Open Access

Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization

Fixed Point Theory and Applications20122012:28

https://doi.org/10.1186/1687-1812-2012-28

Received: 6 February 2012

Accepted: 27 February 2012

Published: 27 February 2012

The original article was published in Fixed Point Theory and Applications 2011 2011:42

We wish to indicate the following corrections to our original paper [1].

(1) The first sentence in Definition 2.2, we delete "i{1,2,···,p}".

(2) The first sentence in Definition 2.2, we replace
f i x + s x | D i f i x 0 + s x 0 | D i
to
f x + s x | D f x 0 + s x 0 | D .

(3) The second sentence in Definition 2.3, we delete "i{1,2,···,p}".

(4) The second sentence in Definition 2.3, we replace
f i x + s x | D i f i x 0 + s x 0 | D i + c i | | x - x 0 | | m
to
f x + s x | D f x 0 + s x 0 | D + c | | x - x 0 | | m .

(5) The second sentence in Definition 2.4, we delete "i{1,2,···,p}".

(6) The second sentence in Definition 2.4, we replace
f i x + s x | D i f i x 0 + s x 0 | D i + c i | | x - x 0 | | m
to
f x + s x | D f x 0 + s x 0 | D + c | | x - x 0 | | m . .

(7) The second sentence in the proof of Theorem 2.1, we replace. ′′ c i > 0, i = 1, ,p ′′ to ′′ c i n t R + p ′′.

(8) The second sentence in the proof of Theorem 2.1, we replace
f i x + s x | D i f i x 0 + s x 0 | D i + c i | | x - x 0 | | m
to
f x + s x | D f x 0 + s x 0 | D + c | | x - x 0 | | m .

(9) In equation (3.8), we replace "c i " to "d i ".

(10) The eighth sentence in the proof of Theorem 3.3, we replace "where c = ae," to "where d = ae".

(11) The ninth sentence in the proof of Theorem 3.3, we replace "c int R p " to "d i > 0, i = 1,···,p,".

(12) The ninth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".

(13) The tenth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".

(14) The tenth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".

(15) In equation (4.8), we replace "c i " to "d i ".

(16) The tenth sentence in the proof of Theorem 4.1, we replace "where c = ae," to "where d = ae".

(17) The eleventh sentence in the proof of Theorem 4.1, we replace "c int R p " to "d i > 0, i = 1,···, p,".

(18) The eleventh sentence in the proof of Theorem 4.1, we replace "c i " to "d i ".

(19) The twelfth sentence in the proof of Theorem 4.1, we replace "c int R p ," to "d i > 0, i = 1,···, p,".

(20) The twelfth sentence in the proof of Theorem 4.1, we replace "c i " to "d i ".

(21) The twelfth sentence in the proof of Theorem 4.1, we replace "i = 1,···, p." to "i = 1,···,p,".

(22) The fourth sentence in the proof of Theorem 4.2, we replace "c i " to "c".

(23) The fourth sentence in the proof of Theorem 4.2, we replace
f i ( x 0 ) + ( x 0 ) T w i 0 + c i | | u - x 0 | | m f i ( u ) + u T w i , i = 1 , , p .
to
f ( x 0 ) + ( x 0 ) T w 0 + c | | u - x 0 | | m f ( u ) + u T w .

Notes

Authors’ Affiliations

(1)
Department of Applied Mathematics, Pukyong National University

References

  1. Bae , Kim : Optimality and Duality Theorems in Nonsmooth Multiobjective Optimization. Fixed Point Theory and Applications 2011, 2011: 42. 10.1186/1687-1812-2011-42MathSciNetView ArticleGoogle Scholar

Copyright

© Deok Bae and Kim; licensee Springer. 2012

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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.