Abstract
In this paper, we show that, unexpectedly, most of the coupled fixed point theorems in the context of (ordered) G-metric spaces are in fact immediate consequences of usual fixed point theorems that are either well known in the literature or can be obtained easily.
MSC:47H10, 54H25.
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1 Introduction
Investigation of the existence and uniqueness of fixed points of certain mappings in the framework of metric spaces is one of the centers of interests in nonlinear functional analysis. The Banach contraction mapping principle [1] is the limelight result in this direction: A self mapping T on a complete metric space has a unique fixed point if there exists such that for all . Fixed point theory has a wide application in almost all fields of quantitative sciences such as economics, biology, physics, chemistry, computer science and many branches of engineering. It is quite natural to consider various generalizations of metric spaces in order to address the needs of these quantitative sciences. In this respect quasi-metric spaces, ultra-metric spaces, uniform spaces, fuzzy metric spaces, partial metric spaces, cone metric spaces and b-metric spaces can be listed as well-known examples (see e.g. [2–6]). Consequently, the concept of a G-metric space was introduced by Mustafa and Sims [7] in 2004. The authors discussed the topological properties of this space and proved the analog of the Banach contraction mapping principle in the context of G-metric spaces (see e.g. [8–15]).
On the other hand, Ran and Reuring [16] proved the existence and uniqueness of a fixed point of a contraction mapping in partially ordered complete metric spaces. Following this initial work, a number of authors have investigated fixed points of various mappings and their applications in the theory of differential equations (see, e.g., [17–35]). Afterwords, Gnana-Bhaskar and Lakshmikantham [22] proved the existence and uniqueness of a coupled fixed point (defined by Guo and Laksmikantham [36]) in the context of partially ordered metric spaces by introducing the notion of mixed monotone property. In this remarkable paper, Gnana-Bhaskar and Lakshmikantham [22] also gave some applications related to the existence and uniqueness of a solution of periodic boundary value problems. Following this trend, many authors have studied the (common) coupled fixed points (see, e.g., [17–28, 31, 37–47]).
In this paper, we show that, unexpectedly, most of the coupled fixed point theorems in the context of (ordered) G-metric spaces are in fact immediate consequences of well-known fixed point theorems in the literature.
2 Preliminaries
We start with basic definitions and a detailed overview of the essential results developed in the interesting works mentioned above. Throughout this paper, ℕ is the set of nonnegative integers, and is the set of positive integers.
Definition 2.1 (See [7])
Let X be a nonempty set and be a function satisfying the following properties:
-
(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 specially, a G-metric on X, and the pair is called a G-metric space.
Every G-metric on X defines a metric on X by
Example 2.1 Let be a metric space. The function , defined as
or
for all , is a G-metric on X.
Definition 2.2 (See [7])
Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if
that is, for any , there exists such that for all . We call x the limit of the sequence and write or .
Proposition 2.1 (See [7])
Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x,
-
(2)
as ,
-
(3)
as ,
-
(4)
as .
Definition 2.3 (See [7])
Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there is such that for all , that is, as .
Proposition 2.2 (See [7])
Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy,
-
(2)
for any , there exists such that for all .
Definition 2.4 (See [7])
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
We will use the following result which can be easily derived from the definition of G-metric space (see, e.g., [7]).
Lemma 2.1 Let be a G-metric space. Then
Definition 2.5 (See [7])
Let be a G-metric space. A mapping is said to be G-continuous if is G-convergent to where is any G-convergent sequence converging to x.
We characterize this definition for a mapping . A mapping is said to be continuous if is G-convergent to where and are any two G-convergent sequences converging to x and y, respectively.
Definition 2.6 Let be a partially ordered set, be a G-metric space and be a mapping. A partially ordered G-metric space, , is called g-ordered complete if for each convergent sequence , the following conditions hold:
-
(OC1) if is a nonincreasing sequence in X such that , then ,
-
(OC2) if is a nondecreasing sequence in X such that , then .
In particular, a partially ordered G-metric space, , is called ordered complete when g is equal to an identity mapping in (OC1) and (OC2).
