Properties Analysis of Fuzzy Variational Inequality Problems
The fuzzy variable was introduced in the variational inequality, the model of the fuzzy variational inequality problem was presented in possibility space. And some properties were investigated. This paper was organized as follow, firstly, we introduced fuzzy variable in variational inequality. Furthermore, the expected residual minimization of the regularized gap function was established. Then, we concentrated on a subclass of fuzzy variable in variational inequality in which the function involved was assumed to be affine and the properties of the variational inequality problem with fuzzy variable were discussed. Introduction The variational inequality problem (VIP for short) is very important in mathematics. Researchers mainly studies three aspects of VIP including solution to the existence and uniqueness, effective solution algorithms and application such as optimization theory, economics, engineering , game theory and networks. Recently, Chang and Zhu [1] introduced the concepts of variational inequalities for fuzzy mappings which was later studied by Noor [2], Chang and Huang [2], and Lee et al. [3]. Our paper first considers the variational inequality problem under possibility space (FVIP for short). Finding a vector * n s x R ∈ ⊂ such that , . , 0 ) , ( * ) ( s a S x x F x x T Ξ ∈ ∈ ∀ ≥ − ξ ξ (1) where ξ is fuzzy variable in the possibility space, : n n F R R ×Ξ → is a mapping, and a.s. is the abbreviation for almost surely under the given possibility measure. There is no doubt that both of them are natural extensions of deterministic VIP. Therefore, In this paper, our focus is talking about the property of FVIP . The remainder of this paper is organized as follows. Some basic concepts will be reviewed in Section 2, In Section 3, we establish the expected model based on possibility measure. In Section4, some properties were investigated. Preliminaries In this section, we will introduce some fundamental concepts and properties concerning possibility measure, membership function and so on. Definition 1. [4]Let Θ is a nonempty set. F is the power set of Θ . The element in F is called an event. If the function [ ] : 0,1 Pos F → satisfies (1) ( ) 1 Pos Θ = , (2) ( ) ( , ), , A A A B Pos A F Pos B ≤ ⊂ ∈ (3) ( ) sup ( ) i i i i Pos Pos A A ⊂ ∪ , for i = 1,2... i A F ∈ . Then, the function is possibility measure. ( , ( ), ) F Pos Θ Θ is possibility measure space. Definition 2 Let x is a fuzzy variable defined in the possibility space. If