On Coincidence and Fixed-Point Theorems in Symmetric Spaces
© Seong-Hoon Cho et al. 2008
Received: 28 August 2007
Accepted: 5 March 2008
Published: 10 June 2008
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).
In , 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 , the author initially proved some common fixed-point theorems for noncompatible mappings. In , 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 , the authors proved some common fixed-point theorems for strict contractive noncompatible mappings in metric spaces. Recently, in  the authors extended the results of [3, 4] to symmetric(semimetric) spaces under tight conditions. In , 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. .
In order to obtain common fixed-point theorems in symmetric spaces, some axioms are needed. In , the authors assumed axiom (W3), and in  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 .
Also the following axiom can be found in .
The following examples show that other relationships in Proposition 2.1 do not hold.
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)  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.
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  if , whenever . Obviously, if and are commuting, the pair is weakly compatible.
which is a contradiction.
Recently, R. P. Pant and V. Pant  obtained the existence of a common fixed point of the pair of in a metric space satisfying the condition
Now, we will extend R. P. Pant and V. Pant's result to symmetric spaces which satisfy additional conditions (H.E) and (C.C).
which is a contradiction.
The following example shows that the axioms (H.E) and (C.C) cannot be dropped in Theorem 3.4.
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.
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.
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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