A Class of Fuzzy Variational Inequality Based on Monotonicity of ...

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 854751, 17 pages http://dx.doi.org/10.1155/2013/854751

Research Article A Class of Fuzzy Variational Inequality Based on Monotonicity of Fuzzy Mappings Zezhong Wu1 and Jiuping Xu2 1 2

College of Mathematics, Chengdu University of Information Technology, Chengdu 610225, China Uncertainty Decision-Making Laboratory, School of Business and Administration, Sichuan University, Chengdu 610064, China

Correspondence should be addressed to Jiuping Xu; [email protected] Received 11 May 2013; Accepted 1 September 2013 Academic Editor: Kanishka Perera Copyright © 2013 Z. Wu and J. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Invex monotonicity and pseudoinvex monotonicity of fuzzy mappings are introduced in this paper, and relations are discussed between invex monotonicity (pseudoinvex monotonicity) and invexity (pseudoinvexity) of fuzzy mappings. The existence of a solution to the fuzzy variational-like inequality is discussed, and the existence theorem can be achieved. Furthermore, some extended properties of the fuzzy variational-like inequality are researched. Finally, method of solution is discussed based on genetic algorithm.

1. Introduction In [1], Chang and Zadeh introduced the concept of fuzzy mapping. Since then, fuzzy mapping has been extensively studied by many authors. Nanda and Kar [2] proposed a concept of convex fuzzy mapping in 1992 and proved that a fuzzy mapping is convex if and only if its epigraph is a convex set. In recent years, there have been increasing attempts to weaken the convexity condition of fuzzy mapping, such as Yan and Xu [3], Panigrahi et al. [4], and Wu and Xu [5, 6]. It is well known that in classical (non-fuzzy) convex analysis, some properties are shared by different kinds of functions that are more general than convex functions, which has given rise to the study of generalized convexity and later to that of generalized monotonicity. Just as convex functions are characterized by a monotone gradient, different kinds of generalized convex functions give rise to gradient maps with certain generalized monotonicity properties which are inherited from generalized convexity of the underlying functions. At the same time, it is a useful method in researching variational inequality by means of monotonicity and generalized monotonicity of functions. For corresponding research of monotonicity and generalized monotonicity of functions, one can refer to [7–9]. Similarly, one can research some properties of fuzzy mappings and fuzzy variational

inequality by studying their monotonicity. On the other hand, it is worth noting that monotonicity has played a very important role in the study of the existence and solution methods of variational inequality problems. Similarly, one can research fuzzy variational inequality by monotonicity of fuzzy mappings. However, very few investigations have appeared to study monotonicity of fuzzy mappings. Based on the above, we give concepts of monotonicity and generalized monotonicity of fuzzy mappings and discuss relations of generalized monotonicity and generalized convexity. Vector variational inequality was first introduced and studied by Cottle et al. [10] in finite-dimensional Euclidean spaces. This is a generalization of a scalar variational inequality to the vector case by virtue of multicriteria considering. Since then, the theory with applications for vector variational inequality and vector complementarity problems have been studied and generalized by many authors (see, e.g., [11–21] and the references therein). In particular, in [12], Guang-Ya and Xiao-Qi discussed the existence of a solution to the vector variational inequality and the existence of the solution of the vector complementary problem and obtained some valuable results. It offers some ideal for research of the existence of a solution of other variational inequality problems. However, facing uncertainty is a constant challenge for optimization and decision making. Treating uncertainty with fuzzy

2 mathematics results in the study of fuzzy optimization and decision making. Recently, Chang and Zhu [22] introduced the concepts of the variational inequality problem for fuzzy mappings which were later developed by Noor [23–25]. For further research of variational inequalities of fuzzy mappings, one can refer to [26–30]. However, very few investigations have appeared to study fuzzy variational inequalities and the existence of a solution by means of fuzzy numbers. In [5], Wu and Xu discussed the relationship between the fuzzy prevariational inequality and the fuzzy vector optimization problem based on nonconvex fuzzy mappings. In [6], Wu and Xu discussed the relationship between the fuzzy variationallike inequality and the fuzzy vector optimization problem based on generalized convex fuzzy mappings and gave an application example of a fuzzy variational-like inequality: the fuzzy variational-like inequality representation of a fuzzy transportation equilibrium problem. On the other hand, the role of generalized monotonicity of the operator in variational inequality problems corresponds to the role of generalized convexity of the objective function in mathematical programming problems. Similarly, the role of monotonicity of fuzzy mappings should also be very important for research of fuzzy variational inequality problems, particularly research of the existence of a solution of a fuzzy variational inequality. Based on the above, we introduce the fuzzy variational-like inequality, discuss some properties of the fuzzy variationallike inequality, and study the existence of a solution of the fuzzy variational-like inequality by virtue of generalized monotonicity of fuzzy mappings. It is well known that we can solve variational inequality by transforming a variational inequality problem into an optimization problem, for example, [18, 31–33]. Similarly, fuzzy variational inequality also can be transformed into a fuzzy optimization problem. It is difficult to solve a fuzzy optimization problem by means of traditional optimization methods, but a special genetic algorithm can be used to solve fuzzy optimization problems. Therefore, in the sixth section, we discuss the solving of the fuzzy variational-like inequality by virtue of genetic algorithm. In this paper, we study invex monotonicity and pseudoinvex monotonicity of fuzzy mappings and discuss relations between invex monotonicity (pseudo-invex monotonicity) and invexity (pseudo-invexity) of fuzzy mappings. We discuss the existence of a solution to the fuzzy variationallike inequality. Furthermore, some extended properties of the fuzzy variational-like inequality are researched. Finally, method of solution is discussed based on genetic algorithm. This paper is organized as follows. Section 2 recalls some definitions and results in reference to fuzzy numbers. Section 3 introduces invex monotonicity and pseudo-invex monotone of fuzzy mapping and discusses some properties. Section 4 defines a class of fuzzy variational inequality: fuzzy variational-like inequality, and discusses the solution existence for the fuzzy variational-like inequality. Section 5 discusses extended qualities of the fuzzy variational-like inequality. Section 6 discusses the method of solving for the fuzzy variational-like inequality and gives a genetic algorithm of a class of the fuzzy variational-like inequality.

