Abstract
In this paper, using the concept of common property, we prove some common fixed point theorems for three pairs of weakly compatible self-maps satisfying a generalized weakly G-contraction condition in the framework of a generalized metric space. Our results do not rely on any commuting or continuity condition of mappings. An example is provided to support our result in nonsymmetric G-metric space.
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1 Introduction and preliminaries
The study of fixed points and common fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. In 2006, Mustafa and Sims [1] introduced the concept of generalized metric spaces or simply G-metric spaces as a generalization of the notion of metric space. Based on the notion of generalized metric spaces, Mustafa et al. [2–5], Obiedat and Mustafa [6], Aydi [7], Gajié and Stojakovié [8], Shatanawi et al. [9], Zhou and Gu [10] obtained some fixed point results for mappings satisfying different contractive conditions. Shatanawi [11] obtained some fixed point results for Φ-maps in G-metric spaces. Chugh et al. [12] obtained some fixed point results for maps satisfying property P in G-metric spaces. Al-khaleel et al. [13] obtained several fixed point results for mappings that satisfy certain contractive conditions in generalized cone metric spaces. The study of common fixed point problems in G-metric spaces was initiated by Abbas and Rhoades [14]. Subsequently, many authors have obtained many common fixed point theorems for the mappings satisfying different contractive conditions; see [15–34] for more details. Recently, Abbas et al. [35] and Mustafa et al. [36] obtained some common fixed point results for a pair of mappings satisfying the property under certain generalized strict contractive conditions in G-metric spaces. Long et al. [37] obtained some common coincidence and common fixed points results of two pairs of mappings when only one pair satisfies the property in G-metric spaces. Very recently, Gu and Yin [38] obtained some common fixed point theorems of three pairs of mappings for which only two pairs need to satisfy the common property in the framework of G-metric spaces.
Now we give preliminaries and basic definitions which are used throughout the paper.
Definition 1.1 (see [1])
Let X be a nonempty set, and let be a function satisfying the following axioms:
-
(G1)
if ;
-
(G2)
for all with ;
-
(G3)
for all with ;
-
(G4)
(symmetry in all three variables);
-
(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 known that the function on a G-metric space X is jointly continuous in all three of its variables, and if and only if ; for more details, see [1] and the references therein.
Definition 1.2 (see [1])
Let be a G-metric space, and let be a sequence of points in X. A point x in X is said to be the limit of the sequence if , and one says that the sequence is G-convergent to x.
Thus, if in a G-metric space , then for any , there exists (throughout this paper we mean by ℕ the set of all natural numbers) such that for all .
Proposition 1.3 (see [1])
Let be a G-metric space, then the following are equivalent:
-
(1)
is G-convergent to x;
-
(2)
as ;
-
(3)
as ;
-
(4)
as .
Definition 1.4 (see [1])
Let be a G-metric space. The sequence is called a G-Cauchy sequence if for each , there exists such that for all ; i.e., if as .
Definition 1.5 (see [1])
A G-metric space is said to be G-complete (or a complete G-metric space) if every G-Cauchy sequence in is G-convergent in X.
Proposition 1.6 (see [1])
Let be a G-metric space. Then the following are equivalent.
-
(1)
The sequence is G-Cauchy.
-
(2)
For every , there exists such that for all .
Proposition 1.7 (see [1])
Let be a G-metric space. Then the function is jointly continuous in all three of its variables.
Proposition 1.8 (see [1])
Let be a G-metric space. Then, for all x, y in X, it follows that .
Definition 1.9 (see [39])
Let f and g be self-maps of a set X. If for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.
Definition 1.10 (see [39])
Two self-mappings f and g on X are said to be weakly compatible if they commute at coincidence points.
Definition 1.11 (see [35])
Let X be a G-metric space. Self-maps f and g on X are said to satisfy the G- property if there exists a sequence in X such that and are G-convergent to some .
Definition 1.12 Let be a G-metric space and A, B, S and T be four self-maps on X. The pairs and are said to satisfy the common property if there exist two sequences and in X such that for some .
Definition 1.13 (see [19])
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 .
Definition 1.14 (see [18])
A pair of self-mappings of a G-metric space is said to be weakly commuting if
Definition 1.15 (see [18])
A pair of self-mappings of a G-metric space is said to be R-weakly commuting if there exists some positive real number R such that
Recently, Shatanawi et al. [9] introduced the following definitions.
Definition 1.16 (see [9])
Let be a G-metric space. A mapping is said to be weakly G-contractive if for all , the following inequality holds:
where is a continuous function with if and only if .
