Oscillation Results for Second Order Neutral Equations with Distributed Deviating Arguments

Abstract The oscillation of second order neutral equations with distributed deviating arguments is studied. By using a class of parameter functions Φ(t, s, l) and the generalized Riccati technique, some new oscillation criteria for the equations are obtained. The obtained results are different from most known ones and can be applied to many cases which are not covered by existing results. Two examples are also included to show the significance of our results.

We restrict our attention to solutions x(t) of Eq. ( 1) which exist on some half-line [t, ∞) and nontrivial for all large t.It is tacitly assumed that such solutions exist.As is customary, a solution x(t) of Eq. ( 1) is called oscillatory if it has arbitrarily large zeros.Otherwise, it is said to be nonoscillatory.Eq. ( 1) is called oscillatory if all its solutions are oscillatory.
As we can see, an important tool in the study of oscillatory behavior of solutions for the equations above is the averaging technique, which involves a function class X which is defined by Philos(1989, pp.482-492) where h 1 , h 2 ∈ L 2 loc (D, R).Following the ideas of Sun(2004, pp.341-351),Sun and Meng(2007, pp.1310-1316), Yang(2007, pp.900-907) and Dubé and Mingarelli(2005, pp.208-220), in this paper, we define another function class Y.We say that a function Φ = Φ(t, s, l) belongs to the function class Y, denoted by In Sections 2 and 3 of this paper, we will establish some new oscillation results for Eq. ( 1) by using the auxiliary function Φ ∈ Y.Our results are different from most known ones in the sense that they are given in the form that lim sup t→∞ [•] is greater than a constant, rather than in the form lim sup t→∞ [•] = ∞ as usual.Thus, our results can be applied to many cases, which are not covered by existing ones.Finally in Section 4, two examples that show the importance of our results are included.

Oscillation criteria of Kamenev type
Theorem 2.1 Suppose that there exist functions q(t, ξ) ∈ C(I × [a, b], R 0 ), which is not eventually zero on any ray and If there exist functions Φ ∈ Y and ρ(t) ∈ C 1 (I, R + ) such that for each l ≥ t 0 , lim sup where then Eq. ( 1) is oscillatory.
If x(t) is an eventually negative solution of Eq. ( 1), let w(t) = −x(t) , then Eq. (1) will transfer the following equation It is easy to see that w(t) is an eventually positive solution of Eq. ( 12).From ( 3) and ( 4), we can obtain Then, Eq. ( 12) satisfies the conditions of Theorem 2.1.Defining ,and using the above-mentioned method, we can also get a contradiction.This completes the proof of Theorem 2.1.
Under the appropriate choices of the functions Φ(t, s, l), we can derive many new oscillation criteria for Eq. ( 1) from Theorem 2.1.For instance, let Φ(t, s, l) = √ H(t, s)H (s, l), where H ∈ X.By Theorem 2.1, we have the following oscillation result.
, where φ(t) ∈ C 1 (I, R + ) and m, n > 1 are constants, then we have the following oscillation theorem by Theorem 2.1.
Theorem 2.3 Suppose that (3) and ( 4) hold.If there exist functions ρ(t), φ(t) ∈ C 1 (I, R + ) and constants m, n > 1 such that for each l ≥ t 0 , lim sup where Q 1 (t) is defined as in ( 6), then Eq. ( 1) is oscillatory. Define and let where φ(t) ∈ C 1 (I, R + ), and m, n > 1 are constants.According to the simple computation, we get the following oscillation criterion by Theorem 2.1.
Theorem 2.5 Suppose that (3) and (4) hold, lim t→∞ R(t) = ∞ and g (t, a) ≥ k > 0 for t ∈ I, where k is a constant.If there exist constants m, n > 1 such that for each l ≥ t 0 , lim sup where then Eq. ( 1) is oscillatory.
Proof Assume that Eq. ( 1) has a nonoscillatory solution x(t) > 0. By using the same arguments as in the proof of Theorem 2.1, we conclude that (11) with ρ(t) ≡ 1 is satisfied, i.e., Since g (t, a) ≥ k > 0 for t ∈ I, we have Letting x = u v , and using the following Euler's Beta function, Substituting back in for v = R(t) − R(l), ( 16) and ( 17) From ( 15) and ( 18), we can easily obtain lim sup which contradicts the assumption (13) .This completes the proof of Theorem 2.5.
Based on Theorem 2.5 we obtain the following corollary.
Corollary 2.1 Suppose that (3) and (4) hold, lim t→∞ R(t) = ∞ and g (t, a) ≥ k > 0 for t ∈ I, where k is a constant.If there exists a constant α > 1 2 such that for each l ≥ t 0 either where Q 2 (t) is defined as in ( 14), then Eq. ( 1) is oscillatory.
(ii) In ( 13), replaced m, n by 2 and 2α, respectively, the remainder of the proof is similar to that of (i) and hence omitted.

Interval oscillation criteria
In this section, we will establish several new interval oscillation criteria for Eq.( 1), that is, criteria given by the behavior of Eq.( 1) only on a sequence of subintervals of [t 0 , ∞) rather than on the whole half-line.
Proof As in the proof of Theorem 2.1, with t and l replaced by d and c, respectively.We can easily see that every solution of Eq. ( 1) has at least one zero in (c, d), i.e., every solution of Eq. ( 1) has arbitrarily large zero on [t 0 , ∞).This completes the proof of Theorem 3.1.
As consequences of Theorem 3.1 we get the following interval oscillation criteria for Eq. ( 1).
Remark 3.1 Theorems 2.1-2.5 and 3.1, Corollaries 2.1 and 3.1-3.3are new because we introduce a new class of kernel functions Φ(t, s, l) which is basically a product H(t, s)H(s, l) for a kernel H(t, s) of Philos' type.
Remark 3.2 Since the integral of Eq. ( 1) is a Stieltjes integral, the criteria in this paper are adapted to the following equation:

Examples
In this section, we will present two examples to illustrate our results.To the best of our knowledge, no previous criteria for oscillation can be applied to these examples.We first give an example to illustrate Corollary 2.1.