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.