Definition 1.17 (see [9])
Let be a G-metric space. A mapping is said to be a weakly G-contractive-type mapping if for all , the following inequality holds:
where is a continuous function with if and only if .
Khan et al. [40] introduced the concept of altering distance function that is a control function employed to alter the metric distance between two points enabling one to deal with relatively new classes of fixed point problems. Here, we consider the following notion.
Definition 1.18 (see [15])
The function is called an altering distance function if the following properties are satisfied:
-
(1)
ψ is continuous and increasing;
-
(2)
if and only if .
In 2011, Aydi et al. [15] introduced the concept of generalized weakly G-contraction mapping of A and B as follows.
Definition 1.19 (see [15])
Let be a G-metric space and be two mappings. We say that f is a generalized weakly G-contraction mapping of type A with respect to g if for all , the following inequality holds:
where
-
(1)
ψ is an altering distance function;
-
(2)
is a continuous function with if and only if .
Definition 1.20 (see [15])
Let be a G-metric space and be two mappings. We say that f is a generalized weakly G-contraction mapping of type B with respect to g if for all , the following inequality holds:
where
-
(1)
ψ is an altering distance function;
-
(2)
is a continuous function with if and only if .
In this paper, using the concept of common property, we prove some common fixed point results for six self-mappings f, g, h, R, S and T, where the triple is a generalized weakly G-contraction mapping of types A and B with respect to the triple . These notions will be given by Definitions 2.1 and 2.5.
2 Main results
We start with the following definition.
Definition 2.1 Let be a G-metric space and be six mappings. We say that the triple is a generalized weakly G-contraction mapping of type A with respect to the triple if for all , the following inequality holds:
where
-
(1)
ψ is an altering distance function;
-
(2)
is a continuous function with if and only if .
Theorem 2.2 Let be a G-metric space and be six mappings such that is a generalized weakly G-contraction mapping of type A with respect to . If one of the following conditions is satisfied, then the pairs , and have a common point of coincidence in X.
-
(i)
The subspace RX is closed in X, , , and two pairs of and satisfy the common property;
-
(ii)
The subspace SX is closed in X, , , and two pairs of and satisfy the common property;
-
(iii)
The subspace TX is closed in X, , , and two pairs of and satisfy the common property.
Moreover, if the pairs , and are weakly compatible, then f, g, h, R, S and T have a unique common fixed point in X.
Proof First, we suppose that the subspace RX is closed in X, , , and two pairs of and satisfy the common property. Then by Definition 1.12 we know that there exist two sequences and in X such that
for some .
Since , there exists a sequence in X such that . Hence . Next, we will show . In fact, from condition (2.1), we can get
On letting and using the continuities of ψ and ϕ, we can obtain
By Proposition 1.8, we have
and hence using the fact that ψ is increasing, (2.2) becomes
which implies that , and so .
Since RX is a closed subspace of X and , there exists p in X such that . We claim that . In fact, by using (2.1), we obtain
Taking on the two sides of the above inequality, using the continuities of ψ and ϕ, Proposition 1.8 and the fact that ψ is increasing, we can get
which implies that , and hence . Therefore, p is the coincidence point of a pair .
By the condition and , there exist a point u in X such that . Now, we claim that . In fact, from (2.1) we have
Letting on the two sides of the above inequality, using the continuities of ψ and ϕ, Proposition 1.8 and the fact that ψ is increasing, we can obtain
which implies that , hence , and so u is the coincidence point of a pair .
Since and , there exist a point v in X such that . We claim that . In fact, from (2.1), using , Proposition 1.8 and the fact that ψ is increasing, we have
This implies that , and so , hence v is the coincidence point of a pair .
Therefore, in all the above cases, we obtain . Now, weak compatibility of the pairs , and gives that , and .
Next, we show that . In fact, using (2.1), Proposition 1.8 and the fact that ψ is increasing, we have
which implies that , and so , that is, , and so . Similarly, it can be shown that and , so we get , which means that t is a common fixed point of f, g, h, R, S and T.
Next, we will show that the common fixed point of f, g, h, R, S and T is unique. Actually, suppose that is another common fixed point of f, g, h, R, S and T, then by condition (2.1), Proposition 1.8 and the fact that ψ is increasing, we have
which implies that , and so , hence , that is, mappings f, g, h, R, S and T have a unique common fixed point.
Finally, if condition (ii) or (iii) holds, then the argument is similar to that above, so we delete it.
This completes the proof of Theorem 2.2. □
Now we introduce an example to support Theorem 2.2.
