In this section, we give and prove the main results of this paper.
First, we will show that the following second-order linear equation:
can be rewritten as (1.1).
Theorem 3.1.
If
and
is continuous, then (3.1) can be written in the form of (1.1), with
Proof.
Multiplying both sides of (3.1) by
, we get
where we used Lemma 2.1. This equation is in the form of (1.1) with
and
as desired.
Lemma 3.2 (Picone Identity).
Let
and
be the nontrivial solutions of (1.1) and (1.2) with
and
for
If
has no generalized zeros on
then the following identity holds:
Proof.
We first divide the left part of (3.4) into two parts
From (1.1) and the product rule (Lemma 2.1(ii), we have
It follows from (1.2), (2.4), product and quotient rules (Lemma 2.1(ii), (iii) and the assumption that
has no generalized zeros on
that
Combining
and
, we get (3.4). This completes the proof.
Now, we turn to proving the main result of this paper.
Theorem 3.3 (Sturm-Picone Comparison Theorem).
Suppose that
and
are the nontrivial solutions of (1.1) and (1.2), and
are two consecutive generalized zeros of
if
then
has at least one generalized zero on 
Proof.
Suppose to the contrary,
has no generalized zeros on
and
for all 
Case 1.
Suppose
are two consecutive zeros of
. Then by Lemma 3.2, (3.4) holds and integrating it from
to
we get
Noting that
we have
Hence, by (3.9) and
we have
which is a contradiction. Therefore, in Case 1,
has at least one generalized zero on 
Case 2.
Suppose
is a zero of
is a node of
and
It follows from the assumption that
has no generalized zeros on
and that
for all
that
Hence by (2.4) and
on
, we have
By integration, it follows from (3.12) and
that
So, from (3.9) and above argument we obtain that
which is a contradiction, too. Hence, in Case 2,
has at least one generalized zero on
.
Case 3.
Suppose
is a node of
and
is a generalized zero of
Similar to the discussion of (3.12), we have
which implies
(i)If
is a node of
then
Hence, we have (3.12), that is,
(ii)If
is a zero of
then
It follows from (3.4) and Lemma 2.3 that
is right-increasing on
Hence, from (i) and (ii) that
which implies
From (3.16), (3.21), and (2.4), we have
Further, it follows from (1.1), (1.2), product rule (Lemma 2.1(ii), and (3.22) that
If
and from
,
, and
we have
This contradicts (3.22). Note that
. It follows from
, (3.23), and (3.24) that
On the other hand, it follows from
and
are solutions of (1.1) and (1.2) that
Combining the above two equations we obtain
It follows from (3.27) and (2.4) that
Hence, from
and (3.21), we get
By referring to
and
it follows that
which contradicts 
It follows from the above discussion that
has at least one generalized zero on
This completes the proof.
Remark 3.4.
If
then Theorem 3.3 reduces to classical Sturm comparison theorem.
Remark 3.5.
In the continuous case:
. This result is the same as Sturm-Picone comparison theorem of second-order differential equations (see Section 1).
Remark 3.6.
In the discrete case:
. This result is the same as Sturm comparison theorem of second-order difference equations (see [8, Chapter 8]).
Example 3.7.
Consider the following three specific cases:
By Theorem 3.3, we have if
and
are the nontrivial solutions of (1.1) and (1.2),
are two consecutive generalized zeros of
and
then
has at least one generalized zero on
Obviously, the above three cases are not continuous and not discrete. So the existing results for the differential and difference equations are not available now.
By Remarks 3.4–3.6 and Example 3.7, the Sturm comparison theorem on time scales not only unifies the results in both the continuous and the discrete cases but also contains more complicated time scales.