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
![](https://www.scirp.org/html/htmlimages\5-7401809x\d111678b-c6b9-457f-86e0-4031e207a299.png)
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
![](https://www.scirp.org/html/htmlimages\5-7401809x\69f259f2-3f20-4986-82fb-51f5b1fe7c68.png)
Since
is normal, we have
,
. Here N is the normality constant.
So
and
are Cauchy sequences. Because E is complete, there exist
such that![](https://www.scirp.org/html/htmlimages\5-7401809x\22ae3cdd-10b1-4456-b8b8-2754c91b1e21.png)
, By (2.4) we know that
and
.
Further
![](https://www.scirp.org/html/htmlimages\5-7401809x\c77743f1-8186-4224-890b-0a18bad83361.png)
And thus
. Let
, we obtain
![](https://www.scirp.org/html/htmlimages\5-7401809x\407a4218-66a3-44ab-81b7-9825423f7412.png)
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
![](https://www.scirp.org/html/htmlimages\5-7401809x\56b2dd54-8718-4f8c-8ad0-76c514c7e6f9.png)
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
![](https://www.scirp.org/html/htmlimages\5-7401809x\176e934a-3090-4bbf-971c-4aa410dce7e9.png)
hold, where
. Then A has exactly one fixed pint x* in
. Moreover, constructing successively the sequence
for any initial
, we have
![](https://www.scirp.org/html/htmlimages\5-7401809x\88c11206-ab28-4e9e-a1ca-eba468106797.png)
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 ![](https://www.scirp.org/html/htmlimages\5-7401809x\8f20c686-d9b8-4eaa-8086-ad36128133a9.png)
for all
, there exist
, such that
hold.
Then
has exactly one fixed pint
in
. Moreover, constructing successively the sequence
for any initial
, we have
![](https://www.scirp.org/html/htmlimages\5-7401809x\f1173a6d-c6a4-47a3-acfe-a549f44ed969.png)
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
![](https://www.scirp.org/html/htmlimages\5-7401809x\90c19f43-d445-4456-944b-39d56b354359.png)
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![](https://www.scirp.org/html/htmlimages\5-7401809x\97d09877-6067-4f0a-9fb3-1b0a94967971.png)
. It follows from (1.1), (1.2) and the mixed monotonicity of A that
![](https://www.scirp.org/html/htmlimages\5-7401809x\e74e9d46-fa05-4f10-b615-66c373f884e7.png)
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
![](https://www.scirp.org/html/htmlimages\5-7401809x\7793e524-1200-41d5-8171-b0fdbb599ecc.png)
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
![](https://www.scirp.org/html/htmlimages\5-7401809x\0dc0b9bf-a893-4fce-b583-1df64a99d87e.png)
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).