Abstract
In this paper, tripled coincidence points of mappings satisfying some nonlinear contractive conditions in the framework of partially ordered -metric spaces are obtained. Our results extend the results of Aydi et al. (Fixed Point Theory Appl., 2012:101, 2012, doi:10.1186/1687-1812-2012-101). Moreover, some examples of the main result are given. Finally, some tripled coincidence point results for mappings satisfying some contractive conditions of integral type in complete partially ordered -metric spaces are deduced.
MSC: 47H10, 54H25.
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1 Introduction and preliminaries
The concepts of mixed monotone mapping and coupled fixed point were introduced in [1] by Bhaskar and Lakshmikantham. Also, they established some coupled fixed point theorems for a mixed monotone mapping in partially ordered metric spaces. For more details on coupled fixed point theorems and related topics in different metric spaces, we refer the reader to [2–13] and [14–25].
Also, Berinde and Borcut [26] introduced a new concept of tripled fixed point and obtained some tripled fixed point theorems for contractive-type mappings in partially ordered metric spaces. For a survey of tripled fixed point theorems and related topics, we refer the reader to [26–32].
Definition 1.1 [26]
An element is called a tripled fixed point of if , and .
Definition 1.2 [27]
An element is called a tripled coincidence point of the mappings and if , and .
Definition 1.3 [27]
An element is called a tripled common fixed point of and if , and .
Definition 1.4 [29]
Let X be a nonempty set. We say that the mappings and are commutative if for all .
The notion of altering distance function was introduced by Khan et al. [10] as follows.
Definition 1.5 The function is called an altering distance function if
-
1.
ψ is continuous and nondecreasing.
-
2.
if and only if .
The concept of generalized metric space, or G-metric space, was introduced by Mustafa and Sims [33]. Mustafa and others studied several fixed point theorems for mappings satisfying different contractive conditions (see [33–45]).
Definition 1.6 (G-metric space, [33])
Let X be a nonempty set and be a function satisfying the following properties:
-
(G1) iff ;
-
(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 G-metric on X and the pair is called a G-metric space.
Example 1.7 If we think that is measuring the perimeter of the triangle with vertices at x, y and z, then (G5) can be interpreted as
where is the ‘length’ of the side x, y. If we take , we have
Thus, (G5) embodies the triangle inequality. And so (G5) can be sharp.
In [46], Aydi et al. established some tripled coincidence point results for mappings and involving nonlinear contractions in the setting of ordered G-metric spaces.
Theorem 1.8 [46]
Let be a partially ordered set and be a G-metric space such that is G-complete. Let and . Assume that there exist such that ψ is an altering distance function and ϕ is a lower-semicontinuous and nondecreasing function with if and only if and for all , with , and , we have
Assume that F and g satisfy the following conditions:
-
(1)
,
-
(2)
F has the mixed g-monotone property,
-
(3)
F is continuous,
-
(4)
g is continuous and commutes with F.
Let there exist such that , and . Then F and g have a tripled coincidence point in X, i.e., there exist such that , and .
Also, they proved that the above theorem is still valid for F not necessarily continuous assuming the following hypothesis (see Theorem 19 of [46]).
-
(I)
If is a nondecreasing sequence with , then for all .
-
(II)
If is a nonincreasing sequence with , then for all .
A partially ordered G-metric space with the above properties is called regular.
In this paper, we obtain some tripled coincidence point theorems for nonlinear -weakly contractive mappings in partially ordered -metric spaces. This results generalize and modify several comparable results in the literature. First, we recall the concept of generalized b-metric spaces, or -metric spaces.
Definition 1.9 [47]
Let X be a nonempty set and be a given real number. Suppose that a mapping satisfies:
-
(G b 1) if ,
-
(G b 2) for all with ,
-
(G b 3) for all with ,
-
(G b 4) , where p is a permutation of x, y, z (symmetry),
-
(G b 5) for all (rectangle inequality).
Then G is called a generalized b-metric and the pair is called a generalized b-metric space or a -metric space.
Obviously, each G-metric space is a -metric space with . But the following example shows that a -metric on X need not be a G-metric on X (see also [48]).
Example 1.10 If we think that is the maximum of the squares of length sides of a triangle with vertices at x, y and z such that:
If , then .
If , then ,
where is the ‘length’ of the side x, y. Then it is easy to see that is a function with .
Since by the triangle inequality we have
hence
Example 1.11 [47]
Let be a G-metric space and , where is a real number. Then is a -metric with .
Also, in the above example, is not necessarily a G-metric space. For example, let and G-metric G be defined by
for all (see [33]). Then is a -metric on ℝ with , but it is not a G-metric on ℝ.