Abstract and Applied Analysis

2. Preliminaries A fuzzy set of 𝑅𝑛 is a mapping 𝜇 : 𝑅𝑛 → [0, 1]. For each such fuzzy set 𝜇, its 𝛼-cut set is denoted by [𝜇]𝛼 = {𝑥 ∈ 𝑅𝑛 : 𝜇(𝑥) ≥ 𝛼} for all 𝛼 ∈ (0, 1]. The support of 𝜇 is denoted by supp 𝜇; that is, {𝑥 ∈ 𝑅𝑛 | 𝜇(𝑥) > 0}. The closure of supp 𝜇 is defined as [𝜇]0 . Definition 1 (see Wu and Xu [6]). A fuzzy number 𝜇 is a fuzzy set with the following properties: (1) 𝜇 is normal; that is, there exists 𝑥0 ∈ 𝑅𝑛 such that 𝜇(𝑥0 ) = 1; (2) 𝜇 is convex fuzzy set; that is, 𝜇(𝜆𝑥 + (1 − 𝜆)𝑦) ≥ min(𝜇(𝑥), 𝜇(𝑦)), 𝑥, 𝑦 ∈ 𝑅𝑛 , 𝜆 ∈ [0, 1]; (3) [𝜇]0 is compact. Let 𝐸 denote the family of fuzzy numbers; that is, 𝐸 denotes the family of compact and convex fuzzy set on 𝑅1 . Obviously, [𝜇]𝛼 is a nonempty compact convex subset of 𝑅1 (denoted by [𝜇∗ (𝛼), 𝜇∗ (𝛼)]) for all 𝜇 ∈ 𝐸 and for all 𝛼 ∈ [0, 1]. A precise number 𝑎 is a special case of fuzzy number encoded as 1, 𝑎̃ (𝑡) = { 0,

if 𝑡 = 𝑎, if 𝑡 ≠ 𝑎.

(1)

However, a precise number will be denoted as usual, in particular, number 0. The fuzzy numbers 𝜇, ] ∈ 𝐸 are represented by (𝜇∗ (𝛼), 𝜇∗ (𝛼)) and (]∗ (𝛼), ]∗ (𝛼)), respectively. ̃ ] and scalar For each real number 𝜆, the addition 𝜇 + multiplication 𝜆𝜇 are defined as follows: ̃ ]) (𝑥) = sup min [𝜇 (𝑦) , ] (𝑧)] , (𝜇 + 𝑦+𝑧=𝑥

−1 { 𝜇 (𝜆 𝑥) if 𝜆 ≠ 0, (𝜆𝜇) (𝑥) = { for 𝜇, ] ∈ 𝐸, 𝜆 ∈ 𝑅. 0 if 𝜆 = 0, {

(2)

It is well known that for all 𝜇, ] ∈ 𝐸 and 𝜆 ∈ 𝑅1 ̃ ])∗ (𝛼) = 𝜇∗ (𝛼) + ]∗ (𝛼) , (𝜇 + ∗

̃ ]) (𝛼) = 𝜇∗ (𝛼) + ]∗ (𝛼) , (𝜇 + (𝜆𝜇)∗ (𝛼) = {

𝜆𝜇∗ (𝛼) if 𝜆 ≥ 0, 𝜆𝜇∗ (𝛼) if 𝜆 < 0,

∗ { 𝜆𝜇 (𝛼) if 𝜆 ≥ 0, for given 𝛼 ∈ [0, 1] . (𝜆𝜇) (𝛼) = { if 𝜆 < 0, 𝜆𝜇 (𝛼) ∗ { ∗

(3)

For 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ), 𝑦 = (𝑦1 , 𝑦2 , . . . , 𝑦𝑛 ) ∈ 𝑅𝑛 , 𝑥 ≤ 𝑦 if and only if 𝑥𝑖 ≤ 𝑦𝑖 (𝑖 = 1, 2, . . . , 𝑛), and 𝑥 < 𝑦 if and only if 𝑥 ≤ 𝑦 and 𝑥 ≠ 𝑦. Definition 2. For 𝜇, ] ∈ 𝐸, 𝜇 ⪯ ], if and only if for every 𝛼 ∈ [0, 1], 𝜇∗ (𝛼) ≤ ]∗ (𝛼) and 𝜇∗ (𝛼) ≤ ]∗ (𝛼).

