For convenience, we list the following definitions which can be found in [1–5].
Definition 2.1.
A time scale
is a nonempty closed subset of real numbers
. For
and
, define the forward jump operator
and backward jump operator
, respectively, by
for all
. If
,
is said to be right scattered, if
,
is said to be left scattered; if
,
is said to be right dense, and if
,
is said to be left dense. If
has a right scattered minimum
, define
; otherwise set
. If
has a left scattered maximum
, define
; otherwise set
.
Definition 2.2.
For
and
, the delta derivative of
at the point
is defined to be the number
(provided that it exists), with the property that for each
, there is a neighborhood
of
such that
for all
.
For
and
, the nabla derivative of
at
, denoted by
(provided it exists) with the property that for each
, there is a neighborhood
of
such that
for all
.
Definition 2.3.
A function
is left-dense continuous (i.e.,
-continuous), if
is continuous at each left-dense point in
and its right-sided limit exists at each right-dense point in
.
Definition 2.4.
If
, then we define the delta integral by
If
, then we define the nabla integral by
Lemma 2.5.
If condition
holds, then for
, the boundary value problem (BVP)
has the unique solution
Proof.
By caculating, we can easily get (2.7). So we omit it.
Lemma 2.6.
If condition
holds, then for
, the boundary value problem (BVP)
has the unique solution
where
.
Proof.
Integrating both sides of equation in (2.8) on
, we have
So,
By boundary value condition
, we have
By (2.10) and (2.12) we know
This together with Lemma 2.5 implies that
where
. The proof is complete.
Lemma 2.7.
Let condition
holds If
and
, then the unique solution
of (2.8) satisfies
Proof.
By
, we can know that the graph of
is concave down on
, and
is nonincreasing on
. This together with the assumption that the boundary condition
implies that
for
. This implies that
So we only prove
By condition
we have
The proof is completed.
Lemma 2.8.
Let condition
hold. If
and
, then the unique positive solution
of (BVP) (2.8) satisfies
where
,
.
Proof.
By
, we can know that the graph of
is concave down on
, and
is nonincreasing on
. This together with the assumption that the boundary condition
implies that
for
. This implies that
For all
, we have from the concavity of
that
that is,
This together with the boundary condition
implies that
This completes the proof.
Let
be endowed with the ordering
if
for all
and
is defined as usual by maximum norm. Clearly, it follows that
is a Banach space.
For the convenience, let
We define two cones by
where
,
is defined in Lemma 2.8 and
Define the operators
and
by setting
where
,
where
,
, and
. Obviously,
is a solution of the BVP(1.6) if and only if
is a fixed point of operator
.
Lemma 2.9.
is completely continuous.
Proof.
It is easy to see that
by
and Lemma 2.8. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operator
is completely continuous.
Lemma 2.10 (see [20, 21]).
Let
be a cone in a Banach space
. Let
be an open bounded subset of
with
and
. Assume that
is a compact map such that
for
. Then the following results hold.
(1)If
,
, then
.
(2)If there exists
such that
for all
and all
, then
.
(3)Let
be open in
such that
. If
and
, then
has a fixed point in
. The same result holds if
and
, where
denotes fixed point index.
We define
Lemma 2.11 (see [20]).
defined above has the following properties:
(a)
(b)
is open relative to K;
(c)
if and only if 
(d)if
, then
for
.
For the convenience, we introduce the following notations:
Remark 2.12.
By
we can know that
, 
Lemma 2.13.
If
satisfies the following condition :
then
Proof.
For
, then from (2.30) we have
So that
Therefore,
This implies that
for
. Hence by Lemma 2.10(1) it follows that
.
Lemma 2.14.
If
satisfies the following condition:
then
Proof.
Let
for
. Then
. We claim that
In fact, if not, there exist
and
such that
. By
, we have
So that
For
, then
This together with Lemma 2.11(c) implies that
a contradiction. Hence by Lemma 2.10(2) it follows that
.