In [48], Mustafa characterized the well-known Banach contraction principle mapping in the context of G-metric spaces in the following ways.
Theorem 2.1 (See [48])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 2.2 (See [48])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Remark 2.1 The condition (2) implies the condition (3). The converse is true only if . For details, see [48].
In 1987, Guo and Lakshmikantham [36] introduced the notion of coupled fixed point. The concept of coupled fixed point was reconsidered by Gnana-Bhaskar and Lakshmikantham [22] in 2006. In this paper, they proved the existence and uniqueness of a coupled fixed point of an operator on a partially ordered metric space under a condition called the mixed monotone property.
Definition 2.7 ([22])
Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if is monotone nondecreasing in x and monotone nonincreasing in y; that is, for any ,
and
Definition 2.8 ([22])
An element is called a coupled fixed point of the mapping if
The results in [22] were extended by Ćirić and Lakshmikantham in [23] by defining the mixed g-monotone property.
Definition 2.9 Let be a partially ordered set, and . The function F is said to have the mixed g-monotone property if is monotone g-nondecreasing in x and is monotone g-nonincreasing in y; that is, for any ,
and
It is clear that Definition 2.9 reduces to Definition 2.7 when g is the identity.
Definition 2.10 An element is called a coupled coincidence point of mappings and if
and is called a coupled common fixed point of F and g if
The mappings F and g are said to commute if
for all .
3 Auxiliary results
We first state the following theorem about the existence and uniqueness of a common fixed point which can be considered as a generalization of Theorem 2.1.
Theorem 3.1 Let be a G-metric space. Let and be two mappings such that
for all x, y, z. Assume that T and g satisfy the following conditions:
-
(A1)
,
-
(A2)
is G-complete,
-
(A3)
g is G-continuous and commutes with T.
If , then there is a unique such that .
Proof Let . By assumption (A1), there exists such that . By the same arguments, there exists such that . Inductively, we define a sequence in the following way:
Due to (6), we have
by taking and . Thus, for each natural number n, we have
We will show that is a Cauchy sequence. By the rectangle inequality, we have for
Letting in (9), we get that . Hence, is a G-Cauchy sequence in . Since is G-complete, then there exists such that . Since g is G-continuous, we have is G-convergent to gz. On the other hand, we have since g and T commute. Thus,
Letting and using the fact that the metric G is continuous, we get that
Hence . The sequence is G-convergent to z since is a subsequence of . So, we have
Letting and using the fact that G is continuous, we obtain that
Hence we have . We will show that z is the unique common fixed point of T and g. Suppose that, contrary to our claim, there exists another common fixed point with . From (6) we have
which is a contradiction since . Hence, the common fixed point of T and g is unique. □
Theorem 3.2 Let be a G-metric space. Let and be two mappings such that
for all x, y. Assume that T and g satisfy the following conditions:
-
(A1)
,
-
(A2)
is G-complete,
-
(A3)
g is G-continuous and commutes with T.
If , then there is a unique such that .
Proof Following the lines of the proof of Theorem 3.1 by taking , one can easily get the result. □
In [16], Ran and Reurings established the following fixed point theorem that extends the Banach contraction principle to the setting of ordered metric spaces.
Theorem 3.3 (Ran and Reurings [16])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
T is continuous and nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
there exists a constant such that for all with ,
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
The result of Ran and Reurings [16] can be also proved in the framework of a G-metric space.
Theorem 3.4 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous and nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
there exists a constant such that for all with ,
(11)
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
Proof Let be a point satisfying (iii), that is, . We define a sequence in X as follows:
Regarding that T is a nondecreasing mapping together with (12), we have implies . Inductively, we obtain
Assume that there exists such that . Since , then is the fixed point of T, which completes the existence part of the proof. Suppose that for all . Thus, by (13) we have
Put and in (11). Then
Then we have
which, upon letting , implies
On the other hand, by Lemma 2.1 we have
The inequality (17) with and becomes
Letting in (18), we get
We will show that the sequence is a Cauchy sequence in the metric space where is given in (1). For we have
and making use of (15) and (18), we obtain
Hence,
that is, the sequence is Cauchy in and hence is G-Cauchy in (see Proposition 9 in [7]). Since the space is G-complete, then is complete (see Proposition 10 in [7]). Thus, is G-convergent to a number, say , that is,
We show now that is a fixed point of T, that is, . By the G-continuity of T, the sequence converges to Tu, that is,
The rectangle inequality on the other hand gives
Passing to limit as in (25), we conclude that . Hence, , that is, u∈ is a fixed point of T.
