Theorem 2.1.

Suppose that the operator satisfies the following.

(S1) is increasing, is decreasing, and is decreasing.

(S2) There exist a constant and a function such that for each and , and

(S3)There exist such that , , and

(S4)There exists a constant such that, for all with ,

Then has a unique fixed point in , that is, .

Proof.

The proof is divided into 4 steps.

Step 1.

Let and

For each , there exists a nonnegative integer such that , that is, . Now, by (S2), we deduce, for all ,

Moreover, by (S3), we get

Hence, in the following proof, one can assume that in (S2) and (S3) without loss.

Step 2.

Fix . Then, there exists such that . Let

Then is an operator from to , and by (S4), is increasing in . Combining (S1)–(S3), we have

provided that . Moreover, it is easy to see that (2.8) holds when . Similarly, one can show that

Then, it follows that

Let

Then, using arguments similar to those in the proof of [7, Theorem ], one can show that has a unique fixed point in , and

We claim that is the unique fixed point of in . In fact, let be a fixed point of in , and such that . By the above proof, has a unique fixed point in , which means that . In addition, it follows from

that .

Step 3.

By Step 2, we can define an operator by

Let with and with . Denote by the corresponding sequences in the proof of Step 2. Then

Next, by induction and being increasing, one can show that for all . So

that is, . Thus, is increasing. By a similar method, one can prove that is decreasing. On the other hand, by (S3), for and ,

Let , and

By choosing in Step 1, we get . Then

As is increasing and is decreasing, it follows immediately that

Next, by making some needed modifications in the proof of [3, Theorem ], one can show that has a fixed point . Suppose that is a fixed point of . It follows from the definition of and that for all . Then, by the normality of , we get . So is the unique fixed point of in .

Step 4.

By Steps 2 and 3, we get

Let such that . Then it follows from Step 2 that , that is, is a fixed point of in . Thus, by Step 3, , which means that is the unique fixed point of in .

Remark 2.2.

Compared with [7,Remark ], the nonlinear operator in Theorem 2.1 is more general, and so Theorem 2.1 may have a wider range of applications.