- Research Article
- Open Access

# Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps

- Lai-Jiu Lin
^{1}Email author and - Sung-Yu Wang
^{1}

**2011**:847170

https://doi.org/10.1155/2011/847170

© Lai-Jiu Lin and Sung-Yu Wang. 2011

**Received:**30 December 2010**Accepted:**20 February 2011**Published:**13 March 2011

## Abstract

We study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.

## Keywords

- Point Theorem
- Fixed Point Theorem
- Existence Theorem
- Cauchy Sequence
- Common Fixed Point

## 1. Introduction and Preliminaries

Let be a metric space and be a multivalued map. We say that is a stationary point of if . The existence theorem of stationary point was first considered by Dancs et al. [1]. If is a self-mapping (multivalued or single valued) defined on , we denote the collection of all the fixed points of . In this paper,we need the following definitions.

Definition 1.1.

A function is called

where .

*weakly contractive*if there exists a lower semicontinuous and nondecreasing function with if and only if such that

for some constant and function .

where .

The result of Zhang and Song [8] generalized the results in [2, 3, 5, 6]. Motivated by Chang [7], Zhang and Song [8], it is natural to ask whether there is a common fixed point of and in satisfy inequality (1.5) with . In this paper, we give a positive answer to this question in complete metric spaces.

Here, is the identity map defined on and . We show that have a unique common fixed point if is complete. As a special case of this result, we give a common fixed point theorem in complete metric spaces under the assumption that inequality (1.5) holds with . One of our results generalized Banach contraction principle, an example is given (Example 2.12) to show that the maps above need not to be continuous. The assumption of continuity is often used in the existence theorems of fixed points [6, 9–14]. We also give an example to show that the family above is not necessary to be commuting, and in contrast that the commutativity assumption is often used in the existence theorems of common fixed points [9, 10, 13, 15, 16]. Finally, we generalize some of our results to the case of multivalued maps.

for some (where denotes the Hausdorff metric). In fact, under the hypothesis that inequality (1.10) holds, we can show that and for all if and have nonempty closed bounded values. Further we give a new stationary point theorem in complete metric spaces and illustrate with examples (Examples 3.4 and 3.8).

## 2. Fixed Point Theorems

Throughout this paper, let be a complete metric space and let be the set of all positive integers. In this section, all the self-maps on are single valued. The following theorem is the main result in this section.

Theorem 2.1.

is the identity map defined on and .

Then, have a unique common fixed point.

Proof.

Then, and, hence, .

We consider the following two cases:

(i)

(ii) .

This yields a contradiction.

Then . This also yields a contradiction.

Therefore, and is a Cauchy sequence in . Since is complete, converges to a point in , say .

In order to show that is the unique common fixed point of . We first claim that , for all .

We consider the following three cases:

(i) ,

(ii) ,

(iii) .

Continuing in this process, we show that . By the same argument as in the case above, we see that .

Then, and . Therefore, is the unique fixed point of and we complete the proof.

- (a)
The sequence approaching to the unique common fixed point in Theorem 2.1 is different from those in [8, 11, 12, 16–19].

- (b)
The finite family of self-mappings in Theorem 2.1 is neither commuting nor continuous, which are often assumed in common fixed point theorems, see [6, 9–16]. In fact, the commuting and continuity assumptions are not needed throughout this paper and we will give examples (Examples 2.12–2.15) to show this fact.

As special cases of Theorem 2.1, we have the following theorems and corollaries.

Theorem 2.3.

for all . Then and have a unique common fixed point.

Proof.

Take , and in Theorem 2.1, then Theorem 2.3 follows from Theorem 2.1.

Corollary 2.4.

Then and have a unique common fixed point.

Corollary 2.5.

Then has a unique fixed point.

Proof.

Take in Theorem 2.3, then Corollary 2.4 follows from Corollary 2.4.

- (b)
Corollary 2.4 is equivalent to Corollary 2.5.

Proof.

- (c)
In Theorem 2.3, the map is not necessary equal , see Example 2.14. In fact, the maps and in Theorem 2.3 are not necessary to be commuting, see Example 2.15.

Theorem 2.7.

and is the identity map defined on and .

Then have a unique common fixed point.

Proof.

Take for all , then Theorem 2.7 follows from Theorem 2.1.

Theorem 2.8.

for all . Then and have unique common fixed point.

Proof.

Take, , and in Theorem 2.7, then Theorem 2.8 follows from Theorem 2.7.

Corollary 2.9.

