Before stating our main results, we begin with the following lemma (Second Mean Value Theorem) which will play an important role in the proof of our main results.
Lemma 2.1 ([[3], Theorem 5.45])
Let h be a bounded function that is integrable on . Let and be the infimum and supremum of the function on respectively. Suppose that g is a nonnegative and nonincreasing function on . Then there is some number Λ with such that
Our first result is stated in terms of an auxiliary function .
Theorem 2.1 Assume that f satisfies (1.3). If there exists a function ϕ such that
(2.1)
where
and
(2.2)
then every solution of equation (1.1) is oscillatory.
Proof Assume (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a solution x of (1.1) and a such that on . Define
Then from the product and quotient rules and the Pötzsche chain rule [[2], Theorem 1.90], we get
where for , . Therefore, from (1.1), we have
(2.3)
Since , , for all and , for all , we get
Integrating the above inequality from T to t (≥T), we obtain
(2.4)
We claim that is bounded above for all . Since and , we have from Lemma 2.1 that for each ,
(2.5)
where , and where m and M denote the infimum and supremum, respectively, of the function for . Define
and so
(2.6)
For a fixed point , we have
and so
Then
And so, from (2.6), we have
Also,
Then, from (1.3), we get
Hence, it follows that
and from (2.5), for all , we have
(2.7)
From (2.4) and (2.7), we get for ,
In view of condition (2.2), it follows from the last inequality that there exists a sufficiently large such that
Also, from (2.2), there exists such that
(2.8)
Indeed, (1.1) yields on integration
Now by the integration by parts, we have
and by the integration by parts and the Pötzsche chain rule and then by (2.8), we get
(2.11)
since and for all and for . Using (2.9) and (2.10) in (2.11), we get
and so
Since , we conclude from (2.2) that , which is a contradiction. This completes the proof. □
In the case , , , and , , Theorem 2.1 is due to Kiguradze [19].
If , , , , , , and , then Theorem 2.1 includes Theorem 4.1 in Hooker and Patula [[20], Theorem 4.1] and Mingarelli [21].
Suppose that there exists a function ϕ such that, for ,
(2.12)
Then, in this case, we have , and so we do not need to assume the superlinearity conditions (1.3), and so the result applies to linear, sublinear, and superlinear case.
Corollary 2.1 If there exists a function ϕ such that (2.2) and (2.12) hold, then every solution of equation (1.1) is oscillatory.
If we do not assume superlinearity condition (1.3) and condition (2.12), then we can conclude that all bounded solutions are oscillatory.
Corollary 2.2 If there exists a function ϕ such that (2.1) and (2.2) hold, then every bounded solution of equation (1.1) is oscillatory.
In the following, we assume that there exists a function ϕ such that
(2.13)
holds and establish some sufficient conditions for equation (1.1).
Theorem 2.2 Assume that f satisfies (1.3). If there exists a function ϕ such that (2.1) and (2.13) hold and if
(2.14)
and
(2.15)
then every solution of equation (1.1) is either oscillatory or tends to zero.
Proof Assume (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a solution x of (1.1) and a such that on . As in the proof of Theorem 2.1, by using (2.14), we have that there exists a sufficiently large such that
and so
which yields
That is,
so integration from to t () yields
(2.16)
Define
Then
(2.17)
Since , we have . If we assume , then
so
and let to get a contradiction to (2.15). This completes the proof. □
Corollary 2.3 If there exists a function ϕ such that (2.12), (2.13), (2.14), and (2.15) hold, then every solution of equation (1.1) is either oscillatory or tends to zero.
Corollary 2.4 If there exists a function ϕ such that (2.1), (2.13), (2.14), and (2.15) hold, then every bounded solution of equation (1.1) is either oscillatory or tends to zero.
Next we present oscillation criteria for equation (1.1) where f satisfies sublinearity condition (1.4).
Theorem 2.3 Assume that f satisfies (1.4). If there exists a function ϕ such that (2.12), (2.13), (2.14), and (2.15), then every solution of equation (1.1) is oscillatory.
Proof Assume (1.1) has a nonoscillatory solution x on . Then, without loss of generality, there is a solution x of (1.1) and a such that on . As in the proof of Theorems 2.1 and 2.2, we have that there exists a sufficiently large such that
and, from (2.16) and (2.17), we get
Since , then and so
Then
which is a contradiction to (2.15). This completes the proof. □
Theorem 2.4 Assume that f satisfies (1.4). If there exists a function ϕ such that (2.1), (2.13), (2.14), and (2.15) hold, then every bounded solution of equation (1.1) is oscillatory.
The results are very general. With appropriate choices of , we can obtain several sufficient conditions for the oscillation of equation (1.1).