On Fixed Point Theorems of Mixed Monotone Operators

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

(2.1)

for any initial , one has

(2.2)

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

(2.3)

It follows from that there exists such that , and hence

(2.4)

By , we know that . So, we can take a positive integer such that

(2.5)

It is clear that

(2.6)

Let . Evidently, and . By the mixed monotonicity of , we have . Further, combining the condition with (2.4) and (2.6), we have

(2.7)

For , from , we get

(2.8)

and hence

(2.9)

Thus, we have

(2.10)

Construct successively the sequences

(2.11)

It follows from (2.7), (2.10), and the mixed monotonicity of that

(2.12)

Note that , so we can get , . Let

(2.13)

Thus, we have , and then

(2.14)

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

(2.15)

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

(2.16)

By the definition of , we have

(2.17)

Let , we get , which is also a contradiction. Thus, . For any natural number , we have

(2.18)

Since is normal, we have

(2.19)

Here, is the normality constant.

So, and are Cauchy sequences. Because is complete, there exist such that . By (2.12), we know that and

(2.20)

Further,

(2.21)

and thus . Let , we obtain

(2.22)

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,

(2.23)

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

(2.24)

Also from , there is such that , and hence

(2.25)

We can choose a sufficiently large integer , such that

(2.26)

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

(3.1)

for any initial , one has

(3.2)

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

(3.3)

for any initial , one has

(3.4)

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

(3.5)

then has exactly one fixed point in . Moreover, constructing successively the sequence

(3.6)

for any initial , one has

(3.7)

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

(3.8)

then has exactly one fixed point in . Moreover, constructing successively the sequence

(3.9)

for any initial , one has

(3.10)

Remark 3.6.

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