Example 2.3 Let be a set with G-metric defined by Table 1.
Note that G is non-symmetric as . Let be defined by Table 2.
Clearly, the subspace RX is closed in X, and with the pairs , and being weakly compatible. Also, two pairs and satisfy the common property, indeed, and for each are the required sequences. The control functions and are defined by
It is easy to show that the triple is a generalized weakly G-contraction mapping of type A with respect to the triple . In fact, contractive condition (2.1) and the following inequality are equivalent:
To check contractive condition (2.3) for all , we consider the following cases.
Note that for Cases (1) , (2) , , (3) , , (4) , , , (5) , , (6) , , , (7) , , (8) , , (9) , , (10) , , (11) , , and (12) , , we have , and hence (2.3) is obviously satisfied.
Case (13) If , , then , , , , hence we have
Case (14) If , , then , , , , hence we have
Case (15) If , , then , , , , , hence we have
Case (16) If , , , then , , , , hence we have
Case (17) If , , then , , , , , hence we have
Case (18) If , , then , , , , , hence we have
Case (19) , then , , , , , hence we have
Case (20) If , , , then , , , , , hence we have
Case (21) If , , then , , , hence we have
Case (22) If , , then , , , , hence we have
Case (23) If , , , then , , , , , hence we have
Case (24) If , , then , , , , , hence we have
Case (25) If , , then , , , , , hence we have
Case (26) , , then , , , hence we have
Case (27) If , then , , , , hence we have
Hence, all of the conditions of Theorem 2.2 are satisfied. Moreover, 0 is the unique common fixed point of f, g, h, R, S and T.
Corollary 2.4 Let be a G-metric space. Suppose that mappings satisfy the following conditions:
for all , where . If one of the following conditions is satisfied, then the pairs , and have a common point of coincidence in X.
-
(i)
The subspace RX is closed in X, , , and two pairs of and satisfy the common property;
-
(ii)
The subspace SX is closed in X, , , and two pairs of and satisfy the common property;
-
(iii)
The subspace TX is closed in X, , , and two pairs of and satisfy the common property.
Moreover, if the pairs , and are weakly compatible, then f, g, h, R, S and T have a unique common fixed point in X.
Proof It suffices to take and in Theorem 2.2. □
Definition 2.5 Let be a G-metric space and be six mappings. We say that the triple is a generalized weakly G-contraction mapping of type B with respect to the triple if for all , the following inequality holds:
where
-
(1)
ψ is an altering distance function;
-
(2)
is a continuous function with if and only if .
Using arguments similar to those in Theorem 2.2, we can prove the following theorem.
Theorem 2.6 Let be a G-metric space and be six mappings such that is a generalized weakly G-contraction mapping of type B with respect to . If one of the following conditions is satisfied, then the pairs , and have a common point of coincidence in X.
-
(i)
The subspace RX is closed in X, , , and two pairs of and satisfy the common property;
-
(ii)
The subspace SX is closed in X, , , and two pairs of and satisfy the common property;
-
(iii)
The subspace TX is closed in X, , , and two pairs of and satisfy the common property.
Moreover, if the pairs , and are weakly compatible, then f, g, h, R, S and T have a unique common fixed point in X.
As in the case of Theorem 2.2, we can deduce the following corollary from Theorem 2.6.
Corollary 2.7 Let be a G-metric space. Suppose that mappings satisfy the following conditions:
for all , where . If one of the following conditions is satisfied, then the pairs , and have a common point of coincidence in X.
-
(i)
The subspace RX is closed in X, , , and two pairs of and satisfy the common property;
-
(ii)
The subspace SX is closed in X, , , and two pairs of and satisfy the common property;
-
(iii)
The subspace TX is closed in X, , , and two pairs of and satisfy the common property.
Moreover, if the pairs , and are weakly compatible, then f, g, h, R, S and T have a unique common fixed point in X.
Remark 2.8 If we take: (1) ; (2) ; (3) (I is an identity mapping); (4) and ; (5) , in Theorems 2.2 and 2.6, Corollaries 2.4 and 2.7, then several new results can be obtained.
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Acknowledgements
The present studies are supported by the National Natural Science Foundation of China (11271105, 11071169), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030).
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Gu, F., Shatanawi, W. Common fixed point for generalized weakly G-contraction mappings satisfying common property in G-metric spaces. Fixed Point Theory Appl 2013, 309 (2013). https://doi.org/10.1186/1687-1812-2013-309
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DOI: https://doi.org/10.1186/1687-1812-2013-309