Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 562130, 9 pages
doi:10.1155/2008/562130

Copyright © 2008 Seong-Hoon Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

A symmetric on a set is a function satisfying the following conditions:

(i) if and only if for , (ii) for all

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) .

The difference of a symmetric and a metric comes from the triangle inequality. Actually a symmetric space need not be Hausdorff. In order to obtain fixed-point theorems on a symmetric space, we need some additional axioms. The following axioms can be found in [7].

(W3) for a sequence in , and imply . (W4) for sequences in and , and imply

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

(1) (W4) (W3), (2) (C.C) (W3).

Proof. Let be a sequence in and with and



(1) By putting for each , we have By (W4), we have . (2) By (C.C), implies 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).

Let and let (2.1)

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).

Let and let (2.2) and

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).

Let , and let ( is odd), ( is even) and (2.3)

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

Example 2.5. (C.C) (H.E).

Let , and let (2.4) 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. Let be a symmetric(semimetric) space that satisfies (W3) and (H.E), and let and be self-mappings of such that

(1) and , (2) the pair satisfies property (E-A) (resp., satisfies property (E-A)), (3) for any , , where (3.1) (4) is a -closed ( -closed) subset of (resp., is a -closed ( -closed) subset of ).

Then, there exist such that .

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 .

From , we have (3.2)

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

Since , there exists a point such that .

We show that From , we have (3.3)Hence, and hence .

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. Let be a symmetric(semimetric) space that satisfies (W3) and (H.E), and let and be self-mappings of such that

(1) and , (2) the pair satisfies property (E-A) (resp., satisfies property (E-A)), (3) the pairs and are weakly compatible, (4) for any (5) is a -closed ( -closed) subset of (resp., is a -closed ( -closed) subset of ).

Then, and have a unique common fixed point in .

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

If , then from (4) we have(3.4)which is a contradiction.

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

For the uniqueness, let be another common fixed point of and . If , then from we get (3.5)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

(P.P) for any ,(3.6)where

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. Let be a symmetric(semimetric) space that satisfies (H.E) and (C.C) and let and be self-mappings of such that

(1) and , (2) the pair satisfies property (E-A) (resp., satisfies property (E-A)), (3) for any , , where (4) is a -closed ( -closed) subset of (resp., is a -closed ( -closed) subset of ).

Then, there exist such that .

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 .

We show From we have (3.7)

In the above inequality, we take , by (C.C) and (H.E), we have (3.8)

Since , we get and hence

Since , there exists a point such that .

We show that From we have(3.9)

Since , we get and hence . Therefore, we have

Theorem 3.5. be a symmetric(semimetric) space that satisfies (H.E) and (C.C) and let and be self-mappings of such that

(1) and , (2) the pair satisfies property (E-A) (resp., satisfies property (E-A)), (3) the pairs and are weakly compatible, (4) for any where (5) is a -closed ( -closed) subset of (resp., is a -closed ( -closed) subset of ).

Then and have a unique common fixed point in .

Proof. From Theorem 3.4, there exist points such that , and

We show that If , then from (4) we have(3.10)which is a contradiction.

Similarly, if , we have a contradiction. Thus

For the uniqueness, let be another common fixed point of and . If , then from we get (3.11)which is a contradiction. Hence

Example 3.6. Let and . Define self-mappings and by and for all . Then, we have the following:

(0) is a symmetric space satisfying the properties (H.E) and (C.C),(1) and ,(2) the pair satisfies property (E-A) for the sequence (3) the pairs and are weakly compatible,(4) for any , (5) is a -closed(-closed) subset of ,(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:

(3.12) 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. (3.13)

Case 2. (3.14)

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].

Let be a function such that

  is nondecreasing on ,  for all Note that from and , we have

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.

Acknowledgments

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

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