Abstract and Applied Analysis

3

If 𝜇 ⪯ ], ] ⪯ 𝜇, then 𝜇 = ]. 𝜇 ≺ ] if and only if 𝜇 ⪯ ] and ∃𝛼0 ∈ [0, 1], such that 𝜇∗ (𝛼0 ) < ]∗ (𝛼0 ) or ]∗ (𝛼0 ) < 𝜇∗ (𝛼0 ). For 𝜇, ] ∈ 𝐸, if either 𝜇 ⪯ ] or ] ⪯ 𝜇, then 𝜇 and ] are comparable; otherwise, they are noncomparable. ̃ 𝜔, then If 𝜇, ] ∈ 𝐸, there exists 𝜔 ∈ 𝐸 such that 𝜇 = ] + we say the Hukuhara difference of 𝜇 and ] exists, call 𝜔 the ̃ ] = 𝜔. H-difference of 𝜇 and ], and denote 𝜇 − ̃ ] exists, then It is obvious that if the H-difference 𝜇 − ̃ ])∗ (𝛼) = 𝜇∗ (𝛼) − ]∗ (𝛼). ̃ ])∗ (𝛼) = 𝜇∗ (𝛼) − ]∗ (𝛼), (𝜇 − (𝜇 − Definition 3. A mapping 𝐹 : 𝐾(⊂ 𝑅𝑛 ) → 𝐸 is said to be a fuzzy mapping. Denote [𝐹(𝑥)](𝛼) = [𝐹(𝑥)∗ (𝛼), 𝐹(𝑥)∗ (𝛼)], for all 𝛼 ∈ [0, 1]. Definition 4 (see Buckley and Feuring [34]). Let 𝐹 be a fuzzy mapping from the set of real numbers 𝑅 to the set of all fuzzy numbers, and let [𝐹(𝑥)](𝛼) = [𝐹∗ (𝑥)(𝛼), 𝐹∗ (𝑥)(𝛼)]. Assume that the partial derivatives of 𝐹∗ (𝑥)(𝛼), 𝐹∗ (𝑥)(𝛼) with respect to 𝑥 ∈ 𝑅 for each 𝛼 ∈ [0, 1] exist and are denoted by 𝐹∗󸀠 (𝑥, 𝛼), 󸀠 󸀠 𝐹∗ (𝑥, 𝛼), respectively. Let Γ(𝑥, 𝛼) = [𝐹∗󸀠 (𝑥, 𝛼), 𝐹∗ (𝑥, 𝛼)] for 𝑥 ∈ 𝑅, 𝛼 ∈ [0, 1]. If Γ(𝑥, 𝛼) defines the 𝛼-cut of a fuzzy number for each 𝑥 ∈ 𝑅, then 𝐹(𝑥) is said to be differentiable and is 󸀠 written as (𝑑𝐹/𝑑𝑥)[𝛼] = [𝐹∗󸀠 (𝑥, 𝛼), 𝐹∗ (𝑥, 𝛼)], for all 𝑥 ∈ 𝑅, 𝛼 ∈ [0, 1]. Definition 5 (see Panigrahi et al. [4]). Let 𝐹 : 𝐾(⊂ 𝑅𝑛 ) → 𝐸 be a fuzzy mapping, where 𝐾 ⊂ 𝑅𝑛 is an open set. Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ 𝐾. Let 𝐷𝑥𝑖 , (𝑖 = 1, 2, . . . , 𝑛) stand for the “partial differentiation” with respect to the 𝑖th variable 𝑥𝑖 . Assume that, for all 𝛼 ∈ [0, 1], 𝐹∗ (𝑥)(𝛼), 𝐹∗ (𝑥)(𝛼) have continuous partial derivatives so that 𝐷𝑥𝑖 𝐹∗ (𝑥, 𝛼), 𝐷𝑥𝑖 𝐹∗ (𝑥, 𝛼) are continuous. Define ∗

𝐷𝑥𝑖 𝐹 (𝑥, 𝛼) = [𝐷𝑥𝑖 𝐹∗ (𝑥, 𝛼) , 𝐷𝑥𝑖 𝐹 (𝑥, 𝛼)] , for 𝑖 = 1, 2, . . . , 𝑛, 𝛼 ∈ [0, 1] .

(4)

If each 𝑖 = 1, 2, . . . , 𝑛, 𝐷𝑥𝑖 𝐹(𝑥, 𝛼) defines the 𝛼-cut of a fuzzy number, then 𝐹 is called differentiable at 𝑥, and it can be represented as

The addition and the scalar multiplication of fuzzy vectors 𝑈 = (𝜇1 , 𝜇2 , . . . , 𝜇𝑛 )𝑇 and 𝑉 = (]1 , ]2 , . . . , ]𝑛 )𝑇 are defined as 𝑇

̃ ]1 , 𝜇2 + ̃ ]2 , . . . , 𝜇𝑛 + ̃ ]𝑛 ) , ̃ 𝑉 = (𝜇1 + 𝑈+ 𝑇

𝑘𝑈 = (𝑘𝜇1 ; 𝑘𝜇2 ; . . . ; 𝑘𝜇𝑛 ) ,

𝑘𝜇𝑖 = {(𝑘𝜇𝑖∗ , 𝑘𝜇𝑖∗ , 𝛼) : 𝛼 ∈ [0, 1]} ,

𝑇

[𝑈]𝜆 = [[𝜇1 ]𝜆 , [𝜇2 ]𝜆 , . . . , [𝜇𝑛 ]𝜆 ] , 𝑇

[𝑈]∗𝜆 = [[𝜇1 ]∗𝜆 , [𝜇2 ]∗𝜆 , . . . , [𝜇𝑛 ]∗𝜆 ] , ∗

∗

∗ 𝑇

[𝑈]∗𝜆 = [[𝜇1 ]𝜆 , [𝜇2 ]𝜆 , . . . , [𝜇𝑛 ]𝜆 ] .

