Generalized vector variational inequality and fuzzy extension

A general vector-valued variational inequality and its fuzzy extension

“…Our existence theorem subsumes Theorem 2.1 of Cottle and Yao [18], the part (i) of Theorem 2.1 of Chen and Yang [6], Theorem 2 of Yang [19] and Theorem 2.1 of Lee et al [7]. Moreover, we obtain the fitzzy extension of our existence theorem.…”

“…Lee et al 13 obtained the fazzy generalizations of new results of Kim and Tan [14], and they [7] established the fuzzy extension of their existence theorem. Our motivation of this paper is to consider the noncompact cases of the existence theorems of variational inequalities for multifimctions with vector values or fuzzy mappings in Banach spaces obtained by Lee et al [7]. Let X and Y be two normed spaces and D a nonempty convex subset of X.…”

“…Our purpose in this paper is to establish the existence theorems for (GVVI) in the noncompact setting, which is the noncompact case of the existence theorem for (GVVI) obtained by Lee et al [7], by using a particular form of the generalized KKM theorems due to Park [15][16][17]. Our existence theorem subsumes Theorem 2.1 of Cottle and Yao [18], the part (i) of Theorem 2.1 of Chen and Yang [6], Theorem 2 of Yang [19] and Theorem 2.1 of Lee et al [7].…”

“…Since then, Chen et al [2][3][4][5][6] have intensively studied variational inequalities for vector-valued mappings in Banach spaces. Lee et al [7] have established the existence theorem of a variational inequality for a multifimction with vector values in a Banach space.…”

Fuzzy General Nonlinear Ordered Random Variational Inequalities in Ordered Banach Spaces

“…Very recently, the problems of random generalized fuzzy variational inclusions involving random nonlinear mapping have been studied by Zhang and Bi [33] in Hilbert spaces. Afterwards, on several kinds of variational inequalities, variational inclusions and complementarity problems for fuzzy mappings were considered and studied by many authors see for instance, Ahmad and Salahuddin [1,2,3], Ahmad and Bazan [4], Agarwal et al [5], Anastassiou et al [7], Chang and Huang [10], Chang and Salahuddin [11], Chang et al [12], Cho et al [14], Ding and Park [15], Huang [20], Lee et al [25,26,27], Salahuddin [28], Salahuddin and Ahmad [30], Salahuddin et al [31] and Salahuddin and Verma [29], etc. Inspired and motivated by recent works, in this communication, fuzzy general nonlinear ordered random variational inequalities and an operator ⊕ is introduced and the qualities of an operator ⊕ is studied in ordered Banach spaces.…”