In this section, we give some new oscillation criteria for (1.1). In order to prove our main results, we will use the formula

where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller's chain rule (see Bohner and Peterson [3, Theorem 1.90]). Also, we need the following auxiliary result.

For convenience, we note that

Lemma 2.1.

Assume that (1.9) holds and assume further that is an eventually positive solution of (1.1). Then there exists a such that

The proof is similar to that of Hassan [7, Lemma 2.1], and so is omitted.

Lemma 2.2 (Chain Rule).

Assume that is strictly increasing and is a time scale, . Let . If , and let exist for , then exist, and

Proof.

Let be given and define Note that According to the assumptions, there exist neighborhoods of and of such that

Put and let Then and and

The proof is completed.

Theorem 2.3.

Assume that (1.9) holds and . Furthermore, assume that there exists a positive function and for all sufficiently large , one has

Then (1.1) is oscillatory on .

Proof.

Suppose that (1.1) has a nonoscillatory solution on . We may assume without loss of generality that and for all . We shall consider only this case, since the proof when is eventually negative is similar. In view of Lemma 2.1, we get (2.3). Define the function by

Then . In view of (1.1) and (2.8) we get

When , using (2.1) and (2.4), we have

So, by (2.9)

hence, we get

and using Lemma 2.1, by we find , by we get and so we obtain

When , using (2.1) and (2.4), we have

So, we get

So by the definition of , we obtain, for ,

where . Define and by

Then, using the inequality

yields

From (2.16), we find

Integrating the inequality (2.20) from to , we obtain

which contradicts (2.7). The proof is completed.

Remark 2.4.

From Theorem 2.3, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of .

Theorem 2.5.

Assume that (1.9) holds and . Furthermore, assume that there exist functions , , where such that

and has a nonpositive continuous -partial derivation with respect to the second variable and satisfies

and for all sufficiently large ,

where is a positive -differentiable function and

Then (1.1) is oscillatory on .

Proof.

Suppose that (1.1) has a nonoscillatory solution on . We may assume without loss of generality that and for all . We proceed as in the proof of Theorem 2.3, and we get (2.16). Then from (2.16) with replaced by , we have

Multiplying both sides of (2.26), with replaced by , by , integrating with respect to from to , we get

Integrating by parts and using (2.22) and (2.23), we obtain

Again, let , define and by

and using the inequality

we find

Therefore, from the definition of , we obtain

and this implies that

which contradicts (2.24). This completes the proof.

Now, we give some sufficient conditions when (1.10) holds, which guarantee that every solution of (1.1) oscillates or converges to zero on .

Theorem 2.6.

Assume (1.10) holds and . Furthermore, assume that there exists a positive function , for all sufficiently large , such that (2.7) or (2.22), (2.23), and (2.24) hold. If there exists a positive function , such that

then every solution of (1.1) is either oscillatory or converges to zero on .

Proof.

We proceed as in Theorem 2.3 or Theorem 2.5, and we assume that (1.1) has a nonoscillatory solution such that , for all .

From the proof of Lemma 2.1, we see that there exist two possible cases for the sign of . The proof when is an eventually positive is similar to that of the proof of Theorem 2.3 or Theorem 2.5, and hence it is omitted.

Next, suppose that for . Then is decreasing and . We assert that . If not, then for . Since , there exists a number such that for .

Defining the function we obtain from (1.1)

Hence, for , we have

because of . So, we have

By condition (2.34), we get as , and this is a contradiction to the fact that for . Thus and then as . The proof is completed.