Example 1.12 [47]
Let and . We know that is a b-metric space with . Let , then is not a -metric space. Indeed, (G b 3) is not true for , and . To see this, we have
and
So, .
However, is a -metric on ℝ with . Similarly, if is selected with , then is a -metric on ℝ with .
Now we present some definitions and propositions in a -metric space.
Definition 1.13 [47]
A -metric G is said to be symmetric if for all .
Definition 1.14 Let be a -metric space. Then, for and , the -ball with center and radius r is
By some straight forward calculations, we can establish the following.
Proposition 1.15 [47]
Let X be a -metric space. Then, for each , it follows that:
-
(1)
if , then ,
-
(2)
,
-
(3)
,
-
(4)
.
Definition 1.16 [47]
Let X be a -metric space. We define for all . It is easy to see that defines a b-metric d on X, which we call the b-metric associated with G.
Proposition 1.17 [47]
Let X be a -metric space. Then, for any and , if , then there exists such that .
From the above proposition, the family of all -balls
is a base of a topology on X, which we call the -metric topology.
Now, we generalize Proposition 5 in [34] for a -metric space as follows.
Proposition 1.18 [47]
Let X be a -metric space. Then, for any and , we have
Thus every -metric space is topologically equivalent to a b-metric space. This allows us to readily transport many concepts and results from b-metric spaces into -metric space setting.
Definition 1.19 [47]
Let X be a -metric space. A sequence in X is said to be:
-
(1)
-Cauchy if for each , there exists a positive integer such that, for all , ;
-
(2)
-convergent to a point if for each , there exists a positive integer such that, for all , .
Proposition 1.20 [47]
Let X be a -metric space. Then the following are equivalent:
-
(1)
the sequence is -Cauchy;
-
(2)
for any , there exists such that for all .
Proposition 1.21 [47]
Let X be a -metric space. The following are equivalent:
-
(1)
is -convergent to x;
-
(2)
as ;
-
(3)
as .
Definition 1.22 [47]
A -metric space X is called complete if every -Cauchy sequence is -convergent in X.
Definition 1.23 [47]
Let and be two -metric spaces. Then a function is -continuous at a point if and only if it is -sequentially continuous at x, that is, whenever is -convergent to x, is -convergent to .
Mustafa and Sims proved that each G-metric function is jointly continuous in all three of its variables (see Proposition 8 in [33]). But, in general, a -metric function for is not jointly continuous in all its variables. Now, we recall an example of a discontinuous -metric.
Example 1.24 [49]
Let and let be defined by
Then it is easy to see that for all , we have
Thus, is a b-metric space with (see corrected Example 3 in [9]).
Let . It is easy to see that G is a -metric with . In [49], it is proved that is not a continuous function.
So, from the above discussion, we need the following simple lemma about the -convergent sequences in the proof of our main result.
Lemma 1.25 [49]
Let be a -metric space with and suppose that , and are -convergent to x, y and z, respectively. Then we have
In particular, if , then we have .
In this paper, we present some tripled coincidence point results in ordered -metric spaces. Our results extend and generalize the results in [46].
2 Main results
Let be an ordered -metric space and and . In the rest of this paper, unless otherwise stated, for all , let
and
Now, the main result is presented as follows.
Theorem 2.1 Let be a partially ordered -metric space and and be such that . Assume that
for every with , and , or , and , where are altering distance functions.
Assume that
-
(1)
F has the mixed g-monotone property.
-
(2)
g is -continuous and commutes with F.
Also suppose that
-
(a)
either F is -continuous and is -complete, or
-
(b)
is regular and is -complete.
If there exist such that , and , then F and g have a tripled coincidence point in X.
Proof Let be such that , and . Define such that , and . Then , and . Similarly, define , and . Since F has the mixed g-monotone property, we have , and .
In this way, we construct the sequences , and as
and
for all .
We will finish the proof in two steps.
Step I. We shall show that , and are -Cauchy.
Let
So, we have
and
As , and , using (2.1) we obtain that
Since ψ is an altering distance function, by (2.2) we deduce that
that is, is a nonincreasing sequence of nonnegative real numbers. Thus, there is such that
Letting in (2.2), from the continuity of ψ and φ, we obtain that
which implies that and hence .
Next, we claim that , and are -Cauchy.
We shall show that for every , there exists such that if ,
Suppose that the above statement is false. Then there exists for which we can find subsequences and of , and of and and of such that and
where is the smallest index with this property, i.e.,
From (2.4), we have
From the rectangle inequality,
Similarly,
and
So,
Letting as , by (2.3) and (2.4), we can conclude that
Since
and
and
we obtain that
If in the above inequality as , from (2.3) we have
As , we have , and . Putting , , , , , , , and in (2.1), we have
Letting in (2.16),
From (2.17), we have
Therefore,
which is a contradiction to (2.10). Consequently, , and are -Cauchy.
