- Research
- Open Access

# Some remarks on ‘Multidimensional fixed point theorems for isotone mappings in partially ordered metric spaces’

- Ravi P Agarwal
^{1, 2}, - Erdal Karapınar
^{3, 4}and - Antonio-Francisco Roldán-López-de-Hierro
^{5}Email author

**2014**:245

https://doi.org/10.1186/1687-1812-2014-245

© Agarwal et al.; licensee Springer. 2014

**Received:**14 August 2014**Accepted:**4 December 2014**Published:**22 December 2014

## Abstract

The main aim of this paper is to advise researchers in the field of Fixed Point Theory against an extended mistake that can be found in some proofs. We illustrate our claim proving that theorems in the very recent paper (Wang in Fixed Point Theory Appl. 2014:137, 2014) are incorrect, and we provide different corrected versions of them.

## Keywords

- fixed point
- contractivity condition
- isotone mapping
- mixed monotone mapping

## 1 Introduction and preliminaries

*k*be a positive integer number, and let ${X}^{k}=X\times X\times \stackrel{(k)}{\cdots}\times X$ be the Cartesian product of

*k*identical copies of

*X*. The function ${\rho}_{k}:{X}^{k}\times {X}^{k}\to [0,\mathrm{\infty})$ defined, for all $({y}_{1},{y}_{2},\dots ,{y}_{k}),({v}_{1},{v}_{2},\dots ,{v}_{k})\in {X}^{k}$, by

is a metric on ${X}^{k}$.

Let Φ denote the set of all continuous and strictly increasing functions $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$, and Ψ denote the set of all functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that ${lim}_{t\to r}\psi (t)>0$, for all $r>0$.

Inspired by Roldán *et al.*’s notion of a multidimensional fixed point (see [1–3]), Wang announced the following results in [4].

**Theorem 1** (Wang [[4], Theorem 3.1])

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:{X}^{k}\to {X}^{k}$

*be an isotone mapping for which there exist*$\phi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*,

*for all*$Y,V\in {X}^{k}$

*with*$Y\u2ab0V$,

*where*${\rho}_{k}$

*is defined by*(1).

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${Z}_{0}\in {X}^{k}$ *such that* ${Z}_{0}\asymp T({Z}_{0})$, *then* *T* *has a fixed point*.

As a consequence, she deduced the following result.

**Theorem 2** (Wang [[4], Corollary 3.2])

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exist*$\phi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*,

*for all*$y,v\in X$

*with*$y\u2ab0v$,

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

**Remark 3** Theorems 1 and 2 are equivalent: Wang interpreted Theorem 2 as a corollary of Theorem 1 (taking $k=1$), but is also true that Theorem 1 is a particularization of Theorem 2 taking $({X}^{k},{\rho}_{k},\u2aaf)$ rather than $(X,d,\u2aaf)$.

We claim that Theorem 2 is false, providing the following counterexample. As a consequence, all results in [4] are not correct.

## 2 A counterexample

*T*is a continuous, nondecreasing mapping. Moreover, it verifies condition (2) because if $y,v\in X=\mathbb{N}$ are natural numbers, then $d(y,v)$ is also a nonnegative integer number, so $\psi (d(y,v))=0$, and this proves that

(it is not necessary to assume that $y\u2ab0v$). Any point ${z}_{0}\in X$ verifies ${z}_{0}\u2aafT({z}_{0})$. However, *T* does not have any fixed point.

**Remark 4** Notice that Wang’s results are valid if we add the assumption $\psi (t)=0\iff t=0$ but, in this case, the results are not as attractive.

## 3 Some considerations of Wang’s paper

We write the following in order to advise researchers against the mistake in Wang’s proofs, which could be found on other papers.