To prove the uniqueness, we assume that is another fixed point of T such that . We examine two cases. For the first case, assume that either or . Then we substitute and in (11) which yields . This is true only for , but by definition. Thus, the fixed point of T is unique.
For the second case, we suppose that neither nor holds. Then by assumption (iv), there exists such that and . Substituting and in (11), we get that . Since T is nondecreasing, . Substitute now and , which implies . Continuing in this way, we conclude . Passing to limit as , we get
Similarly, if we take and in (11), then we obtain
From (26) and (27), we deduce and . The uniqueness of the limit implies that . Hence, the fixed point of T is unique. □
Corollary 3.1 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous and nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
there exists a constant such that for all with ,
(28)
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
Proof It is sufficient to take in the proof of Theorem 3.4. □
Nieto and López [49] extended the result of Ran and Reurings [16] for a mapping T not necessarily continuous by assuming an additional hypothesis on .
Theorem 3.5 (Nieto and López [49])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
X is ordered complete;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exists a constant such that for all with ,
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
The result of Nieto and López [49] can also be proved in the framework of G-metric space.
Theorem 3.6 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is ordered complete;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exists a constant such that for all with ,
(29)
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
Proof Following the lines in the proof of Theorem 3.4, we have a sequence which is G-convergent to . Due to (ii), we have for all n. We will show that u is a fixed point of T. Suppose on the contrary that , that is, . Regarding (1) and (29) with , , we have
Passing to limit as , we get , which is a contradiction. Hence, . Uniqueness of u can be observed as in the proof of Theorem 3.4. □
Corollary 3.2 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is ordered complete;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exists a constant such that for all with ,
(31)
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
Proof It is sufficient to take in the proof of Theorem 3.6. □
Denote by Ψ the set of functions satisfying the following conditions:
() ,
() for all ;
() .
Following the work of Ćirić et al. [50], we generalize the above-mentioned results by means of introducing a function g. More specifically, we modify the definitions and theorems according to the presence of the function g.
Definition 3.1 (See [50])
Let be an ordered set and and be given mappings. The mapping T is called g-nondecreasing if for every ,
Theorem 3.7 Let be an ordered set endowed with a G-metric and and be given mappings. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous;
-
(iii)
T is g-nondecreasing;
-
(iv)
there exists such that ;
-
(v)
and g is G-continuous and commutes with T;
-
(vi)
there exists a function such that for all with ,
(32)
Then T and g have a coincidence point, that is, there exists such that .
Proof Let such that . Since , we can choose such that . Again, by , we can choose such that . By repeating the same argument, we construct the sequence in the following way:
Regarding that T is a g-nondecreasing mapping together with (33), we observe that
Inductively, we obtain
If there exists such that , then , that is, T and g have a coincidence point which completes the proof. Assume that for all .
