### 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.