Step II. We shall show that F and g have a tripled coincidence point.
First, let (a) hold, that is, F is -continuous and is -complete.
Since X is -complete and is -Cauchy, there exists such that
Similarly, there exist such that
and
Now, we prove that is a tripled coincidence point of F and g.
Continuity of g and Lemma 1.25 yields that
Hence,
and similarly,
and
Since , and , the commutativity of F and g yields that
and
From the continuity of F and (2.24), (2.25) and (2.26) and Lemma 1.25, is -convergent to , is -convergent to and is -convergent to . From (2.21), (2.22) and (2.23) and uniqueness of the limit, we have , and , that is, g and F have a tripled coincidence point.
In what follows, suppose that assumption (b) holds.
Following the proof of the previous step, there exist such that
and
as is -complete.
Now, we prove that , and . From regularity of X and using (2.1), we have
As is -convergent to gu, from Lemma 1.25, we have . Analogously, .
As ψ and φ are continuous, from (2.30) we have
or, equivalently,
Similarly,
On the other hand,
Taking the limit when and using (2.27) and (2.31), we get
that is, .
Analogously, we can show that and .
Thus, we have proved that g and F have a tripled coincidence point. This completes the proof of the theorem. □
Let
Taking (the identity mapping on X) in Theorem 2.1, we obtain the following tripled fixed point result.
Corollary 2.2 Let be a -complete partially ordered -metric space, and let be a mapping with the mixed monotone property. Assume that
for every with , and , or , and , where are altering distance functions.
Also suppose that
-
(a)
either F is -continuous, or
-
(b)
is regular.
If there exist such that , and , then F has a tripled fixed point in X.
Taking and for all in Corollary 2.2, we obtain the following tripled fixed point result.
Corollary 2.3 Let be a -complete partially ordered -metric space and with the mixed monotone property. Assume that
for every with , and , or , and .
Also suppose that
-
(a)
either F is -continuous, or
-
(b)
is regular.
If there exist such that , and , then F has a tripled fixed point in X.
Remark 2.4 Theorem 1.8 is a special case of Theorem 2.1.
Remark 2.5 Theorem 2.1 part (a) holds if we replace the commutativity assumption of F and g by compatibility assumption (also see Remark 2.2 of [30]).
The following corollary can be deduced from our previously obtained results.
Corollary 2.6 Let be a partially ordered set and be a -complete -metric space. Let be a mapping with the mixed monotone property such that
for every with , and , or , and .
Also suppose that
-
(a)
either F is -continuous, or
-
(b)
is regular.
If there exist such that , and , then F has a tripled fixed point in X.
Proof If F satisfies (2.37), then F satisfies (2.35). So, the result follows from Theorem 2.1. □
In Theorem 2.1, if we take and for all , where , we obtain the following result.
Corollary 2.7 Let be a partially ordered set and be a -complete -metric space. Let be a mapping having the mixed monotone property and
for every with , and , or , and .
Also suppose that
-
(a)
either F is -continuous, or
-
(b)
is regular.
If there exist such that , and , then F has a tripled fixed point in X.
Corollary 2.8 Let be a partially ordered set and be a -complete -metric space. Let be a mapping with the mixed monotone property such that
for every with , and , or , and .
Also suppose that
-
(a)
either F is -continuous, or
-
(b)
is regular.
If there exist such that , and , then F has a tripled fixed point in X.
Proof If F satisfies (2.39), then F satisfies (2.38). □
Note that if is a partially ordered set, then we can endow with the following partial order relation:
for all (see [26]).
In the following theorem, we give a sufficient condition for the uniqueness of the common tripled fixed point (also see, e.g., [4, 46, 50] and [51]).
Theorem 2.9 In addition to the hypotheses of Theorem 2.1, suppose that for every and , there exists such that is comparable with and . Then F and g have a unique common tripled fixed point.
Proof From Theorem 2.1 the set of tripled coincidence points of F and g is nonempty. We shall show that if and are tripled coincidence points, that is,
and
then and and .
Choose an element such that is comparable with
and
Let , and and choose , and so that , and . Then, similarly as in the proof of Theorem 2.1, we can inductively define sequences , and such that , and . Since and are comparable, we may assume that . Then , and . Using the mathematical induction, it is easy to prove that , and for all .
Applying (2.1), as , and , one obtains that
From the properties of ψ, we deduce that
is nonincreasing.
Hence, if we proceed as in Theorem 2.1, we can show that
that is, , and are -convergent to gx, gy and gz, respectively.
Similarly, we can show that
that is, , and are -convergent to , and , respectively. Finally, since the limit is unique, , and .
