Abstract
Saadati et al. (Math. Comput. Model. 52:797-801, 2010) introduced the concept of Ω-distance in generalized metric spaces and studied some nice fixed point theorems. Very recently, Jleli and Samet (Fixed Point Theory Appl. 2012:210, 2012) showed that some of the fixed point theorems in G-metric spaces can be obtained from quasi-metric space. In this paper, we utilize the concept of Ω-distance in the sense of Saadati et al. to establish some common coupled fixed point results. Also, we introduce an example to support the useability of our results. Note that the method of Jleli and Samet cannot be used in our results.
MSC:47H10, 54H25.
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1 Introduction
In 2006, Mustafa and Sims [1] introduced a generalization of metric spaces, the G-metric spaces, which assigns to each triple of elements a non-negative real number. Very recently, Jleli and Samet [2] showed that some of the fixed point theorems in G-metric spaces can be obtained from quasi-metric spaces. For some works in G-metric spaces, see [3–36]. In 2010, Saadati et al. [26] introduced the concept of Ω-distance and studied some nice fixed point theorems (also, see [13]). Meanwhile, Bhaskar and Lakshmikantam [37] introduced the concept of coupled fixed point and proved several fixed point theorems. Lakshmikantam and Ćirić [38] generalized the concept of coupled fixed point to the the concept of coupled coincidence point of two mappings [39]. After that, many authors established coupled fixed point results (please, see [19–44]). In the present paper, we utilize the concept of Ω-distance to establish some coupled fixed point results. Also, we introduce an example to support the useability of our study.
2 Preliminaries
Definition 2.1 ([1])
Let X be an nonempty set. The mapping is called G-metric if the following axioms are fulfilled:
-
(1)
if (the coincidence);
-
(2)
for all , ;
-
(3)
for each triple from with ;
-
(4)
for each permutation of (the symmetry);
-
(5)
for each x, y, z and a in X (the rectangle inequality).
Definition 2.2 ([1])
Consider X a G-metric space and a sequence in G.
-
(1)
is called G-Cauchy sequence if for each there is a positive integer so that for all , .
-
(2)
is said to be G-convergent to if for each there is a positive integer such that for each .
Definition 2.3 ([26])
Consider a G-metric space and . The mapping Ω is called an Ω-distance on X if it satisfies the three conditions in the following:
-
(1)
for all x, y, z, a from X.
-
(2)
For each x, y from X, are lower semi-continuous.
-
(3)
for each there is , so that and imply .
The following lemma [13, 26] is going to be very helpful in computing the limits of several sequences.
Lemma 2.1 Let X be a metric space, endowed with metric G, and let Ω be an Ω-distance on X. , are sequences in X, and are sequences in with . If x, y, z and , then
-
(1)
If and for , then , and, by consequence, .
-
(2)
Inequalities and for imply , hence .
-
(3)
If for with , then is a G-Cauchy sequence.
-
(4)
If , , then is a G-Cauchy sequence.
Definition 2.4 ([37])
Consider X a nonempty set. A pair is called coupled fixed point of mapping if
Definition 2.5 ([38])
Let X be a nonempty set. The element is a coupled coincidence point of mappings and if
3 Main results
Theorem 3.1 Let be a G-metric space and Ω an Ω-distance on X such that X is Ω-bounded. and are mappings. Suppose there exists such that for each x, y, z, , and in X
Consider also that the following conditions hold true:
-
(1)
;
-
(2)
gX is a complete subspace of X with respect to the topology, induced by G;
-
(3)
If or , then
Then, F and g have a unique coupled coincidence point . Moreover, .
Proof Consider and . Because , there exist and in X such that and . By continuing the process, we obtain two sequences, and , with the properties
Using the contraction condition, we obtain
By applying the contraction inequality repeatedly, we get that
where .
Since X is Ω-bounded, there is such that for each triple . Hence, relation (1) becomes
Consider now , . The following relations hold true:
and, also
Making the sum of relations (2) and (3), and using inequality (1), it follows that
Lemma 2.1, part (3), implies that and are G-Cauchy sequences. Since gX is a complete G-subspace of X, there are gu and gv in gX such that and .
Let . From the lower semi-continuity of Ω, we get
Suppose that or . Applying hypotheses (3) of the theorem, and using inequalities (4)-(7), we obtain
for each , which is a contradiction.
Therefore, and .
Using the contraction condition from the hypotheses, we get
We apply repeatedly the contraction inequality, and we obtain
Since X is Ω-bounded, it follows that
In a similar manner, it can be proved that
Taking into account (8), (9) and the first statement of Lemma 2.1, we get .
