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Báo cáo hóa học: " Erratum Erratum for ”Higher-Order Weakly Generalized Adjacent Epiderivatives and Applications to Duality of Set-Valued Optimization”

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Erratum Erratum for ”Higher-Order Weakly Generalized Adjacent Epiderivatives and Applications to Duality of Set-Valued Optimization” | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 817965 3 pages doi 10.1155 2011 817965 Erratum Erratum for Higher-Order Weakly Generalized Adjacent Epiderivatives and Applications to Duality of Set-Valued Optimization Qi-Lin Wang College of Sciences Chongqing Jiaotong University Chongqing 400074 China Correspondence should be addressed to Qi-Lin Wang wangql97@126.com Received 24 September 2010 Accepted 27 January 2011 Copyright 2011 Qi-Lin Wang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. An important property is established for higher-order weakly generalized adjacent epiderivatives. This corrects an earlier result by Wang and Li 2009 . 1. Introduction The concept of higher-order weakly generalized adjacent epiderivatives is introduced and an important property is given for the derivatives in 1 Proposition 1.1. Let E be a nonempty convex subset of X X X0 e E y0 e F x0 . Let F - y0 is C-convexlike on E Ui e E Vi e Ffui C i 1 2 . m - 1. If the set q x - X0 y e Y m m x - X0 y e G - Tepi F X0 y0 U1 - X0 V1 - y0 . . Um-1 - X0 Vm-1 - y0 fulfills the weak domination property for all X e E then F x - y0 c dWmF x0 y0 U1 - X0 V1 - y0 . Um-1 - X0 Vm-1 - y0 X - X0 C. 1 1 For other notations and definitions one may refer to 1 While proving Proposition 1 1 in 1 the authors used the assumption that the F - y0 is C-convexlike see 2 3 on a convex set E which implies cone epi F - x0 y0 is a convex cone In fact the assumption may not hold The following example shows that the case and Proposition 1 1 may not hold where one only takes m 2 2 Journal of Inequalities and Applications Example 1.2. Let X Y R C R E -1 2 c R. Consider a set-valued map F E 2Y defined by F x y e Y I y 0 if x e -1 2 if x -1. 1.2 Take x0 y0 0 0 e graph F u 1 v 0 e F 1 C. Naturally F - yo be .