On invex fuzzy mappings and fuzzy variational-like inequalities

In this paper, we first show the need for introducing invex fuzzy mappings. After that, we show that the concept of invex fuzzy mapping previously given by Wu and Xu are very restrictive and the examples presented there are not correct. Then, we present more general concepts of invex and incave fuzzy mappings involving strongly generalized differentiable fuzzy mapping. Finally, we show that the results obtained by Wu and Xu on the relationship between fuzzy variational-like inequalities and fuzzy optimization problems are still valid using these new concepts.

[1]  S. Nanda,et al.  Convex fuzzy mappings , 1992 .

[2]  H. Román-Flores,et al.  On new solutions of fuzzy differential equations , 2008 .

[3]  Hsien-Chung Wu,et al.  The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions , 2009, Eur. J. Oper. Res..

[4]  S. Seikkala.  On the fuzzy initial value problem , 1987 .

[5]  Luciano Stefanini,et al.  Some notes on generalized Hukuhara differentiability of interval-valued functions and interval differential equations , 2012 .

[6]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[7]  Didier Dubois,et al.  On several definitions of the differential of a fuzzy mapping , 1987 .

[8]  Muhammad Aslam Noor,et al.  Fuzzy preinvex functions , 1994 .

[9]  M. A. Hanson.  On sufficiency of the Kuhn-Tucker conditions , 1981 .

[10]  J. Ramík,et al.  Generalized Concavity in Fuzzy Optimization and Decision Analysis , 2001 .

[11]  Sudarsan Nanda,et al.  On integration of fuzzy mappings , 1989 .

[12]  James J. Buckley,et al.  Fuzzy differential equations , 2000, Fuzzy Sets Syst..

[13]  M. Puri,et al.  Differentials of fuzzy functions , 1983 .

[14]  Sudarsan Nanda,et al.  Convex fuzzy mapping with differentiability and its application in fuzzy optimization , 2008, Eur. J. Oper. Res..

[15]  Jiuping Xu,et al.  Nonconvex fuzzy mappings and the fuzzy pre-variational inequality , 2008, Fuzzy Sets Syst..

[16]  Yu-Ru Syau,et al.  On convex and concave fuzzy mappings , 1999, Fuzzy Sets Syst..

[17]  Nagata Furukawa.  Convexity and local Lipschitz continuity of fuzzy-valued mappings , 1998, Fuzzy Sets Syst..

[18]  Congxin Wu,et al.  Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming , 2003, Fuzzy Sets Syst..

[19]  Hsien-Chung Wu,et al.  Evaluate fuzzy optimization problems based on biobjective programming problems , 2004 .

[20]  Hsien-Chung Wu,et al.  The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function , 2007, Eur. J. Oper. Res..

[21]  Yuanguo Zhu,et al.  On variational inequalities for fuzzy mappings , 1989 .

[22]  Jiuping Xu,et al.  Generalized convex fuzzy mappings and fuzzy variational-like inequality , 2009, Fuzzy Sets Syst..

[23]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[24]  Hong Yan,et al.  A class of convex fuzzy mappings , 2002, Fuzzy Sets Syst..

[25]  M. Hukuhara.  INTEGRATION DES APPLICAITONS MESURABLES DONT LA VALEUR EST UN COMPACT CONVEXE , 1967 .

[26]  Yu-Ru Syau,et al.  Invex and generalized convex fuzzy mappings , 2000, Fuzzy Sets Syst..

[27]  R. Goetschel,et al.  Elementary fuzzy calculus , 1986 .

[28]  Yurilev Chalco-Cano,et al.  Generalized derivative and π-derivative for set-valued functions , 2011, Inf. Sci..

[29]  D. Dubois,et al.  Towards fuzzy differential calculus part 3: Differentiation , 1982 .

[30]  P. Kloeden,et al.  Metric spaces of fuzzy sets , 1990 .

[31]  Y Chalco Cano,et al.  GENERALIZED DERIVATIVE AND -DERIVATIVE FOR SET-VALUED FUNCTIONS , 2011 .