# On Coincidence and Fixed-Point Theorems in Symmetric Spaces

- Seong-Hoon Cho
^{1}Email author, - Gwang-Yeon Lee
^{1}and - Jong-Sook Bae
^{2}

**2008**:562130

https://doi.org/10.1155/2008/562130

© Seong-Hoon Cho et al. 2008

**Received: **28 August 2007

**Accepted: **5 March 2008

**Published: **10 June 2008

## Abstract

We give an axiom (C.C) in symmetric spaces and investigate the relationships between (C.C) and axioms (W3), (W4), and (H.E). We give some results on coinsidence and fixed-point theorems in symmetric spaces, and also, we give some examples for the results of Imdad et al. (2006).

## 1. Introduction

In [1], the author introduced the notion of compatible mappings in metric spaces and proved some fixed-point theorems. This concept of compatible mappings was frequently used to show the existence of common fixed points. However, the study of the existence of common fixed points for noncompatible mappings is, also, very interesting. In [2], the author initially proved some common fixed-point theorems for noncompatible mappings. In [3], the authors gave a notion (E-A) which generalizes the concept of noncompatible mappings in metric spaces, and they proved some common fixed-point theorems for noncompatible mappings under strict contractive conditions. In [4], the authors proved some common fixed-point theorems for strict contractive noncompatible mappings in metric spaces. Recently, in [5] the authors extended the results of [3, 4] to symmetric(semimetric) spaces under tight conditions. In [6], the author gave a common fixed-point theorem for noncompatible self-mappings in a symmetric spaces under a contractive condition of integral type.

In this paper, we give some common fixed-point theorems in symmetric(semimetric) spaces and give counterexamples for the results of Imdad et al. [5].

In order to obtain common fixed-point theorems in symmetric spaces, some axioms are needed. In [5], the authors assumed axiom (W3), and in [6] the author assumed axioms (W3), (W4), and (H.E); see Section 2 for definitions.

We give another axiom for symmetric spaces and study their relationships in Section 2. We give common fixed-point theorems of four mappings in symmetric spaces and give some examples which justifies the necessity of axioms in Section 3.

## 2. Axioms on Symmetric Spaces

Let
be a symmetric on a set
. For
and
, let
. A topology
on
defined as follows:
if and only if for each
, there exists an
such that
. A subset
of
is a neighbourhood of
if there exists
such that
. A symmetric
is a *semimetric* if for each
and each
,
is a neighbourhood of
in the topology
.

A *symmetric* (resp., *semimetric*) *space*
is a topological space whose topology
on
is induced by symmetric(resp., semi-metric)
.

- (W3)
- (W4)

Also the following axiom can be found in [6].

(H.E) for sequences in and , and imply .

Now, we add a new axiom which is related to the continuity of the symmetric .

(C.C) for sequences in and , implies .

Note that if is a metric, then (W3), (W4), (H.E), and (C.C) are automatically satisfied. And if is Hausdorff, then (W3) is satisfied.

Proposition 2.1.

For axioms in symmetric space , one has

Proof.

Let be a sequence in and with and

The following examples show that other relationships in Proposition 2.1 do not hold.

Example 2.2.

(W4) (H.E) and (W4) (C.C) and so (W3) (H.E) and (W3) (C.C) by Proposition 2.1 (1).

Then, is a symmetric space which satisfies (W4) but does not satisfy (H.E) for . Also does not satisfy (C.C).

Example 2.3.

(H.E) (W3), and so (H.E) (W4) and (H.E) (C.C).

Then, is a symmetric space which satisfies (H.E). Let . Then, But and hence the symmetric space does not satisfy (W3).

Example 2.4.

(C.C) (W4) and so (W3) (W4) by Proposition 2.1 (2).

Then, the symmetric space satisfies (C.C) but does not satisfy (W4) for and .

Example 2.5.

and . Then, is a symmetric space which satisfies (C.C). Let . Then, But Hence, the symmetric space does not satisfy (H.E).

## 3. Common Fixed Points of Four Mappings

Let be a symmetric(or semimetric) space and let be self-mappings of . Then, we say that the pair satisfies property (E-A) [3] if there exists a sequence in and a point such that

A subset of a symmetric space is said to be -closed if for a sequence in and a point , implies . For a symmetric space , -closedness implies -closedness, and if is a semimetric, the converse is also true.

