Abstract
Very recently, Hussain et al. (Fixed Point Theory Appl. 2014:62, 2014) announced the existence and uniqueness of some coupled coincidence point. In this short note we remark that the announced results can be derived from the coincidence point results in the literature.
MSC: 47H10, 54H25.
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1 Introduction
Recently, a number of studies related to fixed points, coupled fixed points and coupled coincidence points of maps defined via auxiliary functions have appeared in the literature. In particular, the so-called weak φ-contractions, contractions defined by means of altering distance functions, -type contractions have been a subject of considerable interest. Studies of this type aim to generalize and improve contractive condition on the maps (see, e.g., [1–15]).
A great deal of these studies investigate contractions on partially ordered metric spaces because of their applicability to initial value problems defined by differential or integral equations. This is the case of the following result.
Theorem 1.1 (Hussain et al. [16], Theorem 15)
Let be a partially ordered set such that there exists a complete metric d on X. Assume that are two generalized compatible mappings such that F is G-increasing with respect to ⪯, G is continuous and has the mixed monotone property, and there exist two elements such that
Suppose that there exist and such that
for all with and . Suppose that for any , there exist such that
Also suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if a ⪯-non-decreasing sequence , then for all ,
-
(ii)
if a ⪯-non-increasing sequence , then for all .
Then F and G have a coupled coincidence point in X.
In this paper we show that the previous result can be easily improved because of the following facts.
-
(1)
The mixed monotone property is not necessary since F is G-increasing with respect to ⪯.
-
(2)
It is possible to consider a pair of mappings satisfying a weaker condition than the generalized compatible property (using monotone sequences).
-
(3)
In fact, Theorem 1.1 is not a true advance because it can be reduced to its corresponding unidimensional coincidence point theorem.
To prove our main claims, we will show a unidimensional proof of the mentioned theorem.
2 Preliminaries
Firstly, we recall some basic definitions and elementary results needed throughout the paper. Some of them can be found in [17]. In the sequel, we denote by X a nonempty set. Given a natural number , let be the nth Cartesian product (n times). We employ mappings and . For simplicity, if , we denote by Tx.
Definition 2.1 (Khan et al. [18])
An altering distance function is a continuous, non-decreasing function such that if and only if . Let denote the family of all altering distance functions.
A function is said to be subadditive if for all . Following [16], we introduce the following families of control functions. Let Φ denote the family of all subadditive altering distance functions, that is, functions which satisfy the following:
() ϕ is continuous and non-decreasing;
() if and only if ;
() for all .
We denote by Ψ the family of all functions which satisfy the following:
-
(1)
for all ;
-
(2)
.
Remark 2.1 Let , and define by for all . Then .
A coincidence point of two mappings is a point such that .
Definition 2.3 (Hussain et al. [16], Definition 10)
A coupled coincidence point of two mappings is a point such that
Definition 2.4 An ordered metric space is a metric space provided with a partial order ⪯.
An ordered metric space is said to be non-decreasing-regular (respectively, non-increasing-regular) if for every sequence such that and (respectively, ) for all m, we have that (respectively, ) for all m. is said to be regular if it is both non-decreasing-regular and non-increasing-regular.
Remark 2.2 Notice that condition (b) in Theorem 1.1 means that is regular.
Definition 2.6 Let be a partially ordered set, and let be two mappings. We say that T is -non-decreasing if for all such that . If g is the identity mapping on X, we say that T is ⪯-non-decreasing.
Remark 2.3 If T is -non-decreasing and , then . It follows that
Definition 2.7 (Hussain et al. [16], Definition 7)
Suppose that are two mappings, and let ⪯ be a partial order on X. The mapping F is said to be G-increasing with respect to ⪯ if for all with we have .
Lemma 2.1 (see [22])
Let be a metric space and define , for all , by
Then is metric on and is complete if and only if is complete.
Consider on the product space the following partial order: for ,
Let be an ordered metric space. Two mappings are said to be O-compatible if
provided that is a sequence in X such that is ⪯-monotone, that is, it is either non-increasing or non-decreasing with respect to ⪯, and
Definition 2.9 (Hussain et al. [16], Definition 12)
Let . We say that the pair is generalized compatible if for all sequences such that
we have that
3 Main results
To start with, we highlight the weakness of Theorem 1.1 using the following example.
Example 3.1 Let endowed with the standard metric for all . Consider the maps defined by
Then, for all with , we have
Thus,
Regarding the properties of the functions in Φ, we derive that
Since the function in the class Ψ takes values on , it is impossible to verify inequality (2). Hence, Theorem 1.1 cannot be applied to get a coupled coincidence point. However, it is easy to see that is a coupled coincidence point of F and G.
Next, we show a unidimensional version of Theorem 1.1. Notice that, indeed, the following result is better than Theorem 1.1 because we reorder the hypotheses obtaining that, in some cases, neither the continuity of, at least, one mapping (T or g) nor the O-compatibility of the pair is necessary. In fact, both hypotheses are omitted in case (c).
Theorem 3.1 Let be an ordered metric space, and let be two mappings such that the following properties are fulfilled:
-
(i)
;
-
(ii)
T is -non-decreasing;
-
(iii)
there exists such that ;
-
(iv)
there exist and verifying
Also assume that, at least, one of the following conditions holds.
