- Research Article
- Open Access

# On Fixed Point Theorems of Mixed Monotone Operators

- Xinsheng Du
^{1}Email author and - Zengqin Zhao
^{1}

**2011**:563136

https://doi.org/10.1155/2011/563136

© X. Du and Z. Zhao. 2011

**Received:**11 November 2010**Accepted:**4 January 2011**Published:**11 January 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.

## Keywords

- Banach Space
- Positive Integer
- Integral Equation
- Natural Number
- Point Theorem

## 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.

Proof.

We divide the proof into 3 steps.

Step 1.

We prove that has a fixed point in .

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.

By the definition of , we get , which is a contradiction.

Case 2.

Here, is the normality constant.

Let , we get . That is, is a fixed point of in .

Step 2.

We prove that is the unique fixed point of in .

Since , this contradicts the definition of . Hence, , thus, . Therefore, has a unique fixed point in .

Step 3.

We prove (2.2).

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.

Corollary 3.2.

When , , , , the conditions and are automatically satisfied. So, one has

Corollary 3.3.

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.

Remark 3.6.

Corollary 3.5 is the main result in [4]. So, our results generalized the result in [4].

## Declarations

### 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).

## Authors’ Affiliations

## References

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## Copyright

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