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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 t + ϕ(r)<t for each r>0’ should be corrected as ‘(ii) lim r t + ϕ(r)<t 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 ‘ δ j ( k ) + 1 + δ l ( k ) + 1 + t k +nϕ( t k n )’ should be corrected as ‘ δ j ( k ) + 1 + δ l ( k ) + 1 +nϕ( t k n )’.

(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 ϕΦ 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 ‘ϕ(F( x 1 , x 2 ,, x n ),F( y 1 , y 2 ,, y n ))’ should be corrected as ‘d(F( x 1 , x 2 ,, x n ),F( y 1 , y 2 ,, y n ))’.

(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:

ϕ( d ( g ( x 1 ) , g ( y 1 ) ) + d ( g ( x 2 ) , g ( y 2 ) ) + + d ( g ( x n ) , g ( y n ) ) n )’ should be corrected as ‘ϕ( d ( x 1 , y 1 ) + d ( x 2 , y 2 ) + + d ( x n , y n ) n )’.

(11) Page 17, line 23:

m n [d(g( x 1 ),g( y 1 ))+d(g( x 2 ),g( y 2 ))++d(g( x n ),g( y n ))]’ should be corrected as ‘ m n [d( x 1 , y 1 )+d( x 2 , y 2 )++d( x n , y n )]’.

(12) Page 4, line 1:

The statement ‘ x 1 , x 2 , x 3 ,, x n X’ should be corrected as ‘ x 1 , x 2 , x 3 ,, x n , y 1 , y 2 , y 3 ,, y n X’.

(13) ‘d(g( x k n ),g( x k + 2 n ))’ must be ‘d(g( x k + 1 n ),g( x k + 2 n ))’.

(14) Page 8, line 16:

δ j ( k ) + 1 + δ l ( k ) + 1 +d(g( x j ( k ) + 1 1 ),g( x l ( k ) + 1 1 ))+d(g( x j ( k ) + 1 2 ),g( x l ( k ) + 1 2 ))’ must be ‘ δ j ( k ) + δ l ( k ) +d(g( x j ( k ) + 1 1 ),g( x l ( k ) + 1 1 ))+d(g( x j ( k ) + 1 2 ),g( x l ( k ) + 1 2 ))’.

(15) Page 9, line 10:

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

(16) Page 11, line 3:

g( x k 1 ) x 1 ,g( x k 2 ) x 2 ,,g( x k n ) x n (If n is odd)’ must be ‘g( x k 1 ) x 1 ,g( x k 2 ) x 2 ,,g( x k n ) 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

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Correspondence to Müzeyyen Ertürk.

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The authors made up the article together.

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

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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

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  • DOI: https://doi.org/10.1186/1029-242X-2013-368