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Tripled fixed point of W-compatible mappings in abstract metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 134 (2012)
Abstract
In this paper, we introduce the concepts of W-compatible mappings for mappings and , where is an abstract metric space. We establish tripled coincidence and common tripled fixed point theorems in such spaces. The presented theorems generalize and extend several well-known comparable results in literature, in particular the results of Abbas, Ali and Radenović (Appl. Math. Comput. 217:195-202, 2010). We also provide an example to illustrate our obtained results.
MSC:54H25, 47H10.
1 Introduction
Fixed point theory has fascinated many researchers since 1922 with the celebrated Banach fixed point theorem. This theorem provides a technique for solving a variety of applied problems in mathematical sciences and engineering. There exists vast literature on this topic and this is a very active area of research at present. Banach contraction principle has been generalized in different directions in different spaces by mathematicians over the years, for more details on this and related topics, we refer to [6, 10, 11, 13, 14, 21, 22, 24, 25, 27–30] and references therein.
Fixed point theory in K-metric and K-normed spaces was developed by Perov et al. [18], Mukhamadijev and Stetsenko [16], Vandergraft [33] and others. For more details on fixed point theory in K-metric and K-normed spaces, we refer the reader to a fine survey paper by Zabreǐko [32]. The main idea is to use an ordered Banach space instead of the set of real numbers, as the codomain for a metric.
In 2007 Huang and Zhang [12] reintroduced such spaces under the name of cone metric spaces and reintroduced the definition of convergent and Cauchy sequences in terms of interior points of the underlying cone. They also proved some fixed point theorems in such spaces in the same work. Afterwards, many papers about fixed point theory in cone metric spaces appeared (see, for example, [1–4, 7, 17, 19, 23, 26, 31]). In 2011, Abbas et al. [1] introduced the concept of w-compatible mappings and obtained a coupled coincidence point and a coupled point of coincidence for mappings satisfying a contractive condition in cone metric spaces. Very recently, Aydi et al. [5] introduced the concepts of -compatible mappings and generalized the results in [1].
The aim of this paper is to introduce the concepts of W-compatible mappings. Based on this notion, a tripled coincidence point and a common tripled fixed point for mappings and are obtained, where is a cone metric space. It is worth mentioning that our results do not rely on the assumption of normality condition of the cone. The presented theorems generalize and extend several well-known comparable results in literature. An example is also given in support of our results.
The following definitions and results will be needed in the sequel.
Let E be a real Banach space. A subset P of E is called a cone if and only if:
-
(a)
P is closed, non-empty and ,
-
(b)
, , imply that ,
-
(c)
,
where is the zero vector of E.
Given a cone, define a partial ordering ⪯ with respect to P by if and only if . We shall write for , where IntP stands for interior of P. Also we will use to indicate that and .
The cone P in the normed space is called normal whenever there is a number such that for all , implies . The least positive number satisfying this norm inequality is called a normal constant of P.
Definition 2 Let X be a non-empty set. Suppose that satisfies:
(d1) for all and if and only if ,
(d2) for all ,
(d3) for all .
Then d is called a cone metric [12] or K-metric [32] on X and is called a cone metric space [12] or K-metric space [32].
The concept of a K-metric space is more general than that of a metric space, because each metric space is a K-metric space where and . For other examples of K-metric spaces, we refer to [32], pp.853 and 854.
Definition 3 ([12])
Let X be a K-metric space, a sequence in X and . For every with , we say that is
(C1) a Cauchy sequence if there is some such that, for all , ,
(C2) a convergent sequence if there is some such that, for all , . This limit is denoted by or as .
Note that every convergent sequence in a K-metric space X is a Cauchy sequence. A K-metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
2 Main results
For simplicity, we denote from now on by where and X is a non-empty set. We start by recalling some definitions.
Definition 4 (Bhashkar and Lakshmikantham [9])
An element is called a coupled fixed point of the mapping if and .
Definition 5 (Lakshmikantham and Ćirić [15])
An element is called
-
(i)
a coupled coincidence point of mappings and if and , and is called a coupled point of coincidence;
-
(ii)
a common coupled fixed point of mappings and if and .
Note that if g is the identity mapping, then Definition 5 reduces to Definition 4.