(6)

(7)

for 𝑘 > 0.

3. The Fuzzy Invex Monotone Mapping In this section, we put forward some definitions of invex monotonicity of fuzzy mappings and discuss the relationship between invexity and invex monotonicity of fuzzy mappings. Definition 6. A differentiable comparable fuzzy mapping 𝐹 : 𝐾 → 𝐸 is said to be (a) fuzzy invex with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , if and only if 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

∀𝑥, 𝑦 ∈ 𝐾;

(8)

(b) fuzzy incave with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , if and only if 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪯ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

∀𝑥, 𝑦 ∈ 𝐾;

(9)

(c) fuzzy strictly invex with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , if and only if 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ≻ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

∀𝑥, 𝑦 ∈ 𝐾;

(10)

(d) fuzzy strictly incave with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , if and only if 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ≺ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

∀𝑥, 𝑦 ∈ 𝐾;

(11)

(e) fuzzy pseudo-invex with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , if and only if 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) ⪰ 0 󳨐⇒ 𝐹 (𝑥) ⪰ 𝐹 (𝑦) ,

̃ (𝑥) = (𝐷𝑥 𝐹 (𝑥, 𝛼) , 𝐷𝑥 𝐹 (𝑥, 𝛼) , . . . , 𝐷𝑥 𝐹 (𝑥, 𝛼)) . (5) ∇𝐹 1 2 𝑛 ̃ ∇𝐹(𝑥) is said to be the gradient of the fuzzy mapping 𝐹 at 𝑥. 𝑈 is said to be an 𝑛-dimensional fuzzy vector if and only if the components of 𝑈 are composed by 𝑛 fuzzy numbers, denoted by 𝑈 = (𝜇1 , 𝜇2 , . . . , 𝜇𝑛 )𝑇 . The set of all 𝑛-dimensional fuzzy vectors is denoted by (𝐸)𝑛 . A 𝜆 level vector of fuzzy vector 𝑈 = (𝜇1 , 𝜇2 , . . . , 𝜇𝑛 )𝑇 is defined as

𝑘 ∈ 𝑅,

∀𝑥, 𝑦 ∈ 𝐾;

(12)

(f) fuzzy strictly pseudo-invex with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , if and only if 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) ⪰ 0 󳨐⇒ 𝐹 (𝑥) ≻ 𝐹 (𝑦) , ∀𝑥, 𝑦 ∈ 𝐾.

(13)

Example 7 (see Wu and Xu [6]). Let 𝐹 : 𝐾(⊂ 𝑅1 ) → 𝐸 represent the reproduction rate of some germ: 𝑡 { 2 { 𝑥 { { 2 𝐹 (𝑥) (𝑡) = {1 − 𝑡 − 𝑥 { 2 { 𝑥 { {0

𝑡 ∈ [0, 𝑥2 ] , 𝑡 ∈ (𝑥2 , 2𝑥2 ] , 𝑡 ∉ [0, 2𝑥2 ] .

(14)

4

Abstract and Applied Analysis

So, [𝐹(𝑥)]𝛼 = [𝛼𝑥2 , (2 − 𝛼)𝑥2 ], 𝛼 ∈ [0, 1]. Then there is 𝜂(𝑥, 𝑦) = 𝑥−𝑦, such that 𝐹(𝑥) is a fuzzy invex mapping, where 𝐾 = (0, ∞), 𝑥, 𝑦 ∈ 𝐾, 𝑥 represents the predicted quantity, and 𝑡 represents the actual reproduction quantity. 𝛼

From ∇𝐹∗ (𝑦)(𝛼) ≥ ∇𝐹∗ (𝑦)(𝛼) > 0, there is 𝑇

[∇𝐹∗ (𝑦) (𝛼)] (∇𝐹∗ (𝑦) (𝛼) − ∇𝐹∗ (𝑦) (𝛼)) ≥ 0; so,

2

Example 8. Consider the fuzzy mapping [𝐹(𝑥)] = [𝛼𝑥 , (2 − 𝛼)𝑥2 ], 𝛼 ∈ [0, 1]. Then, there is an 𝜂(𝑥, 𝑦) = 𝑥5 − 𝑦5 such that 𝐹(𝑥) is a fuzzy pseudoinvex mapping, where 𝐾 = (0, ∞), 𝑥, 𝑦 ∈ 𝐾. Remark 9. For an invex fuzzy mapping, there must exist 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , such that 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂(𝑥, 𝑦) ∇𝐹 (𝑦) +

∀𝑥, 𝑦 ∈ 𝐾,

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

≥ [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) . 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) =

holds.

𝐹 (𝑥) ⪯ 𝐹 (𝑦) .

(17)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

Proof. Since 𝐹 is a comparable fuzzy mapping, then for all 𝑥, 𝑦 ∈ 𝑅𝑛 , there is (16)

(24)

Hence,

(15)

𝐹 (𝑥) ⪰ 𝐹 (𝑦)

(23)

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) ≤

or

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

(25)

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) = 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

Without loss of generality, suppose that 𝐹 (𝑥) ⪰ 𝐹 (𝑦) .

(18)

That is, 𝑇

𝐹∗ (𝑥) (𝛼) ≥ 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝐹∗ (𝑦) (𝛼) .

