A note on ‘Nfixed point theorems for nonlinear contractions in partially ordered metric spaces’
 Erdal Karapınar^{1}Email author,
 Antonio Roldán^{1},
 Concepción Roldán^{1} and
 Juan MartínezMoreno^{1}
https://doi.org/10.1186/168718122013310
© Karapınar et al.; licensee Springer. 2013
Received: 23 August 2013
Accepted: 29 October 2013
Published: 22 November 2013
Abstract
In this note we prove that a kind of mappings depending on k arguments introduced in (Paknazar et al. in Fixed Point Theory Appl. 2013:111, 2013) only depend on their first argument. Therefore, results in that paper reduce to the unidimensional case. We also include some commentaries about the different notions of multidimensional fixed point.
MSC:46T99, 47H10, 47H09, 54H25.
Keywords
partially ordered set fixed point contractive mapping mixed monotone propertyRecently, Paknazar et al. [1] introduced the concept of new gmonotone property for a mapping $F:{X}^{k}\to X$ as follows.
Definition 1 (Paknazar et al. [1], Definition 2.2)
We prove that there is not a wide range of mappings verifying this condition.
Therefore, F only depends on its first variable and all results in [1] can be reduced to the unidimensional case.
which means that $F({x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{k})=F({x}_{1},{z}_{2},{z}_{3},\dots ,{z}_{k})=f({x}_{1})$.
that is, f is a gnondecreasing mapping. The unicity of f is obvious. □
The main result in Paknazar et al. [1] is the following theorem.
Theorem 3 (Paknazar et al. [1], Theorem 2.5)
 (i)
$\phi (t)<t$ for $t>0$ and $\phi (0)=0$,
 (ii)
${lim}_{r\to {t}^{+}}\phi (r)<t$ for each $t>0$,
 (a)
F is continuous, or
 (b)X has the following property:
 (i)
If a nondecreasing sequence $\{{x}_{n}\}\to x$, then ${x}_{n}\preccurlyeq x$ for all n;
 (ii)
If a nonincreasing sequence $\{{y}_{n}\}\to y$, then ${y}_{n}\succcurlyeq y$ for all n.
Then there exist ${x}_{1},{x}_{2},\dots ,{x}_{k}\in X$ such that$g({x}_{i})=F({x}_{i},{x}_{i+1},\dots ,{x}_{k},{x}_{1},{x}_{2},\dots ,{x}_{i1})\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}i\in \{1,2,\dots k\}.$  (i)
That is, F and g have a kcoincidence point.
Firstly, notice that if φ is continuous and $\phi (t)<t$ for all $t>0$, then $\phi (0)=0$, so the condition $\phi (0)=0$ in (i) can be avoided. But mainly, we claim that this theorem is an immediate consequence of the following unidimensional result (a twodimensional version of the following result can be found in [2], a multidimensional version is in [3]), which can be seen as a natural extension of Ran and Reurings’ theorem [4] and Nieto and RodríguezLópez’s theorem [5].
Also assume that T is continuous or $(X,d,\preccurlyeq )$ is regular (that is, it verifies condition (b) of the previous theorem). If there exists a point ${x}_{0}\in X$ such that $g({x}_{0})\preccurlyeq T({x}_{0})$, then T and g have a coincidence point, that is, a point $x\in X$ such that $g(x)=f(x)$.
When f is continuous, then f is continuous; since F and g commute, then f and g commute; since $F({X}^{k})\subset g(X)$, then $f(X)\subset g(X)$; $g({x}_{1}^{0})\preccurlyeq F({x}_{1}^{0},{x}_{2}^{0},\dots ,{x}_{k}^{0})=f({x}_{1}^{0})$ and similarly $g({x}_{2}^{0})\succcurlyeq f({x}_{1}^{0})$; furthermore, the contractivity condition is similarly proved. Classical techniques assure that f and g have a coincidence point, that is, there is $x\in X$ such that $g(x)=f(x)$. Therefore $(x,x,\dots ,x)\in {X}^{k}$ is a kcoincidence point between F and g, that is, $g(x)=f(x)=F(x,x,\dots ,x)$.
some authors have paid attention to the multidimensional case. A first attempt to generalize this notion was given by Berzig and Samet in [10]. We have to distinguish between two kinds of definitions.