Then, and have a unique common fixed point.

Corollary 2.10.

Then has a unique fixed point.

Proof.

Take in Corollary 2.9, then Corollary 2.10 follows from Corollary 2.9.

- (i)
If is contractive, then there exists such that for all , but the converse is not true. It is obvious that Corollary 2.9 is a special case of Corollaries 2.4 and 2.10 is a generalization of Banach contraction principle. Further we see that Corollary 2.9 is equivalent to Corollary 2.10 by the same argument as in Remark 2.6.

- (ii)
A map satisfies for all and for some is neither continuous nor nonexpansive. We give an example (Example 2.12) to show this fact.

Example 2.12.

Then, for all and is not continuous.

Proof.

We consider the following three cases:

(i) ,

(ii) ,

(iii) .

and, hence . It is obvious that is not continuous at but for all .

Example 2.13.

Let be the same as in Example 2.12. and take for all . Since for all . By Example 2.12 and Corollary 2.10, we see that has a unique common fixed point. But for each , is not continuous, the results in [13, 16] do not work in this example. Further it is obvious that the family have a unique common fixed point 0.

Example 2.14.

Let and define maps by and . Then and we see that and have a unique common fixed point.

Proof.

for all .

We have to consider the following two cases:

(i) ,

(ii) .

If we take , then by Theorem 2.8, we see that and have a unique common fixed point. In fact, 0 is the unique common fixed point of and .

By the same argument as in Example 2.14, we give the following example to show that the maps and in Theorem 2.8 are not necessary to be commuting.

Example 2.15.

Let and define maps by and . Then and are not commuting and we see that and have a unique common fixed point.

Proof.

for all .

We have to consider the following two cases:

(i) ,

(ii) .

If we take , then by Theorem 2.8, we see and have a unique common fixed point. In fact, 0 is the unique common fixed point of and .

## 3. A Common Fixed Point Theorem of Set-Valued Maps and a Stationary Point Theorem

In this section, we study a fixed point theorem and a stationary point theorem which generalize a fixed point theorem in Section 2.

In this section, let be the class of all nonempty bounded closed subsets of and for , let be the Hausdorff metric of and and let for all .

Lemma 3.1 (see [20]).

For all , and , there exists such that .

Theorem 3.2.

Then and for all .

Proof.

and , . Therefore and .

To complete the proof, it suffices to show the following four cases:

(i) and for all ,

(ii) ,

(iii) for all ,

(iv) .

and .

This shows that . Till now, we see that and for all .

Hence .

It remains to show that .

and . Then and .

- (a)
If one of and in Theorem 3.2 is single valued, then the set is singleton and the maps and have a unique common fixed point in . Therefore, Theorem 3.2 is a generalization of Corollary 2.9, but Theorem 3.2 is not a generalization of Theorem 5 Nadler [20].

- (b)
The sequence approaches to the common fixed point of and in Theorem 3.2 is different from those in [20–25].

- (c)
By Example 2.12, we see that both and in Theorem 3.2 are neither to be upper semicontinuous nor to be lower semicontinuous (multivalued maps). Further the maps and are not necessary to be commuting. We give an example below.

Example 3.4.

Let and let maps be defined by and . Then we see that and have a unique common fixed point.

Proof.

We have to consider the following two cases:

(i)

(ii)

If we take , then by Theorem 3.2, we see and have a unique common fixed point. In fact, 0 is the unique common fixed point of and .

Corollary 3.5.

Then and for all .

Similarly, we have the following existence theorem of stationary points.

Theorem 3.6.

Then has a unique stationary point, say . In fact, and .

Proof.

Then, and hence .

This yields a contradiction. Therefore and is a Cauchy sequence in . By the completeness of , converges to a point in , say .

It follows that .

Then, and .

Then, and hence .

- (a)
The single valued map in Theorem 3.6 is not necessary to be continuous (see Example 2.12), but the continuity assumption is used in Theorem 3.2 [21, 22, 25] and Theorem 2.1 Ćirić and Ume [23]. We give an example to show that and in Theorem 3.6 are not necessary to be commuting.

- (b)
Theorems 3.2 and 3.6 are different and Theorem 3.6 is also a generalization of Corollary 2.9.

Example 3.8.

Let and let maps and be defined by and . Then we see that has a unique stationary point.

Proof.

We have to consider the following two cases:

(i) ,

(ii) .

If we take . Then by Theorem 3.6, we see has a unique stationary point. In fact, 0 is the unique stationary point of .

## Authors’ Affiliations

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