We will prove now the uniqueness of the coupled coincidence point of F and g.
Suppose and are coupled coincidence points of F and g. Using the contraction condition, we obtain
hence .
On the other hand,
and
Relations (10) and (11) imply that and also . Lemma 2.1 imposes that and , and the uniqueness is proved. □
If we take in Theorem 3.1, we easily get the following.
Corollary 3.1 Let be a complete G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. Suppose is a mapping for which there exists such that for each x, y, z, , and in X
Consider also that if or , then
Then, F has a unique coupled fixed point . Moreover, .
Corollary 3.2 Let be a G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. and are mappings. Suppose that there exists with such that for each x, y, z, , and in X
Consider also that the following conditions hold true:
-
(1)
;
-
(2)
gX is a complete subspace of X with respect to the topology induced by G;
-
(3)
If or , then
Then, F and g have a unique coupled coincidence point . Moreover, .
Proof Follows from Theorem 3.1 by noting that
□
If we take in Corollary 3.2, we easily get the following.
Corollary 3.3 Let be a complete G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. Suppose is a mapping, for which there exists with such that for each x, y, z, , and in X
Consider also that if or , then
Then, F has a unique coupled fixed point . Moreover, .
By modifying the contraction condition, we get the following theorem.
Theorem 3.2 Let be a G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. and are mappings. Suppose that there exist with such that for each x, y, z, , and in X
and the conditions (1)-(3) from Theorem 3.1 hold.
Then, F and g have a unique coupled coincidence point . Moreover, .
Proof Let and be elements of X. Since , there exist and in X such that and . Repeating this procedure, we obtain two sequences, and , with the properties
The contraction condition implies that
which leads us to
where .
Following the same steps, as we did in Theorem 3.1, the conclusion is straightforward. □
Theorem 3.2 leads us to a coupled fixed point property, by considering .
Corollary 3.4 Let be a complete G-metric space, and let Ω be an Ω-distance on X such that X is Ω-bounded. Suppose that is a mapping, for which there exist with such that for each x, y, z, , and in X
and if or , then
Then, F has a coupled fixed point . Moreover, .
Theorem 3.3 Let be a G-metric space, and let Ω be an Ω-distance on X. Consider , and such that
for all . Suppose that the following conditions are fulfilled:
-
(1)
.
-
(2)
gX is a complete subspace of X with respect to the topology, induced by G.
-
(3)
There exists such that holds for all .
-
(4)
If or , then
Then F and g have a coupled coincidence point .
Proof Consider a pair from . As , there exist so that , .
We continue the process, and we obtain two sequences , from X, having the properties that
Using the contraction condition, we get
For , the first part of the definition Ω-distance and (12) yields
Let
According to (12),
Thus, is an increasing bounded sequence, so
exists.
Now, we shall show that and are G-Cauchy sequences in gX. Consider . By part (3) of the definition of an Ω-distance, we choose such that if and , then . Let .
Using the fact that
and letting in (13), we choose such that
for all .
Thus,
and
for all . Also we have
for all . Thus,
and
for all . Thus, by part (3) of the definition of Ω-distance, we have
and
for .
Therefore, and are G-Cauchy sequences. As gX is G-complete, it follows that there are u, so that and .
Since Ω is lower semi-continuous in its second and third variable, we obtain, for
We make the sum of inequalities (14), (15), (16) and (17). It follows that
for each , which contradicts the hypothesis.
Hence, and , that is, is a coupled coincidence point of F and g. □
By considering , we get the following corrolary.
Corollary 3.5 Let be a complete G-metric space, and let Ω be an Ω-distance on X. Consider and such that
for all . Suppose that the following conditions are fulfilled:
-
(1)
There exists such that holds for all .
-
(2)
If or , then
Then F has coupled fixed point .
Now, we introduce the following example to support the useability of our result.
Example 3.1 Let . Define
and
Also define
Then,
-
(1)
is a complete G-metric space.
-
(2)
Ω is Ω-distance.
-
(3)
for all .
-
(4)
.
-
(5)
For we have
-
(6)
If or , then
Proof The proof of (1), (2), (3) and (4) is clear. To prove (5) given .
To prove (6), let or . Then or . Thus,
So, F and g satisfy all the hypotheses of Corollary 3.5. Hence the mappings F and g have a coupled coincidence point, Here is the coupled coincidence point of F and g. □
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Shatanawi, W., Pitea, A. Ω-Distance and coupled fixed point in G-metric spaces. Fixed Point Theory Appl 2013, 208 (2013). https://doi.org/10.1186/1687-1812-2013-208
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DOI: https://doi.org/10.1186/1687-1812-2013-208