Definition 6 (Abbas, Khan and Radenović [1])
The mappings and are called w-compatible if whenever and .
In 2010, Samet and Vetro [20] introduced a fixed point of order . In particular, for , we have the following definition.
Definition 7 (Samet and Vetro [20])
An element is called a tripled fixed point of a given mapping if , and .
Note that, Berinde and Borcut [8] defined differently the notion of a tripled fixed point in the case of ordered sets in order to keep true the mixed monotone property. For more details, see [8].
Now, we introduce the following definitions.
Definition 8 An element is called
-
(i)
a tripled coincidence point of mappings and if , and . In this case is called a tripled point of coincidence;
-
(ii)
a common tripled fixed point of mappings and if , and .
Example 1 Let . We define and as follows:
for all . Then is a tripled coincidence point of F and g, and is a tripled point of coincidence.
Definition 9 Mappings and are called W-compatible if
whenever , and .
Example 2 Let . Define and as follows:
for all . One can show that is a tripled coincidence point of F and g if and only if . Here is a common tripled fixed point of F and g. Note that F and g are W-compatible.
Now we prove our first result.
Theorem 1 Let be a K-metric space with a cone P having non-empty interior and and be mappings such that . Suppose that for any , the following condition
holds, where , are nonnegative real numbers such that . Then F and g have a tripled coincidence point provided that is a complete subspace of X.
Proof Let , and be three arbitrary points in X. By given assumption, there exists such that
Continuing this process, we can construct three sequences , and in X such that
Now, taking and in the considered contractive condition and using (1), we have
Then, using the triangular inequality one can write for any
Similarly, following similar arguments to those given above, we obtain
and
Denote
Adding (2) to (4), we have
On the other hand, we have
Thus
Similarly,
and
Adding (6) to (8), we obtain that
Finally, from (5) and (9), for any , we have
that is
where
Consequently, we have
If , we get , that is, , and . Therefore, from (1) we have
that is, is a tripled coincidence point of F and g. Now, assume that . If , we have
and
Adding above inequalities and using (12), we obtain
As , we have . Hence for any with , there exists such that for any we have . Furthermore, for any , we get
This implies that , and are Cauchy sequences in . By completeness of , there exist such that
Now, we prove that , and . Note that
On the other hand, applying given contractive condition, we obtain
Combining above inequality with (14), and using triangular inequality, we have
Therefore,
Similarly, we obtain
and
Adding (15) and (17), we get
Therefore, we have
where
From (12) and (13), for any there exists such that
for all . Thus, for all , we have
It follows that , that is , and . □
As consequences of Theorem 1, we give the following corollaries.
Corollary 1 Let be a K-metric space with a cone P having non-empty interior. Let and be mappings such that and for any , the following condition
holds, where , are nonnegative real numbers such that . Then F and g have a tripled coincidence point provided that is a complete subspace of X.
Proof It suffices to take , , , and in Theorem 1 with . □
Corollary 2 (Abbas, Khan and Radenović [1])
Let be a K-metric space with a cone P having non-empty interior. Let and be mappings satisfying , is a complete subspace of X and for any ,
where , are nonnegative real numbers such that . Then and g have a coupled coincidence point , that is, and .
Proof Consider the mappings defined by for all . Then, the contractive condition (18) implies that, for all
Then F and g satisfy the contractive condition of Theorem 1. Clearly other conditions of Theorem 1 are also satisfied as and is a complete subspace of X. Therefore, from Theorem 1, F and g have a tripled fixed point such that , and , that is, and . This makes end to the proof. □
Now, we are ready to state and prove the result of a common tripled coupled fixed point.
Theorem 2 Let and be two mappings which satisfy all the conditions of Theorem 1. If F and g are W-compatible, then F and g have a unique common tripled fixed point. Moreover, a common tripled fixed point of F and g is of the form for some .
Proof First, we will show that the tripled point of coincidence is unique. Suppose that and with
Using the contractive condition in Theorem 1, we obtain
Similarly, we have
and
Adding above three inequalities, we get
Since , we obtain
which implies that
which implies the uniqueness of the tripled point of coincidence of F and g, that is, . Note that
Similarly,
and
Adding above inequalities, we obtain
The fact that yields that
In view of (19) and (20), one can assert that
That is, the unique tripled point of coincidence of F and g is .