Thus, for all 𝛼 ∈ [0, 1], 𝐹∗ (𝑥) (𝛼) ≥ 𝐹∗ (𝑦) (𝛼) , 𝐹∗ (𝑥) (𝛼) ≥ 𝐹∗ (𝑦) (𝛼) .

From (22) and (26), it follows that (19)

𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

𝜂 (𝑥, 𝑦) =

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

(28)

Thus, (20)

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) =

Thus, ∗

𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) (𝛼) =

(27)

× ∇𝐹∗ (𝑦) (𝛼) .

× ∇𝐹∗ (𝑦) (𝛼) . 𝑇

∀𝑥, 𝑦 ∈ 𝐾.

(ii) If ∇𝐹∗ (𝑦)(𝛼) < 0, then take

If ∇𝐹(𝑦) = 0, for any 𝜂(𝑥, 𝑦) : 𝐾 × 𝐾 → 𝑅𝑛 , the result holds. If ∇𝐹(𝑦) ≠ 0, for given 𝛼 ∈ [0, 1]. (1) When 𝐹∗ (𝑥)(𝛼) − 𝐹∗ (𝑦)(𝛼) ≥ 𝐹∗ (𝑥)(𝛼) − 𝐹∗ (𝑦)(𝛼). (i) If ∇𝐹∗ (𝑦)(𝛼) > 0, then take 𝜂 (𝑥, 𝑦) =

(26)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

(29)

𝑇

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

(21)

From ∇𝐹∗ (𝑦)(𝛼) < 0 and ∇𝐹∗ (𝑦)(𝛼) ≥ ∇𝐹∗ (𝑦)(𝛼), there is 𝑇

[∇𝐹∗ (𝑦) (𝛼)] (∇𝐹∗ (𝑦) (𝛼) − ∇𝐹∗ (𝑦) (𝛼)) ≤ 0;

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) ≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) ;

(30)

so, 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

that is, 𝑇

𝐹∗ (𝑥) (𝛼) ≥ 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝐹∗ (𝑦) (𝛼) .

(22)

𝑇

≤ [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) .

(31)

Abstract and Applied Analysis

5 From ∇𝐹∗ (𝑦)(𝛼) + 𝐶 ≤ 0 and ∇𝐹∗ (𝑦)(𝛼) ≥ ∇𝐹∗ (𝑦)(𝛼), there is

Hence, 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) =

𝑇 [∇𝐹∗ (𝑦) (𝛼) + 𝐶] (∇𝐹∗ (𝑦) (𝛼) − ∇𝐹∗ (𝑦) (𝛼)) ≤ 0; (39)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

so,

𝑇

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) ≤

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

𝑇

[∇𝐹∗ (𝑦) (𝛼) + 𝐶] ∇𝐹∗ (𝑦) (𝛼) (32)

𝑇

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) Hence,

𝑇

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼)

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) ≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

=

That is, 𝑇

𝐹∗ (𝑥) (𝛼) ≥ 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝐹∗ (𝑦) (𝛼) .

∀𝑥, 𝑦 ∈ 𝐾.

(34)

𝑇

𝐶 ∇𝐹∗ (𝑦) (𝛼) ≤ 0

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼) + 𝐶] ∇𝐹∗ (𝑦) (𝛼)

(iii) If ∇𝐹∗ (𝑦)(𝛼) is indefinite, there is a vector 𝐶 ∈ 𝑅𝑛 , such that ∇𝐹∗ (𝑦) (𝛼) + 𝐶 ≤ 0,

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

≤

̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) × [∇𝐹∗ (𝑦) (𝛼) + 𝐶] ∇𝐹∗ (𝑦) (𝛼)

(33)

From (29) and (33), it follows that 𝑇

(40)

≤ [∇𝐹∗ (𝑦) (𝛼) + 𝐶] ∇𝐹∗ (𝑦) (𝛼) .

(41)

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) +

(35)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

× 𝐶𝑇 ∇𝐹∗ (𝑦) (𝛼)

holds. Take 𝜂 (𝑥, 𝑦) =

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

(36)

× (∇𝐹∗ (𝑦) (𝛼) + 𝐶) .

That is,

Thus,

𝑇

𝐹∗ (𝑥) (𝛼) ≥ 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝐹∗ (𝑦) (𝛼) .

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) =

From (38) and (42), it follows that

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼) + 𝐶] ∇𝐹∗ (𝑦) (𝛼) = 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) +

(42)

(37)

∀𝑥, 𝑦 ∈ 𝐾.

(43)

(2) When 𝐹∗ (𝑥)(𝛼) − 𝐹∗ (𝑦)(𝛼) ≤ 𝐹∗ (𝑥)(𝛼) − 𝐹∗ (𝑦)(𝛼). (i) If ∇𝐹∗ (𝑦)(𝛼) > 0, then take

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

𝜂 (𝑥, 𝑦) =

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

(44)

∗

× ∇𝐹 (𝑦) (𝛼) .

× 𝐶𝑇 ∇𝐹∗ (𝑦) (𝛼) Thus,

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼)

That is, 𝑇

𝐹∗ (𝑥) (𝛼) ≥ 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝐹∗ (𝑦) (𝛼) .