*φ*is strictly increasing, we have $d({x}_{n+1},{x}_{n+2})\le d({x}_{n},{x}_{n+1})$, for all $n\in \mathbb{N}$, which means that ${\{d({x}_{n},{x}_{n+1})\}}_{n\in \mathbb{N}}$ is a nonincreasing sequence of nonnegative real numbers. As a consequence, it is convergent. Let $L\ge 0$ be its limit. In order to prove that $L=0$, we reason by contradiction assuming that $L>0$. Taking into account that

*φ*is continuous and ${lim}_{t\to L}\psi (t)>0$ by hypothesis, the author tried to deduce that

can be false: as in the previous counterexample, the sequence ${\{\psi (d({x}_{n},{x}_{n+1}))\}}_{n\in \mathbb{N}}$ can be identically zero but the limit ${lim}_{t\to L}\psi (t)$ must take a positive value.

## 4 A suggestion to correct Wang’s paper

In this section, inspired by Wang’s paper [4], we suggest a new theorem in the context of partially ordered metric spaces. We also underline that one can easily state the same theorem in the frame of multidimensional fixed points. In fact, they are equivalent as is mentioned in Section 1.

Firstly, we give a modified version of the collection of auxiliary functions Ψ in the following way. Let ${\mathrm{\Psi}}^{\prime}$ denote the set of all lower semi-continuous functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that $\psi (t)=0\iff t=0$. The following two lemmas will be useful in the proofs of the upcoming theorems.

**Lemma 5** (See, *e.g.*, [5])

*Let*$(X,d)$

*be a metric space and let*$\{{x}_{n}\}$

*be a sequence in*

*X*

*such that*

*If*$\{{x}_{n}\}$

*is not a Cauchy sequence in*$(X,d)$,

*then there exist*$\epsilon >0$

*and two sequences*$\{n(k)\}$

*and*$\{m(k)\}$

*of positive integers such that*,

*for all*$k\in \mathbb{N}$,

*Furthermore*,

**Lemma 6**

*Let*$\phi \in \mathrm{\Phi}$, $\psi \in {\mathrm{\Psi}}^{\prime}$,

*and let*$\{{t}_{n}\},\{{s}_{n}\},\{{a}_{n}\}\subset [0,\mathrm{\infty})$

*be three sequences of nonnegative real numbers such that*

*If there exists* $\alpha \in [0,\mathrm{\infty})$ *such that* $\{{t}_{n}\}\to \alpha $, $\{{s}_{n}\}\to \alpha $, $\{{a}_{n}\}\to 0$, *and* ${s}_{n}>\alpha $, *for all* $n\in \mathbb{N}$, *then* $\alpha =0$.

*Proof*Notice that

*ψ*is continuous, we deduce that

*ψ*is lower semi-continuous, we deduce that

Therefore, $\psi (\alpha )=0$. As $\psi \in {\mathrm{\Psi}}^{\prime}$, we conclude that $\alpha =0$. □

**Theorem 7**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exist*$\phi \in \mathrm{\Phi}$, $\psi \in {\mathrm{\Psi}}^{\prime}$

*and*$L\ge 0$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$,

*where*

*and*

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

*Proof*By assumption, there exists ${z}_{0}\in X$ such that ${z}_{0}\asymp T({z}_{0})$. Without loss of generality, we assume that ${z}_{0}\u2aafT{z}_{0}$ (the case ${z}_{0}\u2ab0T{z}_{0}$ can be treated analogously). Define an iterative sequence $\{{z}_{n}\}$ as ${z}_{n+1}=T{z}_{n}$ for each $n\in \{0,1,2,\dots \}$. If ${z}_{{n}_{0}}=T{z}_{{n}_{0}}$ for some ${n}_{0}$, then we conclude the assertion of the theorem. So, assume that ${z}_{n+1}\ne {z}_{n}$ for each $n\in \{0,1,2,\dots \}$. Since

*T*is nondecreasing, ${z}_{0}\u2aafT{z}_{0}={z}_{1}$ implies that ${z}_{1}=T{z}_{0}\u2aafT{z}_{1}={z}_{2}$. Recursively, we derive that

*n*such that ${M}_{d}({z}_{n},{z}_{n+1})=d({z}_{n+1},{z}_{n+2})$, then (4) turns into

*φ*is a strictly increasing function, we have, for all $n\in \mathbb{N}$,

*φ*,

*ϕ*, we derive that

If *T* is continuous, it is clear that *z* is a fixed point of *T*.

