In what follows,
denotes the set of real numbers,
,
denotes the set of integers,
denotes the set of nonnegative integers,
denotes the class of all continuous functions defined on set
with range in the set
,
is an arbitrary time scale,
denotes the set of rd-continuous functions,
denotes the set of all regressive and rd-continuous functions, and
. We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper, we always assume that
,
,
, and
are real constants, and
.
We firstly introduce the following lemmas, which are useful in our main results.
Lemma 2.1 ([15] (Bernoulli's inequality)).
Let
and
. Then 
Lemma 2.2 ([2]).
Let
and
be continuous at
, where
with
. Assume that
is rd-continuous on
. If for any
, there exists a neighborhood
of
, independent of
, such that
where
denotes the derivative of
with respect to the first variable, then
implies
Lemma 2.3 ([2] (Comparison Theorem)).
Suppose
,
. Then
implies
Lemma 2.4 (see [13]).
Let
,
,
, and
be nonnegative. If
is nondecreasing, then
implies
Next, we establish our main results.
Theorem 2.5.
Assume that
,
,
,
,
and
are nonnegative. Then
implies
where
Proof.
Define a function
by
Then (2.8) can be restated as
Using Lemma 2.1, from the above inequality, we easily obtain
It follows from (2.12) and (2.15) that
where
and
are defined as in (2.10) and (2.11), respectively. Using Lemma 2.3 and noting
, from (2.16) we have
Therefore, the desired inequality (2.9) follows from (2.14) and (2.17). This completes the proof of Theorem 2.5.
Corollary 2.6.
Assume that
,
, and
are nonnegative. If
is a constant, then
implies
where
Proof.
Letting
,
, and
in Theorem 2.5, we obtain
Therefore,
The proof of Corollary 2.6 is complete.
Remark 2.7.
The result of Theorem 2.5 holds for an arbitrary time scale. Therefore, using Theorem 2.5, we can obtain many results for some peculiar time scales. For example, letting
and
, respectively, we have the following two results.
Corollary 2.8.
Let
and assume that
, and
. Then the inequality
implies
where
and
are defined as in Theorem 2.5.
Corollary 2.9.
Let
and assume that
,
,
,
, and
are nonnegative functions defined for
. Then the inequality
implies
where
and
are defined as in Theorem 2.5.
Investigating the proof procedure of Theorem 2.5 carefully, we easily obtain the following more general result.
Theorem 2.10.
Assume that
,
,
,
,
, and
are nonnegative,
If there exists a series of positive real numbers
such that
,
then
implies
where
Theorem 2.11.
Assume that
,
,
,
,
,
and
are nonnegative. If
is defined as in Lemma 2.2 such that
and
for
with
, then
implies
where
Proof.
Define a function
by
Then
. As in the proof of Theorem 2.5, we easily obtain (2.14) and (2.15). Using Lemma 2.2 and combining (2.34) and (2.15), we have
where
and
are defined as in (2.32) and (2.33), respectively. Therefore, in the above inequality, using Lemma 2.3 and noting
, we get
It is easy to see that the desired inequality (2.31) follows from (2.14) and (2.36). This completes the proof of Theorem 2.11.
Corollary 2.12.
Let
and assume that
,
. If
and its partial derivative
are real–valued nonnegative continuous functions for
with
, then the inequality
implies
where
Corollary 2.13.
Let
and assume that
,
,
,
,
and
are nonnegative functions defined for
. If
and
are real-valued nonnegative functions for
with
, then the inequality
implies
where
for
with
,
Corollary 2.14.
Suppose that
, and
are defined as in Theorem 2.11. If
is nondecreasing for
, then
implies
where
Proof.
Letting
,
,
and
in Theorem 2.11, we obtain
where the inequality holds because
is nondecreasing for
. Therefore, using Theorem 2.11 and noting (2.46), we easily have
The proof of Corollary 2.14 is complete.
By Theorem 2.11, we can establish the following more general result.
Theorem 2.15.
Assume that
,
,
,
,
,
, and
are nonnegative,
and there exists a series of positive real numbers
such that
,
. If
is defined as in Lemma 2.2 such that
and
for
with
, then
implies
where
Theorem 2.16.
Let
,
and
be nonnegative,
, and
be nondecreasing. Assume that there exists a series of positive real numbers
such that
,
. If
is a continuous function such that
for
and
,
where
is a nonnegative continuous function,
, then
implies
where
Proof.
Let
Then (2.52) can be restated as
It is easy to see that
,
, and
is nondecreasing. Using Lemma 2.4, from (2.58), we have
where
is defined as in (2.54). It follows from (2.57) and (2.59) that
Using Lemma 2.1 to the above inequality, we obtain
Noting the hypotheses on
, from (2.62), we get
where
is defined by (2.55). Clearly,
and
are nondecreasing. Therefore, for any
, from (2.63), we obtain
Let
and define
by the right hand of (2.64). Then
,
,
, and
where
is defined by (2.56). Using Lemma 2.3 and noting
, from (2.66), we have
Therefore,
It follows from (2.61) and (2.68) that
Letting
in (2.69), we immediately obtain the desired inequality (2.53). This completes the proof of Theorem 2.16.
Corollary 2.17.
Let
,
,
, and
be nondecreasing. Assume that there exists a series of positive real numbers
such that
,
. If
is a continuous function such that
for
and
,
where
is a continuous function,
, then
implies
where
is defined as in (2.56),
Corollary 2.18.
Let
,
,
be nondecreasing,
and
be nonnegative functions defined for
. Assume that there exists a series of positive real numbers
such that
,
. If
such that
for
and
,
where
,
, then
implies
where
is defined as in (2.56),
Remark 2.19.
Using our main results, we can obtain many dynamic inequalities for some peculiar time scales. Due to limited space, their statements are omitted here.