Regarding (34), we set and in (32). Then we get
which is equivalent to
since for all . Let . Then is a positive nonincreasing sequence. Thus, there exists such that
We will show that . Suppose that contrary to our claim, . Letting in (35), we get
which is a contradiction. Hence, we have
We will show that is a G-Cauchy sequence. Suppose on the contrary that the sequence is not G-Cauchy. Then there exists and sequences of natural numbers , such that for each natural number k,
and we have
Corresponding to , the number is chosen to be the smallest number for which (38) holds. Hence, we have
By using (G5), we obtain that
Regarding (37) and letting in the previous inequality, we deduce
Again by the rectangle inequality (G5), together with (G4) and Lemma 2.1, we get that
Setting and , the inequality (32) implies
Combining the inequalities (41) and (42), we find
Taking (37) and (40) into account and letting in (43), we obtain that
which is a contradiction. Hence, is a G-Cauchy sequence in the G-metric space . Since is G-complete, there exists such that is G-convergent to w. By Proposition 2.1, we have
The G-continuity of g implies that the sequence is G-convergent to gw, that is,
On the other hand, due to the commutativity of T and g, we can write
and the G-continuity of T implies that the sequence G-converges to Tw so that
By the uniqueness of the limit, the expressions (45) and (47) yield that . Indeed, from the rectangle inequality, we get
which implies upon letting . Hence, . □
In the next theorem, G-continuity of T is no longer required. However, we require the g-ordered completeness of X.
Theorem 3.8 Let be an ordered set endowed with a G-metric and and be given mappings. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is g-ordered complete;
-
(iii)
T is g-nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
and g is G-continuous and commutes with T;
-
(vi)
there exists a function such that for all with ,
(49)
Then T and g have a coincidence point, that is, there exists such that .
Proof Following the lines of the proof of Theorem 3.7, we define a sequence and conclude that it is a G-Cauchy sequence in the G-complete, G-metric space . Thus, there exists such that is G-convergent to gw. Since is nondecreasing and X is g-ordered complete, we have . If for some natural number n, then T and g have a coincidence point. Indeed, and hence . Suppose that . By the rectangle inequality together with the inequality (49) and the property (), we have
Letting in the inequality above, we get that . Hence, . □
If we take , where in Theorem 3.7 and Theorem 3.8, we deduce the following corollaries, respectively.
Corollary 3.3 Let be an ordered set endowed with a G-metric and and be given mappings. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous;
-
(iii)
T is g-nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
and g is G-continuous and commutes with T;
-
(vi)
there exists such that for all with ,
(50)
Then T and g have a coincidence point, that is, there exists such that .
Corollary 3.4 Let be an ordered set endowed with a G-metric and and be given mappings. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is g-ordered complete;
-
(iii)
T is g-nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
and g is G-continuous and commutes with T;
-
(vi)
there exists such that for all with ,
(51)
Then T and g have a coincidence point, that is, there exists such that .
If we take in Theorem 3.7 and Theorem 3.8, we obtain the following particular cases.
Corollary 3.5 Let be an ordered set endowed with a G-metric and and be given mappings. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous;
-
(iii)
T is g-nondecreasing;
-
(iv)
there exists such that ;
-
(v)
and g is G-continuous and commutes with T;
-
(vi)
there exists a function such that for all with ,
(52)
Then T and g have a coincidence point, that is, there exists such that .
Corollary 3.6 Let be an ordered set endowed with a G-metric and and be given mappings. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is g-ordered complete;
-
(iii)
T is g-nondecreasing;
-
(iv)
there exists such that ;
-
(v)
and g is G-continuous and commutes with T;
-
(vi)
there exists a function such that for all with ,
(53)
Then T and g have a coincidence point, that is, there exists such that .
Finally, we let in the Theorem 3.7 and Theorem 3.8 and conclude the following theorems.
Theorem 3.9 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous;
-
(iii)
T is nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
there exists a function such that for all with ,
(54)
Then T has a fixed point.
Theorem 3.10 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is ordered complete;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exists a function such that for all with ,
(55)
Then T has a fixed point.
We next consider some equivalence conditions and their implementation on G-metric spaces. Let denote the set of functions satisfying the condition
In 2007, Jachymski and Jóźwik [51] proved that the classes and Ψ are equivalent. Regarding this result, we state the following fixed point theorems on G-metric spaces.
Theorem 3.11 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous;
-
(iii)
T is nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
there exists a function such that for all with ,
(56)
Then T has a fixed point.
Theorem 3.12 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is ordered complete;
-
(iii)
T is nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
there exists a function such that for all with ,
(57)
Then T has a fixed point.
The two corollaries below are immediate consequences of Theorem 3.11 and Theorem 3.12.
