A note on ‘n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’
© Karapınar and Roldán; licensee Springer. 2013
Received: 23 August 2013
Accepted: 4 November 2013
Published: 27 November 2013
In this note, we show that multidimensional fixed point theorems established in the recent report [M. Ertürk and V. Karakaya, n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces, Journal of Inequalities and Applications 2013, 2013:196] have gaps. Furthermore, the results of the mentioned paper can be reduced to unidimensional (existing) fixed point theorems.
Keywordsmultidimensional fixed point partially ordered metric space
1 Introduction and preliminaries
Throughout this manuscript, X will be a non-empty set and ⪯ will denote a partial order on X. Given with , let us denote by the product space of n identical copies of X.
The study of multidimensional fixed point theorems was initiated by Guo and Lakshmikantham in  in the coupled case.
Definition 1.1 (Guo and Lakshmikantham )
In 2006, Bhaskar and Lakshmikantham  proved some coupled fixed point theorems for a mapping (where X is a partially ordered metric space) by introducing the notion of mixed monotone mapping.
Definition 1.2 (See )
Following this paper, Lakshmikantham and Ćirić  established coupled fixed/coincidence point theorems for mappings and by defining the concept of the mixed g-monotone property. Later, Berinde and Borcut studied the tripled case.
Definition 1.3 (Berinde and Borcut )
Definition 1.4 (See )
Karapınar and Luong studied the quadruple case.
Definition 1.6 (See )
When a mapping is involved, we have the notion of coincidence point. We will only recall the corresponding definitions in the quadruple case since they are similar in other dimensions.
Definition 1.7 (See )
Definition 1.8 (See )
It is very natural to extend the definition of two-dimensional fixed point (coupled fixed point), three-dimensional fixed point (tripled fixed point) and so on to multidimensional fixed point (n-tuple fixed point) (see, e.g., [9–17]). In this paper, we give some remarks on the notion of n-tuple fixed point given by Ertürk and Karakaya in [18, 19]. Notice that the authors preferred to say ‘n-tuplet fixed point’ instead of ‘n-tuple fixed point’.
Definition 1.9 (See )
Definition 1.10 (See )
Definition 1.11 (See )
Definition 1.12 (See )
2 Some remarks
do not extend the notion of tripled coincidence point by Berinde and Borcut . Therefore, their results are not extensions of the well-known results in the tripled case. This fact shows that the odd case is not well posed by Definitions 1.9 and 1.11 or, more precisely, the mixed monotone property is not useful to ensure the existence of coincidence points. In this sense, we have the following result.
Theorem 2.1 Theorem 1 in is not valid if n is odd.
Since other possibilities yield similar incomparable cases, we cannot get the inequality . □
For completeness and to conclude this paper, instead of Definitions 1.9 and 1.11, we recall here the concept of multidimensional fixed/coincidence point introduced by Roldán et al. in  (see also [12–14]), which is an extension of Berzig and Samet’s notion given in .
Let be n mappings from into itself, and let Φ be the n-tuple .
Let and be two mappings.
Definition 2.2 
If g is the identity mapping on X, then is called a Φ-fixed point of the mapping F.
Definition 2.3 
Notice that, in fact, when n is even, Definitions 1.11 and 1.12 can be seen as particular cases of the previous definitions, when A is the set of all odd numbers and B is the family of all even numbers in , and the mappings are appropriate permutations of the variables.
Finally, to be fair, we remark that most of multidimensional fixed point theorems can be reduced to one-dimensional (usual) fixed point results (see, e.g., [14, 20]). More precisely, for instance in , the authors proved that the first coupled fixed point result, Theorem 2.1 in , is a consequence of Theorem 2.1 in . In , the authors proved that the initial multidimensional fixed point result, Theorem 9 in , can be derived from Theorem 2.1 in  either.
The first author was supported by the Research Center, College of Science, King Saud University. The second author was supported by Junta the Andalucía through Projects FQM-268 of the Andalusian CICYE.
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