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.