Corollary 3.7 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is G-continuous;
-
(iii)
T is nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
there exists a function such that for all with ,
(58)
Then T has a fixed point.
Corollary 3.8 Let be an ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
X is ordered complete;
-
(iii)
T is nondecreasing (with respect to ⪯);
-
(iv)
there exists such that ;
-
(v)
there exists a function such that for all with ,
(59)
Then T has a fixed point.
Denote by Φ the set of functions satisfying the conditions () and (). Jachymski [52] proved the equivalence of the so-called distance functions (see Lemma 1 in [52]). Inspired by this result, we can state the following theorem.
Theorem 3.13 Let be an ordered set endowed with a G-metric and T be a self-map on a G-complete partially ordered G-metric space . The following statements are equivalent:
-
(i)
there exist functions such that
(60) -
(ii)
there exist and a function such that
(61) -
(iii)
there exists a continuous and nondecreasing function such that for all such that
(62) -
(iv)
there exist a function and a nondecreasing function with such that
(63) -
(iv)
there exist a function and a lower semi-continuous function with and such that
(64)
for any with .
As a consequence of Theorem 3.13, we state the next corollary.
Corollary 3.9 Let be an ordered set endowed with a G-metric and T be a self-map on a G-complete partially ordered G-metric space . The following statements are equivalent:
-
(i)
there exist functions such that
(65) -
(ii)
there exist and a function such that
(66) -
(iii)
there exists a continuous and nondecreasing function such that for all such that
(67) -
(iv)
there exist a function and a nondecreasing function with such that
(68) -
(iv)
there exist a function and a lower semi-continuous function with and such that
(69)
for any with .
4 Remarks on coupled fixed point theorems in G-metric spaces
In this section, we prove that most of the coupled fixed point theorems on a G-metric space X can be derived from the well-known fixed point theorems on G-metric spaces in the literature provided that is a symmetric G-metric space. In the rest this paper, we shall assume that represents a symmetric G-metric space.
Let be a partially ordered set endowed with a metric G and and be given mappings. We define a partial order ⪯2 on the product set as follows:
Definition 4.1 F is said to have the mixed g-monotone property if is monotone nondecreasing in x and is monotone nonincreasing in y; that is, for any ,
If g is an identity mapping, then F is said to have the mixed monotone property, that is,
Let . It is easy to show that the mappings defined by
for all , are G-metrics on Y.
Now, define the mapping by
The following lemma is obvious.
Lemma 4.1 The following properties hold:
-
(a)
If is G-complete, then and are Λ-complete and Δ-complete, respectively;
-
(b)
F has the mixed (g-mixed) monotone property if and only if T is monotone nondecreasing (g-nondecreasing) with respect to ⪯2;
-
(c)
is a coupled fixed point of F if and only if is a fixed point of T;
-
(d)
is a coupled coincidence point of F and g if and only if is a coupled coincidence point of T and g.
4.1 Shatanawi’s coupled fixed point results in a G-metric space
In [31], Shatanawi proved the following theorems.
Theorem 4.1 (cf. [31])
Let be a G-complete G-metric space. Let be a mapping such that
for all . If , then there exists a unique such that .
In what follows, we prove the following theorem.
Theorem 4.2 Theorem 4.1 follows from Theorem 2.1.
Proof From (73), for all with and , we have
and
which implies that
that is,
for all , where Λ is defined in (70). From Lemma 4.1, since is G-complete, is Λ-complete. In this case, regarding Theorem 2.1, we conclude that T has a fixed point, which due to Lemma 4.1 implies that F has a coupled fixed point.
Analogously, Theorem 4.3 is obtained from Theorem 2.2. □
The following theorem can be derived easily from Theorem 4.1.
Theorem 4.3 Let be a G-complete G-metric space. Let be a mapping such that
If , then there is a unique such that .
We note that Theorem 4.3 above is not stated in [31].
Theorem 4.4 Theorem 4.3 follows from Theorem 2.2.
