We now state and prove several lemmas before stating our main results.

Lemma 2.1.

Let . Then for , the problem

has a unique solution

Proof.

From (2.1), we have

For , integration from to , gives

For , integration from to yields that

that is,

So,

Integrating (2.7) from to , where , we have

From (2.2), we obtain that

Thus,

Therefore, (2.1)-(2.2) has a unique solution

Lemma 2.2.

Let . If and on , then the unique solution of (2.1)-(2.2) satisfies for .

Proof.

If , then, by the concavity of and the fact that , we have for .

Moreover, we know that the graph of is concave down on , we get

where is the area of triangle under the curve from to for .

Assume that . From (2.2), we have

By concavity of and , it implies that .

Hence,

which contradicts the concavity of .

Lemma 2.3.

Let . If and for , then (2.1)-(2.2) has no positive solution.

Proof.

Assume (2.1)-(2.2) has a positive solution .

If , then , it implies that and

which contradicts the concavity of .

If , then , this is for all . If there exists such that , then , which contradicts the concavity of . Therefore, no positive solutions exist.

In the rest of the paper, we assume that . Moreover, we will work in the Banach space , and only the sup norm is used.

Lemma 2.4.

Let . If and , then the unique solution of the problem (2.1)-(2.2) satisfies

where

Proof.

Set . We divide the proof into three cases.

Case 1.

If and , then the concavity of implies that

Thus,

Case 2.

If and , then (2.2), (2.13), and the concavity of implies

Therefore,

Case 3.

If , then . Using the concavity of and (2.2), (2.13), we have

This implies that

This completes the proof.