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Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization

  • 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 ′′ cint 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 Rp" 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 Rp" 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 Rp," 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 .

References

  1. 1.

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

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Correspondence to Do Sang Kim.

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The online version of the original article can be found at 10.1186/1687-1812-2011-42

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Deok Bae, K., Kim, D.S. Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization. Fixed Point Theory Appl 2012, 28 (2012). https://doi.org/10.1186/1687-1812-2012-28

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