(38)

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

(45)

6

Abstract and Applied Analysis

On the other hand,

On the other hand, 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼)

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) ∗

=

∗

𝐹 (𝑥) (𝛼) − 𝐹 (𝑦) (𝛼)

=

[∇𝐹∗

(𝑦) (𝛼)]

𝑇

∇𝐹∗

(𝑦) (𝛼)

∗

𝐹 (𝑥) (𝛼) − 𝐹 (𝑦) (𝛼)

≤

[∇𝐹∗

(𝑦) (𝛼)]

𝑇

∇𝐹∗

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

𝑇

× [∇𝐹∗ (𝑦) (𝛼)] 𝐹∗ (𝑦) (𝛼) ∗

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

≤

(46)

(𝑦) (𝛼)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼)] 𝐹∗ (𝑦) (𝛼)

𝑇

× [∇𝐹∗ (𝑦) (𝛼)] 𝐹∗ (𝑦) (𝛼)

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) ;

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) that is,

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼)

That is,

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) . 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) ≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

(47)

∀𝑥, 𝑦 ∈ 𝐾.

∀𝑥, 𝑦 ∈ 𝐾.

(48)

∇𝐹∗ (𝑦) (𝛼) + 𝐶 ≤ 0, 𝐶𝑇 ∇𝐹∗ (𝑦) (𝛼) ≤ 0

𝑇

(55)

holds. Take

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

(54)

(iii) If ∇𝐹∗ (𝑦)(𝛼) is indefinite, there is a vector 𝐶 ∈ 𝑅𝑛 , such that

(ii) If ∇𝐹∗ (𝑦)(𝛼) < 0, then take 𝜂 (𝑥, 𝑦) =

(53)

From (51) and (53), it follows that 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

From (45) and (47), it follows that 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

(52)

(49)

𝜂 (𝑥, 𝑦) =

× ∇𝐹∗ (𝑦) (𝛼) .

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

(56)

× (∇𝐹∗ (𝑦) (𝛼) + 𝐶) .

Thus, Thus, 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) =

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

=

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

(50)

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

× [∇𝐹∗ (𝑦) (𝛼) + 𝐶] ∇𝐹∗ (𝑦) (𝛼)

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

= 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

+

That is,

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) 𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

× 𝐶𝑇 ∇𝐹∗ (𝑦) (𝛼) 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) ≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

(51)

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) ≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

(57)

Abstract and Applied Analysis

7 Definition 11. The function 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 is said to be a skew function if

That is, 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) ≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

𝜂 (𝑥, 𝑦) = −𝜂 (𝑦, 𝑥) ,

(58)

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼)

𝑦 + 𝜆𝜂 (𝑥, 𝑦) ∈ 𝐾.

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) [∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼) 𝑇

≤

𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼)

(59)

𝑇

[∇𝐹∗ (𝑦) (𝛼)] ∇𝐹∗ (𝑦) (𝛼)

Theorem 13. If a differentiable fuzzy mapping 𝐹 : 𝐾 → 𝐸 is invex on 𝐾 with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 and 𝜂 is a skew function. Then, ∇𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy invex monotone with respect to the same 𝜂. Proof. Let 𝐹 be invex on 𝐾, then there exists 𝜂(𝑥, 𝑦) ∈ 𝑅𝑛 , such that

𝑇

× [∇𝐹∗ (𝑦) (𝛼) + 𝐶] 𝐹∗ (𝑦) (𝛼) < 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) ;

𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

∀𝑥, 𝑦 ∈ 𝐾.

that is, 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼)

(60)

𝑇

𝐹∗ (𝑥) (𝛼) ≥ 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝐹∗ (𝑦) (𝛼) , 𝑇

𝐹∗ (𝑥) (𝛼) ≥ 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝐹∗ (𝑦) (𝛼) ,

From (58) and (60), it follows that 𝑇 ̃ 𝐹 (𝑦) , 𝐹 (𝑥) ⪰ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

∀𝑥, 𝑦 ∈ 𝐾.

(61)

𝑇 ̃ 𝐹 (𝑥) . 𝐹 (𝑦) ⪰ 𝜂 (𝑦, 𝑥) ∇𝐹 (𝑥) +

Definition 10. A comparable fuzzy mapping 𝐹 : 𝐾 → (𝐸)𝑛 is said to be (a) fuzzy invex monotone on 𝐾, if ∃𝜂 : 𝐾×𝐾 → 𝑅𝑛 such that for any 𝑥, 𝑦 ∈ 𝐾, 𝑇

𝜂 (𝑦, 𝑥) 𝐹 (𝑦) ⪰ 𝜂 (𝑦, 𝑥) 𝐹 (𝑥) ;

(62)

(b) fuzzy pseudo-invex monotone on 𝐾, if ∃𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 such that for any 𝑥, 𝑦 ∈ 𝐾, 𝑇

𝑇

𝜂 (𝑦, 𝑥) 𝐹 (𝑦) ⪰ 0 󳨐⇒ 𝜂 (𝑦, 𝑥) 𝐹 (𝑥) ⪰ 0;

(63)

𝜂 (𝑦, 𝑥) 𝐹 (𝑦) ≻ 𝜂 (𝑦, 𝑥) 𝐹 (𝑥) ;

(64)

(d) fuzzy strictly pseudo-invex monotone if and only if ∃𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 such that for any 𝑥, 𝑦 ∈ 𝐾, 𝑥 ≠ 𝑦, 𝑇

𝑇

𝜂 (𝑦, 𝑥) 𝐹 (𝑦) ⪰ 0 󳨐⇒ 𝜂 (𝑦, 𝑥) 𝐹 (𝑥) ≻ 0.