Since , and , by commutativity of F and g, we have , and . Let , and . Then , and . Thus, is another tripled coincidence point of F and g. Then , and . Therefore, is a tripled common fixed point of F and g.
To prove the uniqueness, assume that is another tripled common fixed point of F and g. Then , and . Since is a tripled coincidence point of F and g, we have , and . Thus, , and . Hence, the tripled common fixed point is unique. □
3 Examples
The following examples support our results.
Example 3.1 Let be endowed with the usual ordering and the -complete -metric
where .
Define as
for all and with for all .
Let be defined by , and let be defined by .
Now, from the fact that for , , we have
Analogously, we can show that
and
Thus,
Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover, is the unique common tripled fixed point of F and g.
The following example has been constructed according to Example 2.12 of [2].
Example 3.2 Let , where with the order ⪯ defined as
Let d be given as
and
where and . is, clearly, a -complete -metric space.
Let and be defined as follows:
and
Let be as in the above example.
According to the order on X and the definition of g, we see that for any element , is comparable only with itself.
By a careful computation, it is easy to see that all of the conditions of Theorem 2.1 are satisfied. Finally, Theorem 2.1 guarantees the existence of a unique common tripled fixed point for F and g, i.e., the point .
4 Applications
In this section, we obtain some tripled coincidence point theorems for a mapping satisfying a contractive condition of integral type in a complete ordered -metric space.
We denote by Λ the set of all functions verifying the following conditions:
-
(I)
μ is a positive Lebesgue integrable mapping on each compact subset of .
-
(II)
For all , .
Corollary 4.1 Replace the contractive condition (2.1) of Theorem 2.1 by the following condition:
There exists such that
If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.
Proof Consider the function . Then (4.1) becomes
Taking and and applying Theorem 2.1, we obtain the proof (it is easy to verify that and are altering distance functions). □
Corollary 4.2 Substitute the contractive condition (2.1) of Theorem 2.1 by the following condition:
There exists such that
If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.
Proof Again, as in Corollary 4.1, define the function . Then (4.2) changes to
Now, if we define and and apply Theorem 2.1, then the proof is completed. □
Corollary 4.3 Replace the contractive condition (2.1) of Theorem 2.1 by the following condition:
There exists such that
for altering distance functions , , and . If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.
Similar to [52], let be fixed. Let be a family of N functions which belong to Λ. For all , we define
We have the following result.
Corollary 4.4 Replace inequality (2.1) of Theorem 2.1 by the following condition:
If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.
Proof Consider and . Then the above inequality becomes
Applying Theorem 2.1, we obtain the desired result (it is easy to verify that and are altering distance functions). □
Another consequence of the main theorem is the following result.
Corollary 4.5 Substitute contractive condition (2.1) of Theorem 2.1 by the following condition:
There exist such that
If the other conditions of Theorem 2.1 are satisfied, then F and g have a tripled coincidence point.
Proof It is clear that the function for is an altering distance function. □
Motivated by [46], we study the existence of solutions for nonlinear integral equations using the results proved in the previous section.
Consider the integral equations in the following system.
We will consider system (4.5) under the following assumptions:
-
(i)
are continuous,
-
(ii)
is continuous,
-
(iii)
is continuous,
-
(iv)
there exists such that for all ,
and
-
(v)
We suppose that
-
(vi)
There exist continuous functions such that
and
We consider the space of continuous functions defined on endowed with the -metric given by
for all , where and (see Example 1.12).
We endow X with the partial ordered ⪯ given by
On the other hand, is regular [53].
Our result is the following.
Theorem 4.6 Under assumptions (i)-(vi), system (4.5) has a solution in where .
Proof As in [46], we consider the operators and defined by
and
for all and .
F has the mixed monotone property (see Theorem 25 of [46]).
Let be such that , and . Since F has the mixed monotone property, we have
On the other hand,
Now, for all from (iv) and the fact that for , , we have
Thus,
Repeating this idea and using the definition of the -metric G, we obtain
and
So, from (4.6), (4.7) and (4.8), we have
Similarly, we can obtain
and
Now, from (4.9), (4.10) and (4.11), we have
But from (v), we have
This proves that the operator F satisfies the contractive condition appearing in Corollary 2.7.
Let α, β, γ be the functions appearing in assumption (vi), then by (vi), we get
Applying Corollary 2.7, we deduce the existence of such that , and . □
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Mustafa, Z., Roshan, J.R. & Parvaneh, V. Existence of a tripled coincidence point in ordered -metric spaces and applications to a system of integral equations. J Inequal Appl 2013, 453 (2013). https://doi.org/10.1186/1029-242X-2013-453
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DOI: https://doi.org/10.1186/1029-242X-2013-453