of integration, we get Hence Then From (35) and (38), we find that However, using Lemma 3, it follows that eventually. Therefore, from (40) in (39), we get That is, which implies by condition (19) that On the other hand, from Lemma 2, we have This is a contradiction with (43). The proof is complete. (46) Observe that Then All conditions of Theorem 5 are satisfied. Then all solutions of (45) oscillate.  (5) and (6). It is easily seen, by direct substituting, that ( ) and ( ) are also solutions of (1) when and are constants; that is By Lemma 1, we have that ( ) is decreasing and ( ) > 0. Also, we have indeed that Then Using (54) in (52) implies that As > −1, we have 1 + > 0. Then In view of the -periodicity of ( ), (56) implies that As ( ) is positive solution, so by Lemma 4, the delay differential equation has an eventually positive solution as well. Let Then ( ) is positive and continuous, and there exists 1 ≥ 0 such that ( 1 ) > 0, and Furthermore, ( ) satisfies the generalized characteristic equation where Let Therefore Applying the inequality (32) to (64), we have The Scientific World Journal 5 or ( ) (∫ Then, for > , we have By interchanging the order of integration, we get Hence Then From (67) and (70), we find that However, using Lemma 3, it follows that eventually. Therefore, from (72) in (71), we get This is a contradiction with (75). The proof is complete.
Then all conditions of Theorem 7 are satisfied and therefore all solutions of (77) oscillate.