At first, we prove coincidence point theorems of four mappings satisfying the property (E-A) under some contractive conditions.

Theorem 3.1.

- (1)
- (2)
- (3)
- (4)

Proof.

From (2), there exist a sequence in and a point such that

From , there exists a sequence in such that and hence . By (H.E),

From , there exists a point such that .

By taking , we have By (W3), we get

Since , there exists a point such that .

For the existence of a common fixed point of four self-mappings of a symmetric space, we need an additional condition, so-called weak compatibility.

Recall that for self-mappings
and
of a set, the pair
is said to be *weakly compatible* [8] if
, whenever
. Obviously, if
and
are commuting, the pair
is weakly compatible.

Theorem 3.2.

- (1)
- (2)
- (3)
- (4)
- (5)

Then, and have a unique common fixed point in .

Proof.

From Theorem 3.1, there exist such that . From , , and

which is a contradiction.

Similarly, if , we have a contradiction. Thus, and is a common fixed point of and .

which is a contradiction. Hence,

Remark 3.3.

In the case of
and
in Theorem 3.1 (resp., Theorem 3.2), we can show that
and
*have a coincidence point(resp.,*
and
have a unique common fixed point) without making the assumption
.

Recently, R. P. Pant and V. Pant [4] obtained the existence of a common fixed point of the pair of in a metric space satisfying the condition

Also in [5], the authors tried to extend the result of [4] to symmetric spaces which satisfy axiom (W3).

Now, we will extend R. P. Pant and V. Pant's result to symmetric spaces which satisfy additional conditions (H.E) and (C.C).

Theorem 3.4.

- (1)
- (2)
- (3)
- (4)

Proof.

As in the proof of Theorem 3.1, there exist sequences in and a point such that and . Hence, .

From , there exists a point such that .

Since , there exists a point such that .

Since , we get and hence . Therefore, we have

Theorem 3.5.

- (1)
- (2)
- (3)
- (4)
- (5)

Then and have a unique common fixed point in .

Proof.

From Theorem 3.4, there exist points such that , and

which is a contradiction.

Similarly, if , we have a contradiction. Thus

which is a contradiction. Hence

Example 3.6.

- (0)
- (1)
- (2)
- (3)
- (4)
- (5)
- (6)

Remark 3.7.

In the case of
and
in Theorem 3.4 (resp., Theorem 3.5), we can show that
and
*have a coincidence point (resp.,*
and
have a unique common fixed point) without the condition
, that is,
.

The following example shows that the axioms (H.E) and (C.C) cannot be dropped in Theorem 3.4.

Example 3.8.

Let be the symmetric space as in Example 2.2. Then, the symmetric does not satisfy both (H.E) and (C.C).

Let and be self-mappings of defined as follows:

Then, the condition (resp., ) of Theorem 3.4 (resp., Theorem 3.5) is satisfied for .

To show this, let . We consider two cases.

Case 1.

Case 2.

Thus, the condition (resp., ) of Theorem 3.4 (resp., Theorem 3.5) is satisfied. Note that is a -closed( -closed) subset of . Also, the pair satisfies property (E-A) for , but the pair has no coincidence points, and also the pair has no common fixed points.

Remark 3.9.

Example 3.6 satisfies all conditions of [5, Theorems 2.1 and 2.2] and satisfies also all conditions of [5, Theorem 2.3].

On the studying of fixed points, various conditions of have been studied by many different authors [3, 5, 6].

Remark 3.10.

The functions in Theorems 3.4 and 3.5 can be generalized to the compositions for .

Example 3.11.

Let be the symmetric space and and be the functions as in Example 3.8. Recall that satisfies (W3) but does not satisfy both (H.E) and (C.C). Let and Then, for any , for . Note that the pairs and satisfy property (E-A), and , and are -closed( -closed).

Therefore, and satisfy all conditions of [5, Theorem 2.4] and satisfy also all conditions of [5, Theorem 2.5]. But the pairs and have no points of coincidence, and also the pairs and have no common fixed points.

## Declarations

### Acknowledgments

The authors are very grateful to the referees for their helpful suggestions. The first author was supported by Hanseo University, 2007.

## Authors’ Affiliations

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