Abstract
By using R-weak commutativity of type and non-compatible conditions of self-mapping pairs in generalized metric space, without the conditions for the completeness of space and the continuity of mappings, we establish some new common fixed point theorems for two self-mappings. Our results differ from other results already known. An example is provided to support our new result.
MSC:47H10, 54H25, 54E50.
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1 Introduction and preliminaries
In 1976, Jungck [1] proved a common fixed point theorem of commuting mappings in a metric space. In 1982, Sessa [2] introduced the concept of weakly commuting mappings, which is a generalization of the concept of commuting mappings, and he has proved some fixed point theorems for weakly commuting mappings. In 1986, Jungck [3] introduced more generalized commuting mappings, called compatible mappings, which are more general than commuting and weakly commuting mappings. These concepts have been useful for obtaining more comprehensive fixed point theorems. In 1992, Dhage [4] introduced the concept of D-metric space. Recently, Mustafa and Sims [5] have shown that most of the results concerning Dhage’s D-metric spaces are invalid. Therefore, they introduced an improved version of the generalized metric space structure and called it a G-metric space [6].
Based on the notion of generalized metric spaces, Mustafa et al. [7–9], Aydi et al. [10], Aydi [11], Gajié and Stojakovié [12], Zhou and Gu [13] obtained some fixed point results for mappings satisfying different contractive conditions. Shatanawi [14] obtained some fixed point results for Φ-maps in G-metric spaces. Chugh et al. [15] obtained some fixed point results for maps satisfying property P in G-metric spaces. In 2010, Manro et al. [16] obtained some fixed point results for expansion mappings in G-metric spaces.
The study of common fixed point problems in G-metric spaces was initiated by Abbas and Rhoades [17]. Subsequently, many authors obtained many common fixed point theorems for the mappings satisfying different contractive conditions; see [18–29] for more details. Recently, some authors have used the property in generalized metric space to prove common fixed point results, such as Abbas et al. [30], Mustafa et al. [31], Long et al. [32], Gu and Yin [33], Gu and Shatanawi [34].
Very recently, Jleli and Samet [35] and Samet et al. [36] noticed that some fixed point theorems in the context of a G-metric space can be concluded by some existing results in the setting of a (quasi-)metric space. In fact, if the contraction condition of the fixed point theorem on a G-metric space can be reduced to two variables instead of three variables, then one can construct an equivalent fixed point theorem in the setting of a usual metric space. More precisely, in [35, 36], the authors noticed that forms a quasi-metric. Therefore, if one can transform the contraction condition of existence results in a G-metric space in such terms, , then the related fixed point results become the well-known fixed point results in the context of a quasi-metric space.
Now we give basic definitions and some basic results [6], which are helpful for improving our main results.
Definition 1.1 [6]
Let X be a nonempty set, and let be a function satisfying the following axioms:
-
(G1) if ;
-
(G2) 0 for all with ;
-
(G3) for all with ;
-
(G4) (symmetry in all three variables); and
-
(G5) for all (rectangle inequality).
Then the function G is called a generalized metric, or more specifically, a G-metric on X and the pair is called a G-metric space.
It is well known that the function on the G-metric space X is jointly continuous in all three of its variables, and if and only if (see [6]).
Definition 1.2 [6]
Let be a G-metric space, be a sequence. Then the sequence is called:
-
(i)
a G-convergent sequence if, for any , there is an and an , such that for all , ; i.e. if ;
-
(ii)
a G-Cauchy sequence if, for any , there is an (the set of natural numbers) such that for all , ; i.e. if as .
A G-metric space is said to be G-complete if every G-Cauchy sequence in is G-convergent in X. It is well known that is G-convergent to .
Proposition 1.1 [6]
Let be a G-metric space, then the following statements are equivalent:
-
(1)
is G-convergent to x;
-
(2)
as ;
-
(3)
as ;
-
(4)
as .
Proposition 1.2 [6]
Let be a G-metric space. Then the function is jointly continuous in all of its three variables.
Definition 1.3 [6]
Let and be G-metric space, and be a function. Then f is said to be G-continuous at a point if and only if for every , there is such that and implies . A function f is G-continuous at X if and only if it is G-continuous at all .
Proposition 1.3 [6]
Let and be a G-metric space. Then is G-continuous at if and only if it is G-sequentially continuous at x, that is, whenever is G-convergent to x, is G-convergent to .
Definition 1.4 [18]
The self-mappings f and g of a G-metric space are said to be compatible if and , whenever is a sequence in X such that , for some .
In 2010, Manro et al. [16] introduced the concepts of weakly commuting, R-weakly commuting mappings, and R-weakly commuting mappings of type , , and in G-metric space as follows.
Definition 1.5 [16]
A pair of self-mappings of a G-metric space are said to be weakly commuting if
Definition 1.6 [16]
A pair of self-mappings of a G-metric space are said to be R-weakly commuting if there exists some positive real number R such that
Remark 1.1 If , then R-weakly commuting mappings are weakly commuting.