(70)

That is, there are 𝑇

𝐹∗ (𝑦) (𝛼) ≥ 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) + 𝐹∗ (𝑥) (𝛼) , 𝑇

𝐹∗ (𝑦) (𝛼) ≥ 𝜂(𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) + 𝐹∗ (𝑥) (𝛼) ,

(71)

for all 𝛼 ∈ [0, 1]. From (69) and (71), it follows that 𝑇

𝑇

𝑇

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) ≤ 0, 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼) + 𝜂(𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) ≤ 0.

(72)

As 𝜂(𝑥, 𝑦) + 𝜂(𝑦, 𝑥) = 0, then

(c) fuzzy strictly invex monotone on 𝐾, if ∃𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 such that for any 𝑥, 𝑦 ∈ 𝐾, 𝑥 ≠ 𝑦, 𝑇

(69)

for all 𝛼 ∈ [0, 1]. By changing 𝑥 for 𝑦,

This completes the proof.

𝑇

(68)

That is, there are

≤ 𝐹∗ (𝑥) (𝛼) − 𝐹∗ (𝑦) (𝛼) .

𝑇

(67)

𝐾 is said to be an invex set with respect to 𝜂 if 𝐾 is invex at each 𝑦 ∈ 𝐾.

𝑇

× [∇𝐹∗ (𝑦) (𝛼) + 𝐶] ∇𝐹∗ (𝑦) (𝛼)

(66)

Definition 12. Let 𝑦 ∈ 𝐾, 𝐾 is said to be invex at 𝑦 with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 if, for each 𝑥 ∈ 𝐾, 𝜆 ∈ [0, 1],

On the other hand,

=

∀𝑥, 𝑦 ∈ 𝐾.

(65)

𝑇

𝑇

𝑇

𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑦) (𝛼) ≥ 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) , 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑦) (𝛼) ≥ 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) .

(73)

Therefore, from (73), there is 𝑇

𝑇

𝜂 (𝑦, 𝑥) ∇𝐹 (𝑦) ⪰ 𝜂 (𝑦, 𝑥) ∇𝐹 (𝑥) .

(74)

8

Abstract and Applied Analysis

Corollary 14. If a differentiable fuzzy mapping 𝐹 : 𝐾 → 𝐸 is invex on 𝐾 with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 and 𝜂 is a skew function. Then, ∇𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy pseudo-invex monotone with respect to the same 𝜂.

On the other hand, for other 𝛼 ∈ [0, 1],

𝑇

𝑇

𝑇

𝑇

𝑇

𝑇

𝑇

𝜂 (𝑦, 𝑥) ∇𝐹 (𝑦) ≻ 𝜂 (𝑦, 𝑥) ∇𝐹 (𝑥) . ∀𝑥, 𝑦 ∈ 𝐾.

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑦) (𝛼) ≥ 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) , 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑦) (𝛼) ≥ 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) ,

(85)

(86)

(75)

Thus, 𝑇

𝑇

Therefore,

Proof. From Theorem 13, it follows that 𝜂 (𝑦, 𝑥) ∇𝐹 (𝑦) ⪰ 𝜂 (𝑦, 𝑥) ∇𝐹 (𝑥) ,

𝑇

𝜂 (𝑦, 𝑥) ∇𝐹 (𝑦) ⪰ 𝜂 (𝑦, 𝑥) ∇𝐹 (𝑥) .

(76)

Theorem 16. If a differentiable fuzzy mapping 𝐹 : 𝐾 → 𝐸 is strictly pseudo-invex on 𝐾 with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 and 𝜂 is a skew function. Then, ∇𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy strictly pseudo-invex monotone on 𝐾 with respect to the same 𝜂. Proof. Let 𝐹 be a fuzzy strictly pseudo-invex, then there exists 𝜂(𝑥, 𝑦) ∈ 𝑅𝑛 , such that for any 𝑥, 𝑦 ∈ 𝐾, 𝑥 ≠ 𝑦,

for all 𝛼 ∈ [0, 1]. If 𝜂(𝑦, 𝑥)𝑇 ∇𝐹(𝑥) ⪰ 0,

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) ⪰ 0 󳨐⇒ 𝐹 (𝑥) ≻ 𝐹 (𝑦) . We need to show that there exists 𝜂(𝑥, 𝑦) ∈ 𝑅 , such that

𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) ≥ 0, 𝑇

(87)

𝑛

(77)

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) ⪰ 0

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) ≥ 0,

𝑇

(88)

󳨐⇒ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑥) ≻ 0,

for all 𝛼 ∈ [0, 1]. Thus, from (76) and (77), 𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑦) (𝛼) ≥ 0, 𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑦) (𝛼) ≥ 0

(78)

for all 𝑥, 𝑦 ∈ 𝐾, 𝑥 ≠ 𝑦. By contradiction, suppose that 𝜂(𝑥, 𝑦)𝑇 ∇𝐹(𝑥) ⪯ 0, then there exists some 𝛼0 ∈ [0, 1], such that 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑥) (𝛼0 ) ≤ 0,

(89)

or

holds. Therefore, 𝜂(𝑦, 𝑥)𝑇 ∇𝐹(𝑦) ⪰ 0.

𝑇

Theorem 15. If a differentiable fuzzy mapping 𝐹 : 𝐾 → 𝐸 is strictly invex on 𝐾 with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 and 𝜂 is a skew function. Then, ∇𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy strictly invex monotone on 𝐾 with respect to the same 𝜂. Proof. Assume that 𝐹 is strictly invex on 𝐾, then there exists 𝜂(𝑥, 𝑦) ∈ 𝑅𝑛 , such that, for any 𝑥, 𝑦 ∈ 𝐾, 𝑥 ≠ 𝑦, ̃ 𝐹 (𝑦) . 𝐹 (𝑥) ≻ 𝜂 (𝑥, 𝑦) ∇𝐹 (𝑦) +

(79)

𝑇

(90)

Without loss of generality, assume that 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑥) (𝛼0 ) ≤ 0.

