In this section, we will first recall some basic definitions and lemmas which are used in what follows.

Definition 2.1 (see [8]).

A time scale

is an arbitrary nonempty closed subset of the real set

with the topology and ordering inherited from

. The forward and backward jump operators

and the graininess

are defined, respectively, by

The point
is called left-dense, left-scattered, right-dense, or right-scattered if
,
, and
or
, respectively. Points that are right-dense and left-dense at the same time are called dense. If
has a left-scattered maximum
, defined
; otherwise, set
. If
has a right-scattered minimum
, defined
; otherwise, set
.

Definition 2.2 (see [8]).

For

and

, then the delta derivative of

at the point

is defined to be the number

(provided it exists) with the property that for each

, there is a neighborhood

of

such that

For

and

, then the nabla derivative of

at the point

is defined to be the number

(provided it exists) with the property that for each

, there is a neighborhood

of

such that

Definition 2.3 (see [8]).

A function
is rd-continuous provided it is continuous at each right-dense point in
and has a left-sided limit at each left-dense point in
. The set of rd-continuous functions
will be denoted by
. A function
is left-dense continuous (i.e., ld-continuous) if
is continuous at each left-dense point in
and its right-sided limit exists (finite) at each right-dense point in
. The set of left-dense continuous functions
will be denoted by
.

Definition 2.4 (see [8]).

If

, then we define the delta integral by

If

, then we define the nabla integral by

Lemma 2.5 (see [8]).

If

and

, then

If

and

, then

be endowed with the norm

, where

and choose a cone

defined by

Lemma 2.6.

If
, then
for all
.

Proof.

If

, then

. It follows that

With

and

, one obtains

From (2.11)–(2.13), we can get that

So Lemma 2.6 is proved.

Lemma 2.7.

is a solution of BVPs (1.1)-(1.2) if and only if

is a solution of the following integral equation:

Proof.

First assume

is a solution of BVPs (1.1)-(1.2); then we have

Integrating (2.18) from

to

, it follows that

Together with (2.19) and

, we obtain

Substituting (2.22) into (2.19), we obtain

The proof of sufficiency is complete.

Conversely, assume

is a solution of the following integral equation:

So

. Furthermore, we have

The proof of Lemma 2.7 is complete.

Define the operator

by

for all
. Obviously,
for all
.

Lemma 2.8.

If
, then
.

Proof.

It is easily obtained from the second part of the proof in Lemma 2.7. The proof is complete.

Lemma 2.9.

is complete continuous.

Proof.

First, we show that

maps bounded set into itself. Assume

is a positive constant and

. Note that the continuity of

guarantees that there is a

such that

for all

. So we get from

and

that

That is,

is uniformly bounded. In addition, notice that

So
is equicontinuous for any
. Using Arzela-Ascoli theorem on time scales [17], we obtain that
is relatively compact. In view of Lebesgue's dominated convergence theorem on time scales [18], it is easy to prove that
is continuous. Hence,
is complete continuous. The proof of this lemma is complete.

Let

and

be nonnegative continuous convex functionals on a pone

,

a nonnegative continuous concave functional on

, and

positive numbers with

we defined the following convex sets:

and introduce two assumptions with regard to the functionals
,
as follows:

(H1) there exists
such that
for all
;

(H2)
for any
and
.

The following fixed point theorem duo to Bai and Ge is crucial in the arguments of our main result.

Lemma 2.10 (see [19]).

Let
be Banach space,
a cone, and
,
. Assume that
and
are nonnegative continuous convex functionals satisfying (H1) and (H2),
is a nonnegative continuous concave functional on
such that
for all
, and
is a complete continuous operator. Suppose

(C1)
,
for
;

(C2)
,
for
;

(C3)
for
with
.

Then

has at least three fixed points

with