Positivity 11 (2007), 477–484
2007 Birkh¨auser Verlag Basel/Switzerland
A Fuzzy Extension of Generalized Implicit
Vector Variational-like Inequalities
M. Kalimuddin Ahmad and Salahuddin
Abstract. In this paper, we study the existence of solutions to the general-
ized implicit vector variational like inequality problems with fuzzy mappings
in Hausdorﬀ topological vector spaces using KKM-Fan Theorem.
Mathematics Subject Classiﬁcation (2000) . 49J40; 47H19.
Keywords. KKM-Fan Theorem, generalized implicit vector variational like
inequalities, fuzzy mappings, upper semicontinuous mappings, Hausdorﬀ topo-
logical vector spaces, cone.
A vector variational inequality in a ﬁnite-dimensional Euclidean space was ﬁrst ini-
tiated by Giannessi , which was the vector valued version of variational inequal-
ity of Hartman and Stampacchia , and shown to be an useful tool in vector
optimization. In the past, the vector equilibrium problems which were the uniﬁed
model for vector variational inequalities, vector variational like inequalities, vector
complementarity problems and vector optimization problems, have been studied
in [11, 14, 17, 21]. Recently, Li et al.  introduced and studied implicit vector
equilibrium problems in Hausdorﬀ topological vector spaces which include implicit
equilibrium problems, implicit vector variational inequalities and implicit vector
complementarity problems as special cases. Chang and Zhu  introduced the
concept of variational inequalities for fuzzy mappings in locally convex Hausdorﬀ
topological vector spaces and investigated the existence theorems for some kind of
variational inequalities for fuzzy mappings, which were the fuzzy extension of some
variational inequality problems . Lee et al.  considered vector variational
inequalities for fuzzy mappings which were the fuzzy extension of vector varia-
tional inequalities, studied by Chen and Yang , and obtained some existence
theorems for solutions of their inequalities for fuzzy mappings. Chang et al. [4, 5]
obtained several kind of existence theorems for vector quasi-variational inequalities
on locally convex Hausdorﬀ topological vector spaces on the basis of Fan, Glick-
berg-Kakutani ﬁxed point theorems and scalarization method of Luc et al. .