*z*, we have ${z}_{n}\u2aafz$, for all $n=\{0,1,2,\dots \}$. Due to (3), we have, for all $n\in \mathbb{N}$,

so $\psi (d(z,Tz))=0$ and $d(z,Tz)=0$. As a consequence, $Tz=z$, which completes the proof. □

The following corollary follows from Theorem 7 using $L=0$.

**Corollary 8**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exist*$\phi \in \mathrm{\Phi}$, $\psi \in {\mathrm{\Psi}}^{\prime}$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$,

*where*

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

In the next result, we take *φ* as the identity mapping on $[0,\mathrm{\infty})$, which clearly belongs to Φ.

**Corollary 9**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exists*$\psi \in {\mathrm{\Psi}}^{\prime}$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$,

*where*

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

The following result is not a direct consequence of Theorem 7, but its proof is verbatim the proof of Theorem 7. Thus, we skip it.

**Theorem 10**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exist*$\phi \in \mathrm{\Phi}$, $\psi \in {\mathrm{\Psi}}^{\prime}$,

*and*$L\ge 0$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$,

*where*

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

If $L=0$ in the previous theorem, we obtain the following result.

**Corollary 11**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exist*$\phi \in \mathrm{\Phi}$, $\psi \in {\mathrm{\Psi}}^{\prime}$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

If we take *φ* as the identity mapping on $[0,\mathrm{\infty})$, we derive the following result.

**Corollary 12**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exists*$\psi \in {\mathrm{\Psi}}^{\prime}$

*such that*

*for all*$x,y\in X$

*with*$x\u2aafy$.

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

In the sequel we suggest another theorem by changing the properties of the auxiliary functions in the following way.

- 1.
$\psi (t)=0\iff t=0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\psi ({t}_{n})>0$ whenever ${lim}_{n\to \mathrm{\infty}}{t}_{n}=t>0$.

- 2.
$\psi (t)>\phi (t)-\phi (t-)$ for any $t>0$, where $\phi (t-)$ is the left limit of

*φ*at*t*, where $\phi \in \mathrm{\Phi}$.

**Theorem 13**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to X$

*be a nondecreasing mapping for which there exist*$\phi \in \mathrm{\Phi}$

*and*$\psi \in {\mathrm{\Psi}}^{\u2033}$, $L\ge 0$

*such that*

*for all*$y,v\in X$

*with*$y\u2aafv$,

*where*

*and*

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${z}_{0}\in X$ *such that* ${z}_{0}\asymp T({z}_{0})$, *then* *T* *has a fixed point*.

The proof is analog to the proof of Theorem 7 and, hence, we skip it.

**Remark 14** As in Theorem 13, by changing the property of the auxiliary function, we get various results (see, *e.g.*, [6–9] and related references therein).

## 5 Fixed point theorems from (one dimensional) fixed point to multidimensional fixed point

As discussed in Remark 3, multidimensional fixed point theorems are equivalent to (one dimensional) fixed point theorems. Thus, Theorem 7 can be translated in a multidimensional case as follows.

**Theorem 15**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there is a metric*

*d*

*on*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:{X}^{k}\to {X}^{k}$

*be a nondecreasing mapping for which there exist*$\phi \in \mathrm{\Phi}$, $\psi \in {\mathrm{\Psi}}^{\prime}$,

*and*$L\ge 0$

*such that*

*for all*$V,Y\in X$

*with*$V\u2aafY$,

*where*

*and*

*Suppose either*

- (a)
*T**is continuous or* - (b)
$(X,d,\u2aaf)$

*is regular*.

*If there exists* ${Z}_{0}\in {X}^{k}$ *such that* ${Z}_{0}\asymp T({Z}_{0})$, *then* *T* *has a fixed point*.

In similar way, we may state the analog of Corollary 8, Corollary 9, Theorem 10, Corollary 11, Corollary 12.

## 6 Applications

Due to [11], we know that $(C[0,T],d,\u2aaf)$ is regular.

In what follows we state the main result of this section.

