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On Fixed Point Theorems of Mixed Monotone Operators
Fixed Point Theory and Applications volume 2011, Article number: 563136 (2011)
Abstract
We obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Some results are new even for increasing or decreasing operators.
1. Introduction
Mixed monotone operators were introduced by Guo and 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, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Some results are new even for increasing or decreasing operators.
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 normal if there exists a positive constant
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
(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 [2, 3]. For more facts about mixed monotone operators and other related concepts, the reader could refer to [4–9] and some of the references therein.
2. Main Results
In this section, we present our main results. To begin with, we give the definition of -
-concave-convex operators.
Definition 2.1.
Let be a real Banach space and
a cone in
. We say an operator
is
-
-concave-convex operator if there exist two positive-valued functions
on interval
such that
is a surjection,
, for all
,
, for all
,
.
Theorem 2.2.
Let be normal cone of
, and let
be a mixed monotone and
-
-concave-convex operator. In addition, suppose that there exists
such that
, then
has exactly one fixed point
in
. Moreover, constructing successively the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ1_HTML.gif)
for any initial , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ2_HTML.gif)
Proof.
We divide the proof into 3 steps.
Step 1.
We prove that has a fixed point in
.
Since , we can choose a sufficiently small number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ3_HTML.gif)
It follows from that there exists
such that
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ4_HTML.gif)
By , we know that
. So, we can take a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ5_HTML.gif)
It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ6_HTML.gif)
Let . Evidently,
and
. By the mixed monotonicity of
, we have
. Further, combining the condition
with (2.4) and (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ7_HTML.gif)
For , from
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ8_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ9_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ10_HTML.gif)
Construct successively the sequences
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ11_HTML.gif)
It follows from (2.7), (2.10), and the mixed monotonicity of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ12_HTML.gif)
Note that , so we can get
,
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ13_HTML.gif)
Thus, we have , and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ14_HTML.gif)
Therefore, , that is,
is increasing with
. Suppose that
as
, then
. Indeed, suppose to the contrary that
. By
, there exists
such that
. We distinguish two cases.
Case 1.
There exists an integer such that
. In this case, we know that
for all
. So, for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ15_HTML.gif)
By the definition of , we get
, which is a contradiction.
Case 2.
If for all integer ,
, then
. By
, there exists
such that
. So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ16_HTML.gif)
By the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ17_HTML.gif)
Let , we get
, which is also a contradiction. Thus,
. For any natural number
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ18_HTML.gif)
Since is normal, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ19_HTML.gif)
Here, is the normality constant.
So, and
are Cauchy sequences. Because
is complete, there exist
such that
. By (2.12), we know that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ20_HTML.gif)
Further,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ21_HTML.gif)
and thus . Let
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ22_HTML.gif)
Let , we get
. That is,
is a fixed point of
in
.
Step 2.
We prove that is the unique fixed point of
in
.
In fact, suppose that is a fixed point of
in
. Since
, there exist positive numbers
such that
. Let
. Evidently,
. We now prove that
. If otherwise,
. From
, there exists
such that
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ23_HTML.gif)
Since , this contradicts the definition of
. Hence,
, thus,
. Therefore,
has a unique fixed point
in
.
Step 3.
We prove (2.2).
For any , we can choose a small number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ24_HTML.gif)
Also from , there is
such that
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ25_HTML.gif)
We can choose a sufficiently large integer , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ26_HTML.gif)
Let . It is easy to see that
and
. Put
,
,
,
,
. Similarly to Step 1, it follows that there exists
such that
,
. By the uniqueness of fixed points of operator
in
, we get
. And by induction,
. Since
is normal, we have
,
.
3. Concerned Remarks and Corollaries
If we suppose that the operator or
with
is a solid cone, then
is automatically satisfied. This proves the following corollaries.
Corollary 3.1.
Let be a normal cone of
, and let
be a mixed monotone and
-
-concave-convex operator, then
has exactly one fixed point
in
. Moreover, constructing successively the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ27_HTML.gif)
for any initial , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ28_HTML.gif)
Corollary 3.2.
Let be a normal solid cone of
, and let
be a mixed monotone and
-
-concave-convex operator, then
has exactly one fixed point
in
. Moreover, constructing successively the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ29_HTML.gif)
for any initial , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ30_HTML.gif)
When ,
,
,
, the conditions
and
are automatically satisfied. So, one has
Corollary 3.3.
Let be a normal cone of a real Banach space
is a mixed monotone operator. In addition, suppose that for all
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ31_HTML.gif)
then has exactly one fixed point
in
. Moreover, constructing successively the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ32_HTML.gif)
for any initial , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ33_HTML.gif)
Remark 3.4.
Corollary 3.3 is the main result in [5]. So, our results generalized the result in [5].
When , the conditions
and
are automatically satisfied. So, we have the following.
Corollary 3.5.
Let be normal solid cone of
, and let
be a mixed monotone operator. In addition, suppose that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ34_HTML.gif)
then has exactly one fixed point
in
. Moreover, constructing successively the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ35_HTML.gif)
for any initial , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F563136/MediaObjects/13663_2010_Article_1409_Equ36_HTML.gif)
Remark 3.6.
Corollary 3.5 is the main result in [4]. So, our results generalized the result in [4].
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Acknowledgments
This research was supported by the National Natural Science Foundation of China (nos. 10871116, 11001151), the Natural Science Foundation of Shandong Province (no. Q2008A03), the Doctoral Program Foundation of Education Ministry of China (no. 20103705120002), and the Youth Foundation of Qufu Normal University (nos. XJ200910, XJ03003).
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Du, X., Zhao, Z. On Fixed Point Theorems of Mixed Monotone Operators. Fixed Point Theory Appl 2011, 563136 (2011). https://doi.org/10.1155/2011/563136
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DOI: https://doi.org/10.1155/2011/563136