Example 4.1 Let . Define by
for all . Then is a G-metric space. Define a map by for all . Then, for all with , we have
and
Then it is easy to see that there is no such that
for all . Indeed, the inequality above holds for . Thus, Theorem 4.1 does not apply to this example. However, it is easy to see that 0 is the unique point such that .
On the other hand, Theorem 2.1 yields the existence of the fixed point. Indeed,
where . Thus, all conditions of Theorem 3.1 are satisfied, which guarantees the existence of the fixed point .
4.2 Choudhury and Maity’s coupled fixed point results in a G-metric space
Choudhury and Maity [53] proved the following coupled fixed point theorems on ordered G-metric spaces.
Theorem 4.5 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a G-continuous mapping having the mixed monotone property on X. Suppose that there exists a such that
for all with and , where either or . If there exist such that and , then F has a coupled fixed point, that is, there exists such that and .
Theorem 4.6 If in the above theorem, instead of G-continuity of F, we assume that X is ordered complete, then F has a coupled fixed point.
We will prove the following result.
Theorem 4.7 Theorem 4.5 and Theorem 4.6 follow from Theorems 3.4 and 3.6, respectively.
Proof From (75), for all with and , we have
and
This implies that for all with and ,
that is,
for all with , where Λ is defined in (70).
It follows from Lemma 4.1 that since is G-complete, then is Λ-complete. Since F has the mixed monotone property, T is a nondecreasing mapping with respect to ⪯2. The assumption that there exist such that and becomes in terms of the order ⪯2. Now, if F is G-continuous, then T is Λ-continuous. In this case, applying Theorem 3.4, we get that T has a fixed point, which due to Lemma 4.1 implies that F has a coupled fixed point. If X is ordered complete, then Y satisfies the following property: if a nondecreasing (with respect to ⪯2) sequence in Y converges to some point , then for all n. Applying Theorem 3.6, we get that T has a fixed point, that is, F has a coupled fixed point. □
Remark 4.1 (Uniqueness)
If, in addition, we suppose that for all , there exists such that and , from the last part of Theorems 3.4 and 3.6, we obtain the uniqueness of the fixed point of T, which implies the uniqueness of the coupled fixed point of F. Now, let be the unique coupled fixed point of F. Since is also a coupled fixed point of F, we get .
4.3 Coupled fixed point results of Aydi et al. in a G-metric space
We consider the following fixed point theorems established by Aydi et al. [23]. The following lemma is trivial.
Lemma 4.2 (See [23]) Let . For all , we have .
Aydi et al. [17] proved the following fixed point theorems.
Theorem 4.8 Let be a partially ordered set and G be a G-metric on X such that is a G-complete G-metric space. Suppose that there exist and such that
for all with and . Suppose also that F is G-continuous and has the mixed monotone property. If there exist such that and , then F has a coupled fixed point, that is, there exists such that and .
Replacing the G-continuity of F by ordered completeness of X yields the next theorem.
Theorem 4.9 Let be a partially ordered set and G be a G-metric on X such that is G-complete. Suppose that there exist and such that
for all with and . Suppose also that F has the mixed monotone property and X is ordered complete. If there exist such that and , then F has a coupled fixed point, that is, there exists such that and .
We will prove the following result.
Theorem 4.10 Theorem 4.8 and Theorem 4.9 follow from Theorems 3.9 and 3.10, respectively.
Proof From (76), for all with and , we have
and
This implies that for all with and ,
holds. Rewrite the above inequality as
for all with . Here, is the G-metric on Y defined by
where Λ is given in (70). Thus, the mapping T satisfies the conditions of Theorem 3.9 (resp. Theorem 3.10). Therefore, T has a fixed point, which implies that F has a coupled fixed point. □
4.4 On coupled fixed point results of Abbas et al. in a G-metric space
Let Θ be the set of functions which satisfy the condition
The following theorems have been given by Abbas et al. [37].
Theorem 4.11 Let be a partially ordered set and G be a G-metric on X. Assume that there is an altering distance function and a map such that
for all with and . Suppose that F is G-continuous and has the mixed monotone property. If there exist such that and , then F has a coupled fixed point, that is, there exists such that and .