**Theorem 16**

*We assume that the following hypotheses hold*:

- (i)
$K:[0,T]\times [0,T]\times \mathbb{R}\to \mathbb{R}$

*and*$g:\mathbb{R}\to \mathbb{R}$*are continuous*, - (ii)
*for all*$s,t\in [0,T]$*and*$u,v\in C[0,T]$*with*$v\u2aafu$,*we have*$K(t,s,v(s))\le K(t,s,u(s)),$ - (iii)
*there exists a continuous function*$G:[0,T]\times [0,T]\to [0,\mathrm{\infty})$*such that*$|K(t,s,x)-K(t,s,y)|\le G(t,s)\sqrt{\frac{{(x-y)}^{4}}{1+{(x-y)}^{2}}},$

*for all*$s,t\in [0,T]$

*and*$x,y\in \mathbb{R}$

*with*$x\ge y$,

- (iv)
${sup}_{t\in [0,T]}{\int}_{0}^{T}G{(t,s)}^{2}\phantom{\rule{0.2em}{0ex}}ds\le \frac{1}{T}$.

*Then the integral* (8) *has a solution* ${u}^{\ast}\in C[0,T]$.

*Proof*We, first, define $T:C[0,T]\to C[0,T]$ by

*T*is nondecreasing. Assume that $v\u2aafu$. From (ii), for all $s,t\in [0,T]$, we have $K(t,s,v(s))\le K(t,s,u(s))$. Thus, we get

for all $u,v\in C[0,T]$ with $v\u2aafu$. Hence, all hypotheses of Corollary 11 are satisfied. Thus, *T* has a fixed point ${u}^{\ast}\in C[0,T]$ which is a solution of (8). □

## Declarations

### Acknowledgements

Antonio-Francisco Roldán-López-de-Hierro has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE.

## Authors’ Affiliations

## References

- Roldán A, Martínez-Moreno J, Roldán C: Multidimensional fixed point theorems in partially ordered metric spaces.
*J. Math. Anal. Appl.*2012, 396: 536–545. 10.1016/j.jmaa.2012.06.049View ArticleMathSciNetGoogle Scholar - Karapınar E, Roldán A, Martínez-Moreno J, Roldán C: Meir-Keeler type multidimensional fixed point theorems in partially ordered metric spaces.
*Abstr. Appl. Anal.*2013., 2013: Article ID 406026Google Scholar - Roldán A, Martínez-Moreno J, Roldán C, Karapınar E:Multidimensional fixed-point theorems in partially ordered complete partial metric spaces under $(\psi ,\phi )$-contractivity conditions.
*Abstr. Appl. Anal.*2013., 2013: Article ID 634371Google Scholar - Wang S: Multidimensional fixed point theorems for isotone mappings in partially ordered metric spaces.
*Fixed Point Theory Appl.*2014., 2014: Article ID 137Google Scholar - Radenović S, Kadelburg Z, Jandrlić D, Jandrlić A: Some results on weakly contractive maps.
*Bull. Iran. Math. Soc.*2012, 38(3):625–645.Google Scholar - Karapınar E: Fixed point theory for cyclic weak
*ϕ*-contraction.*Appl. Math. Lett.*2011, 24(6):822–825. 10.1016/j.aml.2010.12.016View ArticleMathSciNetGoogle Scholar - Popescu O:Fixed points for $(\psi ,\varphi )$-weak contractions.
*Appl. Math. Lett.*2011, 24(1):1–4. 10.1016/j.aml.2010.06.024View ArticleMathSciNetGoogle Scholar - Zhang Q, Song Y: Fixed point theory for generalized
*ϕ*-weak contractions.*Appl. Math. Lett.*2009, 22: 75–78. 10.1016/j.aml.2008.02.007View ArticleMathSciNetGoogle Scholar - Rouhani BD, Moradi S:Common fixed point of multivalued generalized $(\psi ,\varphi )$-weak contractive mappings.
*Fixed Point Theory Appl.*2010., 2010: Article ID 708984Google Scholar - Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations.
*Nonlinear Anal.*2010, 72(3–4):1188–1197. 10.1016/j.na.2009.08.003View ArticleMathSciNetGoogle Scholar - Nieto JJ, Pouso RL, Rodríguez-Lopez R: Fixed point theorems in partially ordered sets.
*Proc. Am. Math. Soc.*2007, 132(8):2505–2517.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.