-
(a)
is complete, T and g are continuous and the pair is O-compatible;
-
(b)
is complete and T and g are continuous and commuting;
-
(c)
is complete and is non-decreasing-regular;
-
(d)
is complete, is closed and is non-decreasing-regular;
-
(e)
is complete, g is continuous and monotone ⪯-non-decreasing, the pair is O-compatible and is non-decreasing-regular.
Then T and g have, at least, a coincidence point.
We omit the proof of the previous result since its proof is similar to the main theorem in [17] and it can be concluded by following, point by point, all of its arguments.
Next, we show how to deduce an appropriate version of Theorem 1.1 from Theorem 3.1. Given the ordered metric space , let us consider the ordered metric space , where was defined in Lemma 2.1 and ⊑ was introduced in (4). We define the mappings , for all , by
Under these conditions, the following properties hold.
Lemma 3.1 Let be an ordered metric space, and let be two mappings. Then the following properties hold.
-
(1)
is complete if and only if is complete.
-
(2)
If is regular, then is also regular.
-
(3)
If F is d-continuous, then is -continuous.
-
(4)
If F is G-increasing with respect to ⪯, then is -non-decreasing.
-
(5)
Condition (1) is equivalent to the existence of a point such that .
-
(6)
Condition (3) is equivalent to .
-
(7)
If there exist and such that (2) holds, then
for all such that , where was defined in Remark 2.1.
-
(8)
If the pair is generalized compatible, then the mappings and are O-compatible in .
-
(9)
A point is a coupled coincidence point of F and G if and only if it is a coincidence point of and .
Proof Item (1) follows from Lemma 2.1 and items (2), (3), (5), (6) and (9) are obvious.
-
(4)
Assume that F is G-increasing with respect to ⪯, and let be such that . Then and . Since F is G-increasing with respect to ⪯, we deduce that and . Therefore, and this means that is -non-decreasing.
-
(7)
Suppose that there exist and such that (2) holds, and let be such that . Therefore and . Using (2), we have that
(5)
Furthermore, taking into account that and , the contractivity condition (2) also guarantees that
Since ϕ is subadditive, it follows from (5) and (6) that
-
(8)
Let be any sequence such that and (notice that we do not need to suppose that is ⊑-monotone). Therefore,
Therefore
Since the pair is generalized compatible, we deduce that
In particular,
Hence, the mappings and are O-compatible in . □
As a consequence, we conclude that Hussain et al.’s result can be deduced from the corresponding unidimensional result. Furthermore, as we have pointed out, it is not necessary for G to have the mixed monotone property because F is G-increasing with respect to ⪯.
Corollary 3.1 Theorem 1.1, even avoiding the assumption that G has the mixed monotone property, is a consequence of Theorem 3.1.
Proof It is only necessary to apply Theorem 3.1 to the mappings and in the ordered metric space , taking into account all items of Lemma 3.1. □
The following result is an improved version of Theorem 1.1 in which the contractivity condition is replaced by a more convenient one, which is symmetric on the variables and .
Corollary 3.2 Let be a partially ordered set such that there exists a complete metric d on X. Assume that are two generalized compatible mappings such that F is G-increasing with respect to ⪯, G is continuous and there exist two elements such that
Suppose that there exist and such that
for all with and . Suppose that for any , there exist such that
Also assume that either
-
(a)
F is continuous, or
-
(b)
is regular.
Then F and G have, at least, a coupled coincidence point, that is, there exist such that and .
Proof It is only necessary to apply Theorem 3.1 to the mappings and in the ordered metric space , where , taking into account all items of Lemma 3.1. □
In the following example we show that Corollary 3.2 is applicable to the mappings of Example 3.1, when Theorem 1.1 is not useful.
Example 3.2 Let endowed with the Euclidean metric for all . Consider the maps defined by
Then, for all with , we have
Thus,
Regarding the properties of the functions in Φ, we derive that
To provide inequality (7), it is sufficient to choose . Hence, Theorem 3.2 can be applied in order to guarantee that F and G have a coupled coincidence point. Indeed, it is easy to check that is a coupled coincidence point of F and G.
To finish the paper, we want to point out a pair of details.
-
(1)
The function in ϕ in Theorem 3.1 is not a true generalization because if , then the mapping , defined by for all , is also a metric on X. For more details, see [26]. Notice also that the assumption of sub-additivity () is superfluous in most of the published results (see, e.g., [27]).
-
(2)
Using the same techniques that can be found in [17, 22, 28–31], it is possible to deduce, from Theorem 3.1, tripled, quadrupled and, in general, multidimensional coincidence point theorems.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support. Antonio Roldán has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE.
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Erhan, İ.M., Karapınar, E., Roldán-López-de-Hierro, AF. et al. Remarks on ‘Coupled coincidence point results for a generalized compatible pair with applications’. Fixed Point Theory Appl 2014, 207 (2014). https://doi.org/10.1186/1687-1812-2014-207
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DOI: https://doi.org/10.1186/1687-1812-2014-207