Theorem 4.12 Let be a partially ordered set and G be a G-metric on X. Assume that there is an altering distance function and a mapping such that
for all with and . Suppose that F has the mixed monotone property and X is ordered complete. If there exist such that and , then F has a coupled fixed point, that is, there exists such that and .
We will prove the following result.
Theorem 4.13 Theorem 4.11 and Theorem 4.12 follow from Theorem 3.11 and Theorem 3.12.
Proof From (78), for all with and , we have
This is equivalent to
for all with , where , which clearly implies that .
From Lemma 4.1, since is G-complete, is Λ-complete. Since F has the mixed monotone property, T is a nondecreasing mapping with respect to ⪯2. According to the assumption of Theorem 4.11, we have . Now, if F is G-continuous, then T is Λ-continuous. In this case, applying Theorem 3.11, we get that T has a fixed point, which implies from Lemma 4.1 that F has a coupled fixed point. If X is ordered complete, then Y satisfies the following property: if a nondecreasing (with respect to ⪯2) then the sequence in Y converges to some point , then for all n. Applying Theorem 3.12, we get that T has a fixed point, which implies that F has a coupled fixed point. □
5 Remarks on common coupled fixed point theorems in G-metric spaces
In this last section, we investigate the similarity between most of the common coupled fixed point theorems and ordinary fixed point theorems in the context of G-metric spaces and we show that the former are immediate consequences of the latter.
5.1 Shatanawi’s common coupled fixed point results in a G-metric space
We start with two theorems by Shatanawi [31].
Theorem 5.1 (cf. [31])
Let be a G-metric space. Let and be two mappings such that
for all . Assume that F and g satisfy the following conditions:
-
(1)
,
-
(2)
is G-complete,
-
(3)
g is G-continuous and commutes with F.
If , then there is a unique such that .
Theorem 5.2 (cf. [31])
Let be a G-metric space. Let and be two mappings such that
for all . Assume that F and g satisfy the following conditions:
-
(1)
,
-
(2)
is G-complete,
-
(3)
g is G-continuous and commutes with F.
If , then there is a unique such that .
Theorem 5.3 Theorem 5.1 and Theorem 5.2 follow from Theorem 3.1 and Theorem 3.2, respectively.
Proof From (81), for all , we have
and
Therefore, for all , we obtain
that is,
for all , where are mappings such that and and Λ is defined in (70). From Lemma 4.1, since is G-complete, is Λ-complete. In this case, applying Theorem 3.1, we get that and have a common fixed point, which implies from Lemma 4.1 that F and g have a common coupled fixed point.
Analogously, Theorem 5.2 is obtained from Theorem 3.2. □
Example 5.1 Let . Define by
for all . Then is a G-metric space. Define a map by and by for all . Then, for all with , we have
and
Then it is easy to see that there is no such that
for all . Thus, Theorem 5.1 does not provide the existence of the common fixed point for the maps on this example. However, it is easy to see that 0 is the unique point such that .
On the other hand, notice that Theorem 3.1 yields the fixed point. Indeed,
and also
Then the condition (6) of Theorem 3.1 holds for . Thus, all conditions of Theorem 3.1 are satisfied, which provides the common coupled fixed point of F and g.
5.2 Nashine’s common coupled fixed point results in a G-metric space
Nashine [54] studied common coupled fixed points on ordered G-metric spaces and proved the following theorems.
Theorem 5.4 Let be a partially ordered G-metric space. Let and be mappings such that F has the mixed g-monotone property. Suppose that there exist such that and . Suppose also that there exists such that
holds for all satisfying and , where either or . We assume the following hypotheses:
-
(i)
,
-
(ii)
F is G-continuous,
-
(iii)
is G-complete,
-
(iv)
g is G-continuous and commutes with F.
Then F and g have a coupled coincidence point, that is, there exists such that and . If and , then F and g have a common fixed point, that is, there exists such that .
Theorem 5.5 If in the above theorem we replace the G-continuity of F by the assumption that X is g-ordered complete, then F and g have a coupled coincidence point.
