# Erratum to: ‘n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’

The Original Article was published on 22 April 2013

## Correction

(1) Page 3, line 21:

The statement ‘(ii) ${lim}_{r\to t+}\varphi \left(r\right) for each $r>0$’ should be corrected as ‘(ii) ${lim}_{r\to t+}\varphi \left(r\right) for each $t>0$’.

(2) Page 4, line 3:

The statement ‘… Condition 1 is satisfied.’ should be rewritten ‘… Condition 1 is satisfied and g is continuous.’

(3) Page 8, line 7:

The statement ‘… and using (2.20)’ should be corrected as ‘… and using (2.19)’.

(4) Page 9, line 14:

The statement ‘$\le {\delta }_{j\left(k\right)+1}+{\delta }_{l\left(k\right)+1}+{t}_{k}+n\cdot \varphi \left(\frac{{t}_{k}}{n}\right)$’ should be corrected as ‘$\le {\delta }_{j\left(k\right)+1}+{\delta }_{l\left(k\right)+1}+n\cdot \varphi \left(\frac{{t}_{k}}{n}\right)$’.

(5) Page 9, line 22:

The statement ‘From (2.10) and by …’ should be corrected as ‘From (2.8) and by … ’.

(6) Page 10, line 26:

‘… now the assumption (b) holds.’ should be corrected as ‘… now the assumption (ii) holds.’

(7) Page 11, line 20 (line 2 in Corollary 2) and Page 17, line 20 (line 2 in Corollary 4):

The statement ‘and there exist $\varphi \in \mathrm{\Phi }$ such that F’ should be deleted.

(8) Page 11, line 22 (line 4 in Corollary 2) and Page 17, line 22 (line 4 in Corollary 4):

The statement ‘$\varphi \left(F\left({x}^{1},{x}^{2},\dots ,{x}^{n}\right),F\left({y}^{1},{y}^{2},\dots ,{y}^{n}\right)\right)$’ should be corrected as ‘$d\left(F\left({x}^{1},{x}^{2},\dots ,{x}^{n}\right),F\left({y}^{1},{y}^{2},\dots ,{y}^{n}\right)\right)$’.

(9) Page 16, line 27 (line 2 in Corollary 3) and Page 17, line 20 (line 2 in Corollary 4):

The statement ‘…F has the mixed g-monotone’ should be corrected as ‘…F has the mixed monotone’. That is, ‘g-’ should be deleted.

(10) Page 17, line 3:

$\le \varphi \left(\frac{d\left(g\left({x}^{1}\right),g\left({y}^{1}\right)\right)+d\left(g\left({x}^{2}\right),g\left({y}^{2}\right)\right)+\cdots +d\left(g\left({x}^{n}\right),g\left({y}^{n}\right)\right)}{n}\right)$’ should be corrected as ‘$\le \varphi \left(\frac{d\left({x}^{1},{y}^{1}\right)+d\left({x}^{2},{y}^{2}\right)+\cdots +d\left({x}^{n},{y}^{n}\right)}{n}\right)$’.

(11) Page 17, line 23:

$\le \frac{m}{n}\left[d\left(g\left({x}^{1}\right),g\left({y}^{1}\right)\right)+d\left(g\left({x}^{2}\right),g\left({y}^{2}\right)\right)+\cdots +d\left(g\left({x}^{n}\right),g\left({y}^{n}\right)\right)\right]$’ should be corrected as ‘$\le \frac{m}{n}\left[d\left({x}^{1},{y}^{1}\right)+d\left({x}^{2},{y}^{2}\right)+\cdots +d\left({x}^{n},{y}^{n}\right)\right]$’.

(12) Page 4, line 1:

The statement ‘${x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}\in X$’ should be corrected as ‘${x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n},{y}_{1},{y}_{2},{y}_{3},\dots ,{y}_{n}\in X$’.

(13) ‘$d\left(g\left({x}_{k}^{n}\right),g\left({x}_{k+2}^{n}\right)\right)$’ must be ‘$d\left(g\left({x}_{k+1}^{n}\right),g\left({x}_{k+2}^{n}\right)\right)$’.

(14) Page 8, line 16:

$\le {\delta }_{j\left(k\right)+1}+{\delta }_{l\left(k\right)+1}+d\left(g\left({x}_{j\left(k\right)+1}^{1}\right),g\left({x}_{l\left(k\right)+1}^{1}\right)\right)+d\left(g\left({x}_{j\left(k\right)+1}^{2}\right),g\left({x}_{l\left(k\right)+1}^{2}\right)\right)$’ must be ‘$\le {\delta }_{j\left(k\right)}+{\delta }_{l\left(k\right)}+d\left(g\left({x}_{j\left(k\right)+1}^{1}\right),g\left({x}_{l\left(k\right)+1}^{1}\right)\right)+d\left(g\left({x}_{j\left(k\right)+1}^{2}\right),g\left({x}_{l\left(k\right)+1}^{2}\right)\right)$’.

(15) Page 9, line 10:

‘… with (2.26)-(2.29)’ must be ‘… with (2.26)-(2.28)’.

(16) Page 11, line 3:

$g\left({x}_{k}^{1}\right)\ge {x}^{1},g\left({x}_{k}^{2}\right)\le {x}^{2},\dots ,g\left({x}_{k}^{n}\right)\le {x}^{n}$ (If n is odd)’ must be ‘$g\left({x}_{k}^{1}\right)\le {x}^{1},g\left({x}_{k}^{2}\right)\ge {x}^{2},\dots ,g\left({x}_{k}^{n}\right)\le {x}^{n}$ (If n is odd)’.

## References

1. Ertürk M, Karakaya V: n -tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 196

## Author information

Authors

### Corresponding author

Correspondence to Müzeyyen Ertürk.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors made up the article together.

The online version of the original article can be found at 10.1186/1029-242X-2013-196

## Rights and permissions

Reprints and Permissions

Ertürk, M., Karakaya, V. Erratum to: ‘n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’. J Inequal Appl 2013, 368 (2013). https://doi.org/10.1186/1029-242X-2013-368 