- Research Article
- Open Access

# Fixed Points for Discontinuous Monotone Operators

- Yujun Cui
^{1}Email author and - Xingqiu Zhang
^{2}

**2010**:926209

https://doi.org/10.1155/2010/926209

© Y. Cui and X. Zhang. 2010

**Received:**24 September 2009**Accepted:**21 November 2009**Published:**24 December 2009

## Abstract

We obtain some new existence theorems of the maximal and minimal fixed points for discontinuous monotone operator on an order interval in an ordered normed space. Moreover, the maximal and minimal fixed points can be achieved by monotone iterative method under some conditions. As an example of the application of our results, we show the existence of extremal solutions to a class of discontinuous initial value problems.

## Keywords

- Operator Equation
- Continuous Operator
- Minimal Element
- Dominate Convergence Theorem
- Integral Inequality

## 1. Introduction

Let be a Banach space. A nonempty convex closed set is said to be a cone if it satisfies the following two conditions: (i) , implies ; (ii) , implies , where denotes the zero element. The cone defines an ordering in given by if and only if . Let be an ordering interval in , and an increasing operator such that , . It is a common knowledge that fixed point theorems on increasing operators are used widely in nonlinear differential equations and other fields in mathematics ([1–7]).

But in most well-known documents, it is assumed generally that increasing operators possess stronger continuity and compactness. Recently, there have been some papers that considered the existence of fixed points of discontinuous operators. For example, Krasnosel'skii and Lusnikov [8] and Chen [9] discussed the fixed point problems for discontinuous monotonically compact operator. They called an operator A to be a monotonically compact operator if ( ) implies that converges to some in norm and that ( ). A monotonically compact operator is referred to as an MMC-operator. A is said to be -monotone if implies , where , , and . They proved the following theorem.

Theorem 1.1 (see [8]).

Let be an -monotone MMC-operator with . Then has at least one fixed point possessing the property of -continuity.

Motivated by the results of [3, 8, 9], in this paper we study the existence of the minimal and maximal fixed points of a discontinuous operator , which is expressed as the form . We do not assume any continuity on . It is only required that (or ) is an MMC-operator and (or ) possesses the quasiseparability, which are satisfied naturally in some spaces. As an example for application, we applied our theorem to study first order discontinuous nonlinear differential equation to conclude our paper.

We give the following definitions.

Definition 1.2 (see [3]).

Let be an Hausdorff topological space with an ordering structure. is called an ordered topological space if for any two sequences and in , and , imply .

Definition 1.3 (see [3]).

Let be an ordered topological space, is said to be a quasi-separable set in if for any totally ordered set in , there exists a countable set such that is dense in (i.e., for any , there exists such that ).

Obviously, the separability implies the quasi-separability.

Definition 1.4 (see [3]).

Let be two ordered topological spaces. An operator is said to be a monotonically compact operator if ( ) implies that converges to some in norm and that .

Remark 1.5.

The definition of the MMC-operator is slightly different from that of [8, 9].

## 2. Main Results

Theorem 2.1.

Let be an ordered topological space, and an order interval in . Let be an operator. Assume that

(i)there exist ordered topological space , increasing operator , and increasing operator such that ;

(ii) is quasiseparable and is an MMC-operator;

Then has at least one fixed point in .

Proof.

It follows from the monotonicity of and condition (iii) that . Set . Since , is nonempty. Suppose that is a totally ordered set in . We now show that has an upper bound in .

Hence , therefore is an upper bound of .

This shows that is an upper bound of in . It follows from Zorn's lemma that has maximal element . Thus . And so , which implies that and . As is a maximal element of , ; that is, is a fixed point of .

Theorem 2.2.

Let be an ordered topological space, and an order interval in . Let be an operator. Assume that

(i)there exist ordered topological space , increasing operator , and increasing operator such that ;

(ii) is quasiseparable and is an MMC-operator;

Then has at least one fixed point in .

Proof.

that is, is a fixed point of the operator in .

Theorem 2.3.

If the conditions in Theorem 2.1 are satisfied, then has the minimal fixed point and the maximal fixed point in ; that is, and are fixed points of , and for any fixed point of in , one has .

Proof.

By (2.22) and (2.23), . Set . By virtue of (2.21), (2.22), and (2.23), . It is easy to see that is a lower bound of in . It follows from Zorn's lemma that has a minimal element.

Since is an increasing operator, this implies that and includes properly . This contradicts that is the minimal element of . Similarly, is a fixed point of . Since , is the minimal fixed point of and is the maximal fixed point of .

Theorem 2.4.

If the conditions in Theorem 2.2 are satisfied, then has the minimal fixed point and the maximal fixed point in ; that is, and are fixed points of , and for any fixed point of in , one has .

Proof.

It is similar to the proof of Theorem 2.4; so we omit it.

Theorem 2.5.

Let be an ordered topological space, and an order interval in . Let be an operator. Assume that

(i)there exist ordered topological space , increasing operator , and increasing operator such that ;

(ii) is an continuous operator;

(iii) is a demicontinuous MMC-operator;

Proof.

By the condition (iii), , that is, . Note that ; we have ; hence ; that is, is a fixed point of . Similarly, there exists such that and is a fixed point of . By the routine standard proof, it is easy to prove that is the minimal fixed point of and is the maximal fixed point of in .

## 3. Applications

As some simple applications of Theorem 2.5, we consider the existence of extremal solutions for a class of discontinuous scalar differential equations.

In the following, stands for the set of real numbers and a compact real interval. Let be the class of continuous functions on . is a normed linear space with the maximum norm and partially ordered by the cone . is a normal cone in .

Then is a Banach space by the norm .

A function is said to be a Carathéodory function if is measurable as a function of for each fixed and continuous as a function of for a.a. (almost all) .

We list for convenience the following assumptions.

(H2) is a Carathéodory function.

(H4)There exists such that is nondecreasing for a.a. .

Consider the differential equation

where . It is a common knowledge that the initial value problem (3.4) is equivalent to the equation

if is continuous. Therefore, when is not continuous, we define the solution of the integral equation (3.5) as the solution of the equation (3.4).

Theorem 3.1.

Proof.

By above discussions we know that and are all increasing. Thus conditions (i) and (ii) in Theorem 2.5 are satisfied.

This implies that in ; that is, is a demicontinuous operator. Since the cone in is regular, it is easy to see that is an MMC-operator. Thus condition (iii) in Theorem 2.5 is satisfied.

This implies that for all , that is, . Similarly we can show that .

Since all conditions in Theorem 2.5 are satisfied, by Theorem 2.5, has the maximal fixed point and the minimal fixed point in . Observing that fixed point of is equivalent to solutions of (3.5), and (3.5) is equivalent to (3.4), the conclusions of Theorem 3.1 hold.

Remark 3.2.

In the proof of Theorem 3.1, we obtain the uniformly convergence of the monotone sequences without the compactness condition.

## Declarations

### Acknowledgment

The project supported by the National Science Foundation of China (10971179).

## Authors’ Affiliations

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