Now, let , then we have . Since F and g are W-compatible, we have
which due to (21) gives that
Consequently, is a tripled coincidence point of F and g, and so is a tripled point of coincidence of F and g, and by its uniqueness, we get . Thus, we obtain
Hence, is the unique common tripled fixed point of F and g. This completes the proof. □
Corollary 3 (Abbas, Khan and Radenović [1])
Let be a metric space with a cone P having non-empty interior. Let and be mappings satisfying , is a complete subspace of X and for any ,
where , are nonnegative real numbers such that . If and g are w-compatible, then and g have a unique common coupled fixed point. Moreover, the common fixed point of and g is of the form for some .
Proof Consider the mappings defined by for all . From the proof of Corollary 2 and the result given by Theorem 2, we have only to show that F and g are W-compatible. Let such that , and . From the definition of F, we get and . Since and g are w-compatible, this implies that
Using (22), we have
Thus, we proved that F and g are W-compatible mappings, and the desired result follows immediately from Theorem 2. □
Remark 1
-
Corollary 1 extends Theorem 2.9 of Samet and Vetro [20] to K-metric spaces (corresponding to the case ).
-
Theorem 2 extends Theorem 2.10 of Samet and Vetro [20] to K-metric spaces (case ).
-
Theorem 2 extends Theorem 2.11 of Samet and Vetro [20] to K-metric spaces (case ).
Similar to Corollaries 2 and 3, by considering for all where , we may state the following consequence of Theorem 2.
Corollary 4 (Olaleru [17])
Let be a K-metric space and be mappings such that
for all , where , and . Suppose that f and g are weakly compatible, and is a complete subspace of X. Then the mappings f and g have a unique common fixed point.
Now, we give an example to illustrate our obtained results.
Example 3 Let . Take endowed with order induced by . The mapping is defined by . In this case is a complete abstract metric space with a non-normal cone having non-empty interior. Define the mappings and by
We will check that all the hypotheses of Theorem 1 are satisfied. Since, for all , we have
Then, the contractive condition is satisfied with for all and . All conditions of Theorem 1 are satisfied. Consequently, F and g have a tripled coincidence point. In this case, is a tripled coincidence point if and only if . This implies that F and g are W-compatible. Applying our Theorem 2, we obtain the existence and uniqueness of a common tripled fixed point of F and g. In this example, is the unique common tripled fixed point.
Example 4 Let and be defined as . Define the mappings and by
Let and , . Now we shall prove that the contractive condition in Theorem 1 holds for all . By its symmetry and without loss of generality, it suffices to prove it for and . Define
There are 16 possibilities which are (I, i), (I, ii), (I, iii), (I, iv), (II, i), (II, ii), (II, iii), (II, iv), (III, i), (III, ii), (III, iii), (III, iv), (IV, i), (IV, ii), (IV, iii) and (IV, iv).
Case 1. If (I, i) holds, we have
Case 2. If (I, ii) holds, we have
Case 3. If (I, iii) holds, we have
Case 4. If (I, iv) holds, we get
Case 5. If (II, i) holds, we have
Case 6. If (II, ii) holds, we have
Case 7. If (II, iii) holds, we have
Case 8. If (II, iv) holds, we get
Case 9 corresponding to (III, i) is as Case 3.
Case 10 corresponding to (III, ii) is as Case 7.
Case 11. If (III, iii) holds, we have
Case 12. If (III, iv) holds, we get
Case 13 corresponding to (IV, i) is as Case 4.
Case 14 corresponding to (IV, ii) is as Case 8.
Case 15. If (IV, iii) holds, we have
Case 16. If (IV, iv) holds, we have
All the conditions of Theorem 1 are fulfilled. Moreover, is a common tripled coincidence point of F and g.
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Acknowledgements
The third author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the fourth author would like to thank the Commission on Higher Education and the Thailand Research Fund (Grant No. MRG550085) for financial support during the preparation of this manuscript.
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Aydi, H., Abbas, M., Sintunavarat, W. et al. Tripled fixed point of W-compatible mappings in abstract metric spaces. Fixed Point Theory Appl 2012, 134 (2012). https://doi.org/10.1186/1687-1812-2012-134
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DOI: https://doi.org/10.1186/1687-1812-2012-134