In some cases, one or more arguments do not have to appear in all equations. For instance, the following notion was given in [11] (and was also mentioned in Paknazar et al., Definition 1.12, although it was not used in that paper):${x}_{i}=F({x}_{i},{x}_{i1},\dots ,{x}_{2},{x}_{1},{x}_{2},\dots ,{x}_{ki+1})\phantom{\rule{1em}{0ex}}\text{for all}i\in \{1,2,\dots ,k\}.$This definition can be interpreted as an extension of the second equation of Berinde and Borcut’s notion, that is, $y=F(y,x,y)$. This case yields the equations systems in which some arguments do not appear. Besides Berinde and Borcut’s second equation in the tripled case, if $k=4$, the corresponding system is$\{\begin{array}{l}x=F(x,y,z,t),\\ y=F(y,x,y,z)\phantom{\rule{1em}{0ex}}(\text{no data for}t),\\ z=F(z,y,x,y)\phantom{\rule{1em}{0ex}}(\text{no data for}t),\\ t=F(t,z,y,x).\end{array}$
This could be a difficulty in proving some results and, furthermore, this case is not possible when a researcher is interested in a system whose equations involve, at the same time, all variables.

In other cases, the arguments are permuted. For instance, the notion of kfixed point introduced in Paknazar et al. (Definition 2.1) is as follows:${x}_{i}=F({x}_{i},{x}_{i1},\dots ,{x}_{n1},{x}_{n},{x}_{1},{x}_{2},\dots ,{x}_{i1})\phantom{\rule{1em}{0ex}}\text{for all}i\in \{1,2,\dots ,k\}$(2)(for simplicity, we do not consider the coincidence case involving a mapping g). This notion generalizes Karapınar and Luong’s quadrupled concept, that is,$\{\begin{array}{l}x=F(x,y,z,t),\\ y=F(y,z,t,x),\\ z=F(z,t,x,y),\\ t=F(t,x,y,z).\end{array}$However, we remark that the equation system (2) is not suitable to work with the classical mixed monotone property when k is odd. For instance, if $k=5$ and F is monotone nondecreasing in its odd arguments and monotone nonincreasing in its even arguments, then the equations${x}_{1}=F({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5})\phantom{\rule{1em}{0ex}}({x}_{1}\text{and}{x}_{5}\text{are placed in nondecreasing arguments of}F)$and$\begin{array}{c}{x}_{2}=F({x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{1})\phantom{\rule{1em}{0ex}}({x}_{1}\text{and}{x}_{5}\text{are placed in arguments}\hfill \\ \phantom{{x}_{2}=F({x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{1})\phantom{\rule{1em}{0ex}}(}\text{of different monotone type of}F)\hfill \end{array}$
do not let us to show the existence of fixed points using the classical mixed monotone property. Berinde and Borcut realized that this definition is not convenient, so they had to consider the equation $y=F(y,x,y)$. System (2) only works when k is even. Obviously, Paknazar et al. succeeded in proving their main results in [1] because, as we have showed before, the new gmixed monotone property is very restrictive.
Finally, we comment that a more convenient notion of kfixed point was given by Roldán et al. in [12]. This definition has three advantages: (1) it describes how the arguments can be reordered or permuted, (2) it works with the classical mixed monotone property, and (3) it extends Guo and Lakshmikantham’s coupled case, Berinde’s tripled case, Karapınar and Luong’s quadrupled case and Berzig and Samet’s multidimensional case. As we can easily see, the main result in [11] (that the reader can find in [[1], Theorem 1.13] as a literature) is weaker than the main result of Roldán et al. [12]. For more publications in this direction, see, e.g., [10, 13–17]. We also notice that Roldán et al. [18] proved that some multidimensional fixed point results can be reduced to the unidimensional case.
Declarations
Acknowledgements
The first author was supported by the Research Center, College of Science, King Saud University. The last three authors were supported by Junta the Andalucía through Projects FQM268, FQM235 and FQM178 of the Andalusian CICYE.
Authors’ Affiliations
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