1. Introduction
Mixed monotone operators were introduced by Dajun Guo and V. Lakshmikantham in [1] in 1987. Thereafter, many authors have investigated these kinds of operators in Banach spaces and obtained a lot of interesting and important results. They are used extensively in nonlinear differential and integral equations. In this paper, without any compact and continuous assumption, we obtained some new existence and uniqueness theorems of positive fixed point of e-concave-convex mixed monotone operators in Banach spaces partially ordered by a cone.
Let the real Banach space 
be partially ordered by a cone 
of
, i.e., 
iff
. 
is said to be a mixed monotone operator if 
is increasing in 
and decreasing in
, i.e., 
, 
implies.
. Element 
is called a fixed point of 
iff
.
Recall that cone 
is said to be solid if the interior 
is nonempty and we denote 
if
. 
is normal if there exists a positive constant N, such that 
implies
, 
is called the normal constant of
.
For all
, the notation 
means that there exist 
and
, such that
. Clearly, 
is an equivalence relation. Given
, 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 [2-4]. For more facts about mixed monotone operators and other related concepts, the reader could refer to [5-7] and some of the reference therein.
2. Main Results
In this section, we present our main results. To begin with, we give the definition of e-concave-convex operators.
Definition 2.1. We say an operator 
is an e-concave-convex operators if there exist one positive function 
such that
(2.1)
Remark 2.1. (2.1) implies that
(2.2)
Theorem 2.1. Let 
be a normal cone of
, and let 
be a mixed monotone and e-concave-convex operator. In addition, suppose that there exist 
(Since
, we can choose a sufficiently small 
such that
) such that
hold, where
. Then 
has exactly one fixed pint x* in
. Moreover, constructing successively the sequence
for any initial
, we have
(2.3)
Proof. We divide the proof into 3 steps.
Step 1. We prove A has a fixed point in
.
Construct successively the sequences
.
It follows from 
and the mixed monotonicity of 
that
. (2.4)
Let
.
Thus we have
, and then
.
Therefore, 
, i.e., 
is increasing with
.
Suppose 
as
. Then
. Indeed, suppose to the contrary that
. By
, (2.1) and the mixed monotonicity of
, we have
.
Therefore
(2.5)
By (2.1) and the mixed monotonicity of
, we know that
, and thus
exists. We distinguish two cases.
Case 1:
. In this case we know that 
So, we have
, which is a contradiction.
Case 2:
. In this case, it is easy to see that
. For convenience’s sake, let
. Since
, there is a nonincreasing subsequence 
of 
such that
. Without loss of generality, we may still use 
to stand for
. From (2.5), we obtain
.
Hence
which is also a contradiction. Thus
.
For any natural number 
we have
Since 
is normal, we have
,
. Here N is the normality constant.
So 
and 
are Cauchy sequences. Because E is complete, there exist 
such that
, By (2.4) we know that 
and
.
Further
And thus
. Let
, we obtain
Let
, we get
. That is, x* is a fixed point of 
in
.
Step 2. We prove that x* is the unique fixed point of 
in
.
In fact, suppose 
is another fixed point of 
in
. Since
, there exist positive numbers 
such that
. Let
. (2.6)
Evidently,
. We now prove
. If otherwise,
. From (2.1), we obtain
Which contradicts the definition of
. Hence
, thus
. Therefore A has a unique fixed point x* in
.
Step 3. We prove (2.3).
For any
, we can choose a small number 
such that
(2.7)
Let
(2.8)
Using (2.7), (2.8) and the mixed monotonicity of A, we have
(2.9)
Let
. (2.10)
From (2.8) and (2.9), we obtain
(2.11)
In what follows, we will prove that
. If not, that is, 
, then from (2.8) and (2.11), we have
(2.12)
. (2.13)
Let
. It follows from (2.11), (2.12) and (2.13) that
, which is a contradiction. That is
.
From (2.9) and (2.11), we have
.
Thus,
.
By using the normality of
, we know that (2.2) holds.
3. Concerned Remarks and Corollaries
Using Theorem 2.1, we have the following corollaries.
Corollary 3.1. Let P be a normal cone of E, and let 
be a mixed monotone and e-concaveconvex operator. In addition, suppose that there exist 
(Since
, we can choose a sufficiently small
, such that
), such that
hold, where
. Then A has exactly one fixed pint x* in
. Moreover, constructing successively the sequence
for any initial
, we have
Corollary 3.2. Let P be a normal cone of E, and let 
be a mixed monotone and e-concaveconvex operator. In addition, suppose that
there exist 
such that 
for all
, there exist
, such that 
hold.
Then 
has exactly one fixed pint 
in
. Moreover, constructing successively the sequence
for any initial
, we have
Corollary 3.3. Let P be a normal cone of
, and let 
be a mixed monotone and e-concaveconvex operator. In addition, suppose that
, 
are monotone on 
and
. Then a necessary and sufficient condition for 
to have exactly one fixed pint 
in 
is that 
holds. Moreover, constructing successively the sequence
for any initial
, we have
Proof. Corollary 3.1 ensures the sufficiency of Corollary 2.3, so we have only to prove the necessity of Corollary 2.3.
Suppose that x* is the unique fixed point of 
in
. For any
, let
. It follows from (1.1), (1.2) and the mixed monotonicity of A that
Therefore, 
holds.
Corollary 3.4. Let 
be a normal cone of
, and let 
be a mixed monotone operator. In addition, suppose that
, there exists 
such that
(3.1)
holds. Then a necessary and sufficient condition for 
to have exactly one fixed pint 
in 
is that 
holds. Moreover, constructing successively the sequence
for any initial
, we have
Corollary 3.5. Let 
be a normal cone of
, and let 
be a mixed monotone operator with property (2.1). In addition, suppose that 
holds. Then 
has exactly one fixed pint x* in
.
Proof. In fact, by corollary 2.4, we have only to prove that 
holds.
For any
, it follows from 
that there exists
, such that
. Let
. Following from (3.1) and the mixed monotonicity of A, we get
The Corollary 3.5 is thus proved.
Acknowledgements
The author thanks the referee for his valuable comments and suggestions. This paper was supported by SRFDP (NO. 20103705120002).