2.1. Time Scales
For the sake of self-consistency, we import some necessary definitions and lemmas on time scales. More details can be found in [4] and reference therein. First of all, a time scale
is any nonempty closed subset of real numbers
with order and topological structure defined in a canonical way, as mentioned above. Then, we have the following definition of the categories of points on time scales.
Definition 2.1.
For
and
, define the forward jump operator
and the backward jump operator
, respectively, by
Then, the graininess operator
is defined as
In addition, if
,
is said to be right scattered, and 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
, denote by
; otherwise, set
. If
has a left scattered maximum
, denote by
; otherwise, set
.
The following definitions describe the categories of functions on time scales and the basic computations of integral and derivative.
Definition 2.2.
Assume that
is a function and that
.
is supposed to be the number (provided it exists) with the property that given any
; there is a neighborhood
of
satisfying
for all
. Then
is said to be the delta derivative of
at
. Similarly, assume that
is a function and that
. Denote by
the number (provided it exists) with the property that given any
, there is a neighborhood
of
such that
for all
. Then
is said to be the nabla derivative of
at
.
Definition 2.3.
A function
is left dense continuous (
-continuous) provided that it is continuous at all left dense points of
, and its right side limits exists (being finite) at right dense points of
. Denote by
the set of all left dense continuous functions on
.
Definition 2.4.
Let
be a function, and
. If there exists a function
such that
for all
, then
is a delta antiderivative of
. In this case the integral is given by the formula
Analogously, let
be a function, and
. If there exists a function
such that
for all
, then
is a nabla antiderivative of
. In this case, the integral is given by the formula
2.2. Main Lemmas
This subsection aims to establish several lemmas which are useful in the proof of the main results in this paper. In particular, these lemmas focus on the following linear boundary value problems:
Lemma 2.5.
If
, then, for
, the linear boundary value problems (2.6) and (2.7) have a unique solution satisfying
. Here,
Proof.
It follows from (2.8) that
Thus, we obtain that
and that
Then,
satisfies (2.6), which verifies that
is a solution of the problems (2.6) and (2.7). Furthermore, in order to show the uniqueness, we suppose that both
and
are the solutions of the problems (2.6) and (2.7). Then, we have
In fact, (2.13) further yields
Hence, from (2.14) and (2.16), the assumption
, and the definition of the
-Laplacian operator, it follows that
This equation, with (2.15), further implies
which consequently leads to the completion of the proof, that is,
specified in (2.8) is the unique solution of the problems (2.6) and (2.7).
Lemma 2.6.
Suppose that
. If
, then the unique solution of the problems (2.6) and (2.7) satisfies
Proof.
Observe that, for any 
Thus, by (2.8) specified in Lemma 2.5, we get
Thus,
is nondecreasing in the interval
. In addition, notice that
The last term in the above estimation is no less than zero owing to those assumptions. Therefore, from the monotonicity of
, we get
which consequently completes the proof.
Lemma 2.7.
Suppose that
. If
, then the unique positive solution of the problems (2.6) and (2.7) satisfies
for
with 
Proof.
Since
is nondecreasing in the interval 
On the other hand,
Hence,
This completes the proof.
Now, denote by
and by
, where
. Then, it is easy to verify that
endowed with
becomes a Banach space. Furthermore, define a cone, denoted by
, through
Also, for a given positive real number
, define a function set
by
Naturally, we denote by
and by
. With these settings and notations, we are in a position to have the following properties.
Lemma 2.8.
If
then (i)
for any
; (ii)
for any pair of
with
.
The proof of this lemma, which could be found in [19, 21], is directly from the specific construction of the set
. Next, let us construct a map
through
for any
. Here,
. Thus, we obtain the following properties on this map.
Lemma 2.9.
Assume that the hypotheses
are all fulfilled. Then,
and
is completely continuous.
Proof.
At first, arbitrarily pick up
. Then it directly follows from Lemma 2.6 that
for all
. Moreover, direct computation yields
for all
, and
for all
. Thus, the latter inequality implies that
is decreasing on
. This implies that
for
. Consequently, we complete the proof of the first part of the conclusion that
for any
.
Secondly, we are to validate the complete continuity of the map
. To approach this aim, we have to verify that
is bounded, where
is obviously bounded. It follows from the proof of Lemma 2.7 that
where
. This manifests the uniform boundedness of the set
. In addition, for any given
with
we have the following estimation:
This validates the equicontinuity of the elements in the set
. Therefore, according to the Arzelà-Ascoli theorem on time scales [2], we conclude that
is relatively compact. Now, let
with
. Then
for all
and
Also,
is uniformly valid on
. These, with the uniform continuity of
on the compact set
, leads to a conclusion that
is uniformly valid on
. Hence, it is easy to verify that
as
tends toward positive infinity. As a consequence, we complete the whole proof.