In this section, we present the expression and properties of Green's function associated with boundary value problem ().
Lemma 2.1.
Assume that Then for any , the unique solution of boundary value problem
is given by
where
Proof.
By Lemma 1.4, we can reduce the equation of problem (2.1) to an equivalent integral equation
By , there is . Thus,
Differentiating (2.7), we have
By (2.8) and we have Similarly, we can obtain that Then
By , we have
Therefore, the unique solution of BVP (2.1) is
where is defined by (2.4).
From (2.11), we have
It follows that
Substituting (2.13) into (2.11), we obtain
where and are defined by (2.3), (2.4), and (2.5), respectively. The proof is complete.
From (2.3), (2.4), and (2.5), we can prove that and have the following properties.
Proposition 2.2.
The function defined by (2.4) satisfies
(i) is continuous for all ;
(ii)for all , one has
where
Proof.

(i)
It is obvious that is continuous on and when .
For , we have
So, by (2.4), we have
Similarly, for , we have .

(ii)
Since , it is clear that is increasing with respect to for .
On the other hand, from the definition of , for given , we have
Let
Then, we have
and so,
Noticing , from (2.22), we have
Then, for given , we have arrives at maximum at when . This together with the fact that is increasing on , we obtain that (2.15) holds.
Remark 2.3.
From Figure 1, we can see that for . If , then
Remark 2.4.
From Figure 2, we can see that is increasing with respect to .
Remark 2.5.
From Figure 3, we can see that for , where .
Remark 2.6.
Let . From (2.15), for , we have
Remark 2.7.
From (2.25), we have
Remark 2.8.
From Figure 4, it is easy to obtain that is decreasing with respect to , and
Proposition 2.9.
There exists such that
Proof.
For , we divide the proof into the following three cases for .
Case 1.
If , then from (i) of Proposition 2.2 and Remark 2.5, we have
It is obvious that and are bounded on . So, there exists a constant such that
Case 2.
If , then from (2.4), we have
On the other hand, from the definition of , we obtain that takes its maximum at . So
Therefore, . Letting , we have
Case 3.
If , from (i) of Proposition 2.2, it is clear that
In view of Remarks 2.6–2.8, we have
From (2.35), there exists a constant such that
Letting and using (2.30), (2.33), and (2.36), it follows that (2.28) holds. This completes the proof.
Let
Proposition 2.10.
If , then one has
(i) is continuous for all ;
(ii).
Proof.
Using the properties of , definition of , it can easily be shown that (i) and (ii) hold.
Theorem 2.11.
If , the function defined by (2.3) satisfies
(i) is continuous for all ;
(ii) for each , and
where
is defined by (2.16), is defined in Proposition 2.9.
Proof.

(i)
From Propositions 2.2 and 2.10, we obtain that is continuous for all , and .

(ii)
From (ii) of Proposition 2.2 and (ii) of Proposition 2.10, we have that for each .
Now, we show that (2.38) holds.
In fact, from Proposition 2.9, we have
Then the proof of Theorem 2.11 is completed.
Remark 2.12.
From the definition of , it is clear that .