Theorem 5.6 Theorem 5.4 and Theorem 5.5 follow from Corollary 3.3 and Corollary 3.4, respectively.
Proof From (83), for all with and , we have
and
This implies that for all with and ,
that is,
for all with .
From Lemma 4.1, since is G-complete, is Λ-complete. Also, since F has the mixed g-monotone property, T is a g-nondecreasing mapping with respect to ⪯2. From the assumption of Theorem 5.4, we have . Now, if F is G-continuous, then T is Λ-continuous. In this case, due to Corollary 3.3, we deduce that T and g have a coincidence point, which from Lemma 4.1 implies that F and g have a coupled coincidence point. If, on the other hand, X is g-ordered complete, then Y satisfies the following property: if a nondecreasing (with respect to ⪯2) sequence in Y converges to a point , then for all n. According to Corollary 3.4, T and g have a coincidence point, which from Lemma 4.1 implies that F and g have a coupled coincidence point. □
5.3 Common coupled fixed point results of Aydi et al. in a G-metric space
Recently, Aydi et al. [17] proved the following theorems on G-metric spaces.
Theorem 5.7 Let be a partially ordered set and G be a G-metric on X such that is a G-complete G-metric space. Suppose that there exist maps and and such that
for all with and . Suppose also that F is G-continuous and has the mixed g-monotone property, , and g is G-continuous and commutes with F. If there exist such that and , then F and g have a coupled coincidence point, that is, there exists such that and .
Theorem 5.8 Let be a partially ordered set and G be a G-metric on X such that is G-complete. Suppose that there exist and and such that
for all with and . Suppose also that X is g-ordered complete and F has the mixed g-monotone property, , and g is G-continuous and commutes with F. If there exist such that and , then F and g have a coupled coincidence point, that is, there exists such that and .
The following result can be also proved easily.
Theorem 5.9 Theorem 4.8 and Theorem 4.9 follow from Theorems 3.7 and 3.8.
Proof We observe from (84) that for all with and , we have
and also
Therefore, for all with and , the following inequality holds:
that is,
for all with , where are mappings such that and . Note that here, is a G-metric on Y defined by
Thus, we proved that the mappings and satisfy the conditions of Theorem 3.7 (resp. Theorem 3.8). Hence, and have a coincidence point, which implies that F and g have a coupled coincidence point. □
5.4 Common coupled fixed point results of Cho et al. in a G-metric space
Finally, we consider the results of Cho et al. [55]. We state their fixed point theorems below.
Theorem 5.10 Let be a partially ordered set and G be a G-metric on X such that is a G-complete G-metric space. Let and be G-continuous mappings such that F has the mixed g-monotone property and g commutes with F. Assume that there exist altering distance functions ϕ and ψ such that
for all with and . Suppose also that . If there exist such that and , then F and g have a coupled coincidence point, that is, there exists such that and .
Theorem 5.11 Let be a partially ordered set and G be a G-metric on X and let and be mappings. Assume that there exist altering distance functions ϕ and ψ such that
for all with and . Suppose that is G-complete, the mapping F has the mixed g-monotone property and . If there exist such that and , then F and g have a coupled coincidence point, that is, there exists such that and .
Theorem 5.12 Theorem 5.10 and Theorem 5.11 follow from Theorems 3.7 and 3.8.
Proof From the assumption, for all with and , we have
and
This implies (since ψ is nondecreasing) that for all with and , we have
that is,
for all with , where such that and and Δ is a G-metric defined in (71). Regarding Theorem 3.13, the conditions (90) and (32) of Theorem 3.7 are equivalent. Therefore, the mappings and satisfy the conditions of Theorem 3.7 (resp. Theorem 3.8) and have a coincidence point. Hence, the maps F and g have a coupled coincidence point. □
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Agarwal, R.P., Karapınar, E. Remarks on some coupled fixed point theorems in G-metric spaces. Fixed Point Theory Appl 2013, 2 (2013). https://doi.org/10.1186/1687-1812-2013-2
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DOI: https://doi.org/10.1186/1687-1812-2013-2