Common Fixed Point Results in Metric-Type Spaces

Abstract

Several fixed point and common fixed point theorems are obtained in the setting of metric-type spaces introduced by M. A. Khamsi in 2010.

1. Introduction

Symmetric spaces were introduced in 1931 by Wilson [1], 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 [5] 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 [6]). Namely, it was observed in [7] that if is a cone metric on the set (in the sense of [5]), 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 [8] 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.

2. Preliminaries

Let be a nonempty set. Suppose that a mapping satisfies the following:

(s1) if and only if ;

(s2), for all .

Then is called a symmetric on , and is called a symmetric space [1].

Let be a real Banach space. A nonempty subset of is called a cone if is closed, if , , and imply , and if . Given a cone , we define the partial ordering with respect to by if and only if .

Let be a nonempty set. Suppose that a mapping satisfies the following:

(co1) for all and if and only if ;

(co2) for all ;

(co3) for all .

Then is called a cone metric on and is called a cone metric space [5].

If is a cone metric space, the function is easily seen to be a symmetric on [7, 8]. Following [7], 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 [8]).

Let be a nonempty set, let be a real number, and let the function satisfy the following properties:

(a) if and only if ;

(b) for all ;

(c) for all .

Then is called a metric-type space.

Obviously, for , metric-type space is simply a metric space.

A metric type space may satisfy some of the following additional properties:

(d) for arbitrary points ;

• (e) function is continuous in two variables; that is,

(2.1)

(The last condition is in the theory of symmetric spaces usually called "property ".)

Condition (d) was used instead of (c) in the original definition of a metric-type space by Khamsi [8]. 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):

(e^') and (in ) imply that

is satisfied in an arbitrary metric type space. It can also be proved easily that the limit of a sequence in a metric type space is unique. Indeed, if and (in ) and , then

(2.2)

for sufficiently large , which is impossible.

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 [11] 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 [14] that it is actually superfluous.

3. Results

We begin with a simple, but useful lemma.

Lemma 3.1.

Let be a sequence in a metric type space such that

(3.1)

for some , , and each . Then is a Cauchy sequence in .

Proof.

Let and . Applying the triangle-type inequality (c) to triples , we obtain

(3.2)

Now (3.1) and imply that

(3.3)

It follows that is a Cauchy sequence.

Remark 3.2.

If, instead of triangle-type inequality (c), we use stronger condition (d), then a weaker condition can be used in the previous lemma to obtain the same conclusion. The proof is similar.

Next is the simplest: Banach-type version of a fixed point result for contractive mappings in a metric type space.

Theorem 3.3.

Let be a complete metric type space, and let be a map such that for some , ,

(3.4)

holds for all . Then has a unique fixed point , and for every , the sequence converges to .

Proof.

Take an arbitrary and denote . Then

(3.5)

for each . Lemma 3.1 implies that is a Cauchy sequence, and since is complete, there exists such that when . Then

(3.6)

when . Hence, and is a fixed point of .

If is another fixed point of , then which is possible only if .

Remark 3.4.

In a standard way we prove that the following estimate holds for the sequence :

(3.7)

for each . Indeed, for ,

(3.8)

and passing to the limit when , we obtain estimate (3.7).

Note that continuity of function (property (e)) was not used.

The first part of the following result was obtained, under the additional assumption of boundedness of the orbit, in [8, Theorem 3.3].

Theorem 3.5.

Let be a complete metric type space. Let be a map such that for every there is such that for all and let . Then has a unique fixed point . Moreover, has the property P.

Proof.

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.

Remark 3.6.

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 .

Indeed, let and . Then

(3.9)

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., [15]) in metric type spaces.

Theorem 3.7.

Let be a metric type space, and let be two mappings such that and one of these subsets of is complete. Suppose that there exist nonnegative coefficients , such that

(3.10)

and that for all

(3.11)

holds. Then and have a unique point of coincidence. If, moreover, the pair is weakly compatible, then and have a unique common fixed point.

Note that condition (3.10) is satisfied, for example, when . Note also that when it reduces to the standard Hardy-Rogers condition in metric spaces.

Proof.

Suppose, for example, that is complete. Take an arbitrary and, using that , construct a Jungck sequence defined by , . Let us prove that this is a Cauchy sequence. Indeed, using (3.11), we get that

(3.12)

Similarly, we conclude that

(3.13)

Adding the last two inequalities, we get that

(3.14)

that is,

(3.15)

The assumption (3.10) implies that

(3.16)

Lemma 3.1 implies that is a Cauchy sequence in and so there is such that when . We will prove that .

Using (3.11) we conclude that

(3.17)

Similarly,

(3.18)

Adding up, one concludes that

(3.19)

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 .

Suppose that is another point of coincidence for and . Then (3.11) implies that

(3.20)

Since (because of (3.10)), the last relation is possible only if . So, the point of coincidence is unique.

