Common Fixed Point Results in Metric-Type Spaces
© Mirko Jovanović et al. 2010
Received: 16 October 2010
Accepted: 8 December 2010
Published: 14 December 2010
Several fixed point and common fixed point theorems are obtained in the setting of metric-type spaces introduced by M. A. Khamsi in 2010.
Symmetric spaces were introduced in 1931 by Wilson , as metric-like spaces lacking the triangle inequality. Several fixed point results in such spaces were obtained, for example, in [2–4]. A new impulse to the theory of such spaces was given by Huang and Zhang  when they reintroduced cone metric spaces replacing the set of real numbers by a cone in a Banach space, as the codomain of a metric (such spaces were known earlier under the name of K-metric spaces, see ). Namely, it was observed in  that if is a cone metric on the set (in the sense of ), then is symmetric with some special properties, particularly in the case when the underlying cone is normal. The space was then called the symmetric space associated with cone metric space .
The last observation also led Khamsi  to introduce a new type of spaces which he called metric-type spaces, satisfying basic properties of the associated space , . Some fixed point results were obtained in metric-type spaces in the papers [7–10].
In this paper we prove several other fixed point and common fixed point results in metric-type spaces. In particular, metric-type versions of very well-known results of Hardy-Rogers, Ćirić, Das-Naik, Fisher, and others are obtained.
Then is called a symmetric on , and is called a symmetric space .
Then is called a cone metric on and is called a cone metric space .
If is a cone metric space, the function is easily seen to be a symmetric on [7, 8]. Following , the space will then be called associated symmetric space with the cone metric space . If the underlying cone of is normal (i.e., if, for some , always implies ), the symmetric satisfies some additional properties. This led M.A. Khamsi to introduce a new type of spaces which he called metric type spaces. We will use the following version of his definition.
Definition 2.1 (see ).
A metric type space may satisfy some of the following additional properties:
Condition (d) was used instead of (c) in the original definition of a metric-type space by Khamsi . Both conditions (d) and (e) are satisfied by the symmetric which is associated with a cone metric (with a normal cone) (see [7–9]).
Note that the weaker version of property (e):
The notions such as convergent sequence, Cauchy sequence, and complete space are defined in an obvious way.
We prove in this paper several versions of fixed point and common fixed point results in metric type spaces. We start with versions of classical Banach, Kannan and Zamfirescu results then proceed with Hardy-Rogers-type theorems, and with quasicontractions of Ćirić and Das-Naik, and results for four mappings of Fisher and finally conclude with a result for strict contractions.
Recall also that a mapping is said to have property P  if for each , where stands for the set of fixed points of .
A point is called a point of coincidence of a pair of self-maps and is its coincidence point if . Mappings and are weakly compatible if for each of their coincidence points [12, 13]. The notion of occasionally weak compatibility is also used in some papers, but it was shown in  that it is actually superfluous.
We begin with a simple, but useful lemma.
Next is the simplest: Banach-type version of a fixed point result for contractive mappings in a metric type space.
The first part of the following result was obtained, under the additional assumption of boundedness of the orbit, in [8, Theorem 3.3].
Take such that . Since , , there exists such that for each . Then for all whenever . In other words, for any , satisfies for all . Theorem 3.3 implies that has a unique fixed point, say . Then , implying that and is a fixed point of . Since the fixed point of is unique, it follows that and is also a fixed point of .
From the given condition we get that for some and each . This property, together with , implies, in the same way as in [11, Theorem 1.1], that has the property P.
If, in addition to the assumptions of previous theorem, we suppose that the series converges and that satisfies property (d), we can prove that, for each , the respective Picard sequence converges to the fixed point .
when (due to the convergence of the given series). So, is a Cauchy sequence and it is convergent. For chosen in the proof of Theorem 3.5 such that , it is and when , but is a subsequence of which is convergent; hence, the latter converges to .
The next is a common fixed point theorem of Hardy-Rogers type (see, e.g., ) in metric type spaces.
The right-hand side of the last inequality tends to 0 when . Since (because of (3.10)), it is , and so also the left-hand side tends to 0, and . Since the limit of a sequence is unique, it follows that and and have a point of coincidence .
If is weakly compatible, then [13, Proposition 1.12] implies that and have a unique common fixed point.
Taking special values for constants , we obtain as special cases Theorem 3.3 as well as metric type versions of some other well-known theorems (Kannan, Zamfirescu, see, e.g., ):
Similarly as in the proof of Theorem 3.7, we get that . Now [11, Theorem 1.1] implies that has property P.
If the metric-type function satisfies both properties (d) and (e), then it is easy to see that condition (3.10) in Theorem 3.7 and the last corollary can be weakened to . In particular, this is the case when for a cone metric on (over a normal cone, see ).
If , then , and it would follow from (3.27) that which is impossible since . (For the same reason the term was omitted in the last row of the previous series of inequalities.) Hence, and (3.27) becomes . Using Lemma 3.1, we conclude that is a Cauchy sequence in . Supposing that, for example, the last subset of is complete, we conclude that when for some .
The rest of conclusion follows as in the proof of Theorem 3.7.
The following cases are possible:
So, relation (3.31) holds for some , and each . Using the mentioned analogue of [11, Theorem 1.1], one obtains that satisfies property P.
We will now prove a generalization and an extension of Fisher's theorem on four mappings from  to metric type spaces. Note that, unlike in , we will not use the case when and , as well as and , commute, neither when and are continuous. Also, function need not be continuous (i.e., we do not use property (e)).
If now these pairs are weakly compatible, then for example, and for example,. Moreover, and implies that . So, we have that . It remains to prove that, for example, . Indeed, , implying that . The proof that this common fixed point of , and is unique is straightforward.
We conclude with a metric type version of a fixed point theorem for strict contractions. The proof is similar to the respective proof, for example, for cone metric spaces in . An example follows showing that additional condition of (sequential) compactness cannot be omitted.
This is a contradiction.
Let , , , and be defined by . Then is a cone metric space over a normal cone with the normal constant (see, e.g., ). The associated symmetric is in this case simply the metric .
The authors are thankful to the Ministry of Science and Technological Development of Serbia.
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