Let the real Banach space
be partially ordered by a cone
of
, that is,
if and only if
.
is said to be a mixed monotone operator if
is increasing in
and decreasing in
, that is,
,
,
implies
. Element
is called a fixed point of
if
.
Recall that cone
is said to be solid if the interior
is nonempty and we denote
if
.
is said to be normal if there exists a positive constant
, such that
, the smallest
is called the normal constant of
. For all
, the notation
means that there exist
and
such that
. Clearly, ~ is an equivalence relation. Given
(i.e.,
and
), we denote by
the set
. It is easy to see that
is convex and
for all
. If
and
, it is clear that
.
All the concepts discussed above can be found in [1, 2, 4]. For more results about mixed monotone operators and other related concepts, the reader is referred to [3, 5–9] and some of the references therein.
In [9], Zhai and Cao introduced the following definition of

concave operators.
Definition 2.1 (see [9]).
Let
be a real Banach space and
be a cone in
. We say an operator
is

concave if there exist two positivevalued functions
on interval
such that
is a surjection;
, for all
;
, for all
,
.
They obtained the following result.
Theorem 2.2 (see [9]).
Let
be a real Banach space and
be a normal cone in
. Suppose that an operator
is increasing and

concave. In addition, suppose that there exists
such that
. Then
(i)there are
and
such that
,
;
(ii)operator
has a unique fixed point
in
;
(iii)for any initial
, constructing successively the sequence
,
, we have
.
We can extend Theorem 2.2 to mixed monotone operators, our main results can be stated as follows.
Theorem 2.3.
Let
be a normal cone in a real Banach space
, and
a mixed monotone operator. Assume that for all
, there exist two positivevalued functions
on interval
such that
is a surjection;
, for all
;
, for all
,
.
In addition, suppose that there exists
such that
. Then
(i)there are
and
such that
,
;
(ii)operator
has a unique fixed point
in
;
(iii)for any initial
, constructing successively the sequences
,
,
, we have
and
as
.
Corollary 2.4.
Let
be a real Banach space,
a normal, solid cone in
. Suppose
is a mixed monotone operator and satisfies the conditions
of Theorem 2.3. Then
(i)there are
and
such that
,
;
(ii)operator
has a unique fixed point
in
;
(iii)for any initial
, constructing successively the sequences
,
,
, we have
and
as
.
Remark 2.5.
In Theorem 2.3, if
with
is a solid cone, we can know that
is automatically satisfied. Therefore, we can deduce that Corollary 2.4 holds from Theorem 2.3. For simplicity, we only present the proof of Theorem 2.3.
Proof of Theorem 2.3.
Note that
, we can find a sufficiently small number
such that
According to
, we can obtain that there exists
such that
, thus
Since
, we can find a positive integer
such that
Let
,
, and construct successively the sequences
It is clear that
and
,
. In general, we obtain
,
.
It follows from
, (2.2), and (2.3) that
From
, we have
Combining (2.2) with (2.3) and (2.6), we have
By induction, it is easy to obtain that
Take any
, then
and
. So we can know that
Thus, we have
,
, and then
Therefore,
; that is,
Set
, we will show that
. In fact, if
, by
, there exists
such that
. Consider the following two cases.
 (i)
There exists an integer
such that
. In this case, we have
and
for all
hold. Hence
By the definition of
, we have
which is a contradiction.
 (ii)
For all integers
. Then, we obtain
. By
, there exist
such that
. Hence
By the definition of
, we have
Let
, we have
which is also a contradiction. Thus,
.
Furthermore, similarly to the proof of Theorem 2.1 in [9], there exits
such that
, and
is the fixed point of operator
.
In the following, we prove that
is the unique fixed point of
in
. In fact, suppose that
is another fixed point of operator
. Let
Clearly,
and
. If
, according to
, there exists
such that
. Then
Hence,
, which is a contradiction. Thus we have
, that is,
. Therefore,
has a unique fixed point
in
. Note that
, so we know that
is the unique fixed point of
in
. For any initial
, we can choose a small number
such that
From
, there is
such that
, thus
We can choose a sufficiently large positive integer
such that
Take
,
. We can find that
constructing successively the sequences
By using the mixed monotone properties of operator
, we have
Similarly to the above proof, we can know that there exists
such that
By the uniqueness of fixed points of operator
in
, we have
. Taking into account that
is normal, we deduce that
. This completes the proof.