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., [15]):

Corollary 3.8.

Let be a metric type space, and let be two mappings such that and one of these subsets of is complete. Suppose that one of the following three conditions holds:

(1°) for some and all ;

(2°) for some and all ;

(3°) for some and all .

Then and have a unique point of coincidence. If, moreover, the pair is weakly compatible, then and have a unique common fixed point.

Putting in Theorem 3.7, we get metric type version of Hardy-Rogers theorem (which is obviously a special case for ).

Corollary 3.9.

Let be a complete metric type space, and let satisfy

(3.21)

for some , satisfying (3.10) and for all . Then has a unique fixed point. Moreover, has property P.

Proof.

We have only to prove the last assertion. For arbitrary , we have that

(3.22)

and similarly

(3.23)

Adding the last two inequalities, we obtain

(3.24)

Similarly as in the proof of Theorem 3.7, we get that . Now [11, Theorem 1.1] implies that has property P.

Remark 3.10.

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 [7]).

The next is a possible metric-type variant of a common fixed point result for Ćirić and Das-Naik quasicontractions [16, 17].

Theorem 3.11.

Let be a metric type space, and let be two mappings such that and one of these subsets of is complete. Suppose that there exists , such that for all

(3.25)

where

(3.26)

Then and have a unique point of coincidence. If, moreover, the pair is weakly compatible, then and have a unique common fixed point.

Proof.

Let be arbitrary and, using condition , construct a Jungck sequence satisfying , . Suppose that for each (otherwise the conclusion follows easily). Using (3.25) we conclude that

(3.27)

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 .

To prove that , put and in (3.25) to get

(3.28)

Note that and when , implying that when . It follows that the only possibilities are the following:

(1°); in this case , and since , it follows that .

(2°); in this case, , so again , .

Since the limit of a sequence is unique, it follows that .

The rest of conclusion follows as in the proof of Theorem 3.7.

Putting , we obtain the first part of the following corollary.

Corollary 3.12.

Let be a complete metric type space, and let be such that for some , , and for all ,

(3.29)

holds. Then has a unique fixed point, say . Moreover, the function is continuous at point and it has the property P.

Proof.

Let when . Then

(3.30)

Since and , the only possibility is that , implying that , . Since , it follows that , , and is continuous at the point .

We will prove that satisfies

(3.31)

for some , and each .

Applying (3.29) to the points and (for any ), we conclude that

(3.32)

The following cases are possible:

(1°), and (3.31) holds with ;

(2°), which is only possible if and then (3.31) obviously holds.

(3°), implying that and , where since .

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 [18] to metric type spaces. Note that, unlike in [18], 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)).

Theorem 3.13.

Let be a metric type space, and let be four mappings such that and , and suppose that at least one of these four subsets of is complete. Let

(3.33)

holds for some , and all . Then pairs and have a unique common point of coincidence. If, moreover, pairs and are weakly compatible, then , , , and have a unique common fixed point.

Proof.

Let be arbitrary and construct sequences and such that

(3.34)

for . We will prove that condition (3.1) holds for . Indeed,

(3.35)

Using Lemma 3.1, we conclude that is a Cauchy sequence. Suppose, for example, that is a complete subset of . Then , , for some . Of course, subsequences and also converge to . Let us prove that . Using (3.33), we get that

(3.36)

hence . Since , we get that there exists such that . Let us prove that also . Using (3.33), again we conclude that

(3.37)

implying that . We have proved that is a common point of coincidence for pairs and .

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 [5]. An example follows showing that additional condition of (sequential) compactness cannot be omitted.

Theorem 3.14.

Let a metric type space be sequentially compact, and let be a continuous function (satisfying property (e)). If is a mapping such that

(3.38)

then has a unique fixed point.

Proof.

According to [9, Theorem 3.1], sequential compactness and compactness are equivalent in metric type spaces, and also continuity is a sequential property. The given condition (3.38) of strict continuity implies that a fixed point of is unique (if it exists) and that both mappings and are continuous. Let be an arbitrary point, and let be the respective Picard sequence (i.e., ). If for some , then is a (unique) fixed point. If for each , then

(3.39)

Hence, there exists , such that for each and , . Using sequential compactness of , choose a subsequence of that converges to some when . The continuity of and implies that

(3.40)

and the continuity of the symmetric implies that

(3.41)

It follows that . It remains to prove that . If , then and (3.41) implies that

(3.42)

This is a contradiction.

Example 3.15.

Let , , , and be defined by . Then is a cone metric space over a normal cone with the normal constant (see, e.g., [5]). The associated symmetric is in this case simply the metric .

Let be defined by . Then

(3.43)

for all . Hence, satisfies condition (3.38) but it has no fixed points. Obviously, is not (sequentially) compact.