Remarks on ‘Coupled coincidence point results for a generalized compatible pair with applications’
© Erhan et al.; licensee Springer. 2014
Received: 19 May 2014
Accepted: 6 August 2014
Published: 13 October 2014
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.
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. , Theorem 15)
F is continuous, or
X has the following property:
if a ⪯-non-decreasing sequence , then for all ,
if a ⪯-non-increasing sequence , then for all .
Then F and G have a coupled coincidence point in X.
The mixed monotone property is not necessary since F is G-increasing with respect to ⪯.
It is possible to consider a pair of mappings satisfying a weaker condition than the generalized compatible property (using monotone sequences).
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.
Firstly, we recall some basic definitions and elementary results needed throughout the paper. Some of them can be found in . 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. )
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 , 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 .
for all ;
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. , Definition 10)
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.
Definition 2.7 (Hussain et al. , 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 )
Then is metric on and is complete if and only if is complete.
Definition 2.9 (Hussain et al. , Definition 12)
3 Main results
To start with, we highlight the weakness of Theorem 1.1 using the following example.
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).
T is -non-decreasing;
there exists such that ;
- (iv)there exist and verifying
is complete, T and g are continuous and the pair is O-compatible;
is complete and T and g are continuous and commuting;
is complete and is non-decreasing-regular;
is complete, is closed and is non-decreasing-regular;
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  and it can be concluded by following, point by point, all of its arguments.
Under these conditions, the following properties hold.
is complete if and only if is complete.
If is regular, then is also regular.
If F is d-continuous, then is -continuous.
If F is G-increasing with respect to ⪯, then is -non-decreasing.
Condition (1) is equivalent to the existence of a point such that .
Condition (3) is equivalent to .
- (7)If there exist and such that (2) holds, then
If the pair is generalized compatible, then the mappings and are O-compatible in .
A point is a coupled coincidence point of F and G if and only if it is a coincidence point of and .
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)
- (8)Let be any sequence such that and (notice that we do not need to suppose that is ⊑-monotone). Therefore,
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 .
F is continuous, or
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.
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.
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 . Notice also that the assumption of sub-additivity () is superfluous in most of the published results (see, e.g., ).
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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