Now, we are in a position to state and prove the main results which guarantee that every solution of Eq. (1.1) oscillates.
Theorem 3.1 Assume that (1.5) holds and . Furthermore, assume that there exists a positive function such that for sufficiently large ,
(3.1)
Then Eq. (1.1) is oscillatory on .
Proof Suppose to the contrary that Eq. (1.1) has a nonoscillatory solution x on . Without loss of generality, we may assume that there exists such that , for all . We shall consider only this case, since the case when x is eventually negative is similar. From (1.1), (2.3) and (C3), we have
(3.2)
Define the function ω by
(3.3)
Then . Using the product rule and the quotient rule, we get
(3.4)
where is shorthand for . Using Lemma 2.3 and (C2), it follows that
(3.5)
and
(3.6)
From (3.2), (3.3) and (3.4), we obtain
(3.7)
It follows from (2.5), (3.5) and (3.7) that
(3.8)
If , from Lemmas 2.2, 2.3 and (2.1), we have
From the above inequality and (3.5), we get
(3.9)
Hence, from (3.6), (3.8) and (3.9), we obtain
(3.10)
If , from Lemmas 2.2, 2.3 and (2.1), we have
Similarly to the proof of the previous case, we still get (3.10). Therefore, when , we obtain
(3.11)
where . Define X and Y by
and
Hence by (2.6), it follows that
(3.12)
In view of (3.11) and (3.12), we have
(3.13)
Integrating (3.13) from to t, we get
which leads to a contradiction with (3.1). This completes the proof. □
Remark 3.1 Note that in the special case when , then , , and , then (3.1) becomes the condition (4.1) in [27], so Theorem 4.1 in [27] is a special case of our results.
Remark 3.2 From Theorem 3.1, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of δ.
Taking and in Theorem 3.1 respectively, we have the following two results.
Corollary 3.1 Assume that (1.5) holds and . If for sufficiently large ,
then Eq. (1.1) is oscillatory on .
Corollary 3.2 Assume that (1.5) holds and . If for sufficiently large ,
then Eq. (1.1) is oscillatory on .
Theorem 3.2 Assume that (1.5) holds and . Furthermore, assume that there exist and a positive function such that for sufficiently large ,
(3.14)
Then Eq. (1.1) is oscillatory on .
Proof Suppose to the contrary that Eq. (1.1) has a nonoscillatory solution x on . Without loss of generality, we may assume that there exists such that , for all . We shall consider only this case, since the case when x is eventually negative is similar. We define the function ω by (3.3) again and proceeding as in the proof of Theorem 3.1, we have (3.13). Multiplying (3.13) by and integrating from to t, we get
(3.15)
Using the integration by parts formula, we arrive at
(3.16)
From the proof of Theorem 2.9 in [17], we obtain
(3.17)
where , . Combining (3.15) with (3.16) and (3.17), we have
that is,
which contradicts (3.14). This completes the proof. □
Theorem 3.3 Assume that (1.5) holds and . Furthermore, assume that there exist and a positive function such that for sufficiently large ,
(3.18)
where
Then Eq. (1.1) is oscillatory on .
Proof Suppose to the contrary that Eq. (1.1) has a nonoscillatory solution x on . Without loss of generality, we may assume that there exists such that , for all . We shall consider only this case, since the case when x is eventually negative is similar. We define the function ω by (3.3) as before and proceed as in the proof of Theorem 3.1 to obtain (3.11). Then from (3.11) we have
Multiplying the above inequality by and integrating from to t, we get
(3.19)
Combining (3.19) with (3.16) and (3.17), we obtain
(3.20)
Define X and Y by
and
Hence by (2.6), it follows that
(3.21)
From (3.20) and (3.21), we have
that is,
This is a contradiction with (3.18). This completes the proof. □
Theorem 3.4 Assume that (1.5) holds and . Furthermore, assume that there exist functions , where such that
(3.22)
and H has a nonpositive continuous Δ-partial derivation with respect to the second variable and satisfies
(3.23)
and for sufficiently large ,
(3.24)
where is a positive function. Then Eq. (1.1) is oscillatory on .
Proof Suppose to the contrary that Eq. (1.1) has a nonoscillatory solution x on . Without loss of generality, we may assume that there exists such that , for all . We shall consider only this case, since the case when x is eventually negative is similar. We define the function ω by (3.3) as before and proceed as in the proof of Theorem 3.1 to obtain (3.11). Multiplying (3.11) by and integrating from to t, we get
Integrating the left side by parts and from (3.22), we obtain
Therefore,
(3.25)
Define X and Y by
and
Hence by (2.6), it follows that
(3.26)
From (3.25) and (3.26), we get
that is,
which is a contradiction with (3.24). This completes the proof. □
Theorem 3.5 Assume that (1.6) holds and . Furthermore, assume that there exists a positive function such that for sufficiently large , (3.1) or (3.14), (3.18) and (3.24) hold, where , P, H and h are defined as in Theorems 3.3 and 3.4. If there exists a positive function , such that
(3.27)
then Eq. (1.1) is oscillatory on .
Proof Suppose to the contrary that Eq. (1.1) has a nonoscillatory solution x on . Without loss of generality, we may assume that there exists such that , for all . We shall consider only this case, since the case when x is eventually negative is similar. Proceeding as in the proof of Lemma 3.5 in [27], we have
(3.28)
By Lemma 2.1, we get is either eventually positive or eventually negative. Hence, there are two possible cases.
Case (I). , .
The proof when is eventually positive is similar to that of Theorems 3.1 or 3.2, 3.3 and 3.4, so we omit the details.
Case (II). , .
From (3.28) and Lemma 2.1, we have
that is,
Hence
(3.29)
Integrating (3.29) from t to ∞, we get
where . Using (3.29) and the above inequality in Eq. (1.1), we obtain
(3.30)
Define the function u by
Then . Using the product rule and from (3.30), we find that
(3.31)
Since , then from Lemma 2.5, it follows
Therefore, for all ,
(3.32)
Letting and using (3.27) in (3.32) yields . This is a contradiction with the fact that . This completes the proof. □
Remark 3.3 We note that (3.27) becomes (4.9) in [27] when , and so Theorem 3.5 in this paper includes Theorem 4.3 given in Zhang [27].