Definition 1.7 [16]
A pair of self-mappings of a G-metric space are said to be
-
(a)
R-weakly commuting mappings of type () if there exists some positive real number R such that , for all x in X.
-
(b)
R-weakly commuting mappings of type if there exists some positive real number R such that , for all x in X.
-
(c)
R-weakly commuting mappings of type if there exists some positive real number R such that , for all x in X.
Proposition 1.4 [6]
Every G-metric on X defines a metric on X by
For a symmetric G-metric space, one obtains
However, if G is not symmetric, then the following inequality holds:
It is also obvious that
2 Main results
Theorem 2.1 Let be a G-metric space and be a pair of non-compatible self-mappings with (here denotes the closure of fX). Assume the following conditions are satisfied:
for all . Here . If are a pair of R-weakly commuting mappings of type , then f and g have a unique common fixed point (say t) and both f and g are not G-continuous at t.
Proof Since f and g are non-compatible mappings, there exists a sequence , such that
but either or does not exist or exists and is different from zero. Since , there must exist a satisfying . We can assert that . If not, from condition (2.1), we get
Letting at both sides, we obtain
Since , we get , and so .
Since are a pair of R-weakly commuting mappings of type , we have
It means .
Next, we prove . In fact, if , from condition (2.1), and , we have
From we have , which implies that , and so is a common fixed point of f and g.
Next we prove that the common fixed point t is unique.
Actually, suppose w is also a common fixed point of f and g and , then using the condition (2.1), we have
which implies that , so that uniqueness is proved.
Now, we prove that f and g are not G-continuous at t. In fact, if f is G-continuous at t, we consider the sequence ; then we have
Since f and g are R-weakly commuting mappings of type , we get
so that we have
it follows that
Hence, we can get
and
This contradicts with f and g being non-compatible, so f is not G-continuous at t.
If g is G-continuous at t, then we have
Since f and g are R-weakly commuting mappings of type , we get
so that we have
and it follows that
This contradicts with f being not G-continuous at t, which implies that g is not G-continuous at t. This completes the proof. □
Next, we give an example to support Theorem 2.1.
Example 2.1 Let be a G-metric space with
We define mappings f and g on X by
Clearly, from the above functions we know that , and the pair are non-compatible self-maps. To see that f and g are non-compatible, consider a sequence . We have , , and . Thus
On the other hand, there exists such that for all x in X, that is, the pair are R-weakly commuting mappings of type .
Now we prove that the mappings f and g satisfy the condition (2.1) of Theorem 2.1 with . For this, let
We consider the following cases:
Case (1) If , then we have , and hence (2.1) is obviously satisfied.
Case (2) If , then we have , and hence (2.1) is obviously satisfied.
Case (3) If , , then we have and . Thus we obtain
Case (4) If , , then we have and . Thus we obtain
Case (5) If , , then we have and . Thus we obtain
Case (6) If , , then we have and . Thus we obtain
Case (7) If , , then we have and . Thus we obtain
Case (8) If , , then we have and . Thus we obtain
Case (9) If , , then we have and
Thus we obtain
Case (10) If , , then we have and
Thus we obtain
Case (11) If , , then we have and
Thus we obtain
Case (12) If , , then we have and
Thus we obtain
Case (13) If , , then we have and
Thus we obtain
Case (14) If , , then we have and
Thus we obtain
Case (15) If , , , then we have and
Thus we obtain
Case (16) If , , , then we have and
Thus we obtain
Case (17) If , , , then we have and
Thus we obtain
Case (18) If , , , then we have and
Thus we obtain
Case (19) If , , , then we have and
Thus we obtain
Case (20) If , , , then we have and
Thus we obtain
Then in all the above cases, the mappings f and g satisfy the condition (2.1) of Theorem 2.1 with , so that all the conditions of Theorem 2.1 are satisfied. Moreover, 2 is the unique common fixed point of f and g.
Theorem 2.2 Let be a G-metric space and let be a pair of non-compatible self-mappings such that
for any . Here . Assume the following conditions hold:
-
(i)
for any sequence that satisfies the condition , we have and ;
-
(ii)
are a pair of R-weakly commuting mappings of type .
Then f and g have a unique common fixed point in X.
Proof Since f and g are non-compatible mappings, there exists a sequence , such that
but, either or does not exist or exists and is different from zero.
From condition (i), we have
Since are a pair of R-weakly commuting mappings of type , we can get
Letting , we have . Thus we know .
From the condition (2.2), we have
Letting , we get
Since , therefore, . Thus, we get .
Suppose w is another fixed point of f and g and . Letting , , under the condition (2.2), we obtain
Since , we find that . Therefore, we have . So, the common fixed point of f and g is unique. Thus, we complete the proof. □
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Yang, Z. Common fixed point theorems for non-compatible self-maps in generalized metric spaces. J Inequal Appl 2014, 275 (2014). https://doi.org/10.1186/1029-242X-2014-275
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DOI: https://doi.org/10.1186/1029-242X-2014-275