Nonconvex fuzzy mappings and the fuzzy pre-variational inequality

In this paper, we introduce the concepts of fuzzy pseudoconvex, fuzzy invex, fuzzy pseudoinvex and fuzzy preinvex mapping from R^n to the set of fuzzy numbers, and put forward the concepts of fuzzy extended directional differential, subgradient and subdifferential. Some properties are obtained for fuzzy pseudoconvex, fuzzy invex, fuzzy pseudoinvex and fuzzy preinvex mapping; relations are discussed between the fuzzy pre-variational inequality and the fuzzy vector optimization problem.

[1]  Jiang Zhu,et al.  Vector-valued variational principle in fuzzy metric space and its applications , 2001, Fuzzy Sets Syst..

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

[3]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[4]  Chenglin Wen,et al.  On fuzzy n-cell numbers and n-dimension fuzzy vectors , 2007, Fuzzy Sets Syst..

[5]  Xiaoqi Yang,et al.  Generalized convex functions and vector variational inequalities , 1993 .

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

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

[8]  D. Dubois,et al.  Towards fuzzy differential calculus part 1: Integration of fuzzy mappings , 1982 .

[9]  Xiaoqi Yang,et al.  Vector variational-like inequality with pseudoinvexity , 2006 .

[10]  Muhammad Aslam Noor,et al.  Nonconvex functions and variational inequalities , 1995 .

[11]  Chen Guang-ya,et al.  The vector complementary problem and its equivalences with the weak minimal element in ordered spaces , 1990 .

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

[13]  Yu-Ru Syau,et al.  Generalization of preinvex and B-vex fuzzy mappings , 2001, Fuzzy Sets Syst..

[14]  Gue-Myung Lee,et al.  On Relations Between Vector Optimization Problems and Vector Variational Inequalities , 2002 .

[15]  Jong Yeoul Park,et al.  Completely generalized strongly quasivariational inequalities for fuzzy mappings , 2000, Fuzzy Sets Syst..

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

[17]  W. Congxin,et al.  Embedding problem of fuzzy number space: part II , 1992 .

[18]  Sang Yeol Joo,et al.  A characterization of compact subsets of fuzzy number space , 2001, Fuzzy Sets Syst..

[19]  Muhammad Aslam Noor,et al.  Variational inequalities for fuzzy mappings (II) , 1998, Fuzzy Sets Syst..

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

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

[22]  Gue Myung Lee,et al.  Remarks on relations between vector variational inequality and vector optimization problem , 2001 .

[23]  Mau-Hsiang Shih,et al.  Generalized quasi-variational inequalities in locally convex topological vector spaces , 1985 .

[24]  Wataru Takahashi,et al.  Nonlinear Variational Inequalities and Fixed Point Theorems , 1975 .

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

[26]  P. Kloeden,et al.  Metric Spaces Of Fuzzy Sets Theory And Applications , 1975 .

[27]  Jong Yeoul Park,et al.  Generalized strongly quasivariational inequalities for fuzzy mappings , 1998, Fuzzy Sets Syst..

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

[29]  Congxin Wu,et al.  Convexity and semicontinuity of fuzzy mappings , 2006, Comput. Math. Appl..

[30]  Yu-Ru Syau.  Preinvex fuzzy mappings , 1999 .

[31]  Cheng Zhang,et al.  Convex fuzzy mapping and operations of convex fuzzy mappings , 2006, Comput. Math. Appl..