(91)

As 𝜂(𝑥, 𝑦) + 𝜂(𝑦, 𝑥) = 0, then 𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼0 ) ≥ 0.

(92)

Since 𝐹 is strictly pseudo-invex on 𝐾, then for 𝛼0 ∈ [0, 1], there is 𝑇

Thus, there exists some 𝛼0 ∈ [0, 1], such that 𝐹∗ (𝑥) (𝛼0 ) > 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼0 ) + 𝐹∗ (𝑦) (𝛼0 ) ,

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑥) (𝛼0 ) ≤ 0.

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼0 ) ≥ 0 (80)

󳨐⇒ 𝐹∗ (𝑥) (𝛼0 ) ≤ 𝐹∗ (𝑦) (𝛼0 ) ,

(93)

which is a contradiction.

or 𝑇

𝐹∗ (𝑥) (𝛼0 ) > 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼0 ) + 𝐹∗ (𝑦) (𝛼0 ) .

(81)

For 𝛼0 ∈ [0, 1], without loss of generality, suppose that 𝑇

𝐹∗ (𝑥) (𝛼0 ) > 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼0 ) + 𝐹∗ (𝑦) (𝛼0 ) .

(82)

By changing 𝑥 for 𝑦,

Proof. As 𝐹 is a strictly pseudo monotone with respect to 𝜂 on 𝐾, then for any 𝛼 ∈ [0, 1], 𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) ≥ 0 𝑇

𝐹∗ (𝑦) (𝛼0 ) > 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼0 ) + 𝐹∗ (𝑥) (𝛼0 ) .

(83)

By (82), (83) and since 𝜂 is a skew function, we have 𝑇

Theorem 17. Let 𝐹 : 𝐾 → (𝐸)𝑛 be a fuzzy strictly pseudo monotone mapping on 𝐾 with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 ; then, 𝐹 is a fuzzy pseudo monotone with respect to 𝜂 on 𝐾.

𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑦) (𝛼0 ) > 𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼0 ) .

󳨐⇒ 𝐹∗ (𝑥) (𝛼) ≤ 𝐹∗ (𝑦) (𝛼) , 𝑇

𝜂 (𝑦, 𝑥) ∇𝐹∗ (𝑥) (𝛼) ≥ 0 (84)

󳨐⇒ 𝐹∗ (𝑥) (𝛼) ≤ 𝐹∗ (𝑦) (𝛼) .

(94)

Abstract and Applied Analysis

9

𝐹∗ (𝑥)(𝛼), 𝐹∗ (𝑥)(𝛼) is strictly pseudo monotone with respect to 𝜂 on 𝐾. Thus, 𝐹∗ (𝑥)(𝛼), 𝐹∗ (𝑥)(𝛼) also is pseudo monotone with respect to 𝜂 on 𝐾; that is, 𝐹 is pseudo monotone with respect to 𝜂 on 𝐾.

̃ ̃ ̃ ∈ (0, 1), (100) From 𝜂(𝑦 + 𝜆𝜂(𝑥, 𝑦), 𝑦) = 𝜆𝜂(𝑥, 𝑦) and 𝜆 becomes

Theorem 18. Let 𝐹 : 𝐾 → 𝐸 be a differentiable mapping, and suppose that

This contradicts 𝜂(𝑥, 𝑦)𝑇 ∇𝐹∗ (𝑦)(𝛼) ⪰ 0.

𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼0 ) < 0.

(101)

Theorem 19. Let 𝐹 : 𝐾 → 𝐸 be a differentiable mapping, and suppose that

(i) 𝜂 satisfies the following conditions: (a) 𝜂(𝑦, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = −𝑡𝜂(𝑥, 𝑦), (b) 𝜂(𝑥, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = (1 − 𝑡)𝜂(𝑥, 𝑦);

(i) 𝜂 satisfies the following conditions: (a) 𝜂(𝑦, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = −𝑡𝜂(𝑥, 𝑦), (b) 𝜂(𝑥, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = (1 − 𝑡)𝜂(𝑥, 𝑦); (a) 𝜂(𝑦, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = −𝑡𝜂(𝑥, 𝑦), (b) 𝜂(𝑥, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = (1 − 𝑡)𝜂(𝑥, 𝑦);

(ii) 𝐾 is an invex set with respect to 𝜂; (iii) for each 𝑥 ≠ 𝑦, some 𝛼0 ∈ [0, 1], (a) 𝐹∗ (𝑦)(𝛼0 ) > 𝐹∗ (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦)𝑇 ∇𝐹∗ (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼 ) < 0, 0

(ii) 𝐾 is an invex set with respect to 𝜂; (iii) for each 𝑥 ≠ 𝑦, some 𝛼0 ∈ [0, 1],

or (b) 𝐹∗ (𝑦)(𝛼0 ) > 𝐹∗ (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦)𝑇 ∇𝐹∗ (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼0 ) < 0;

(a) 𝐹∗ (𝑦)(𝛼0 ) > 𝐹∗ (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦)𝑇 ∇𝐹∗ (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼0 ) < 0,

(iv) ∇𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy pseudo-invex monotone with respect to 𝜂 on 𝐾.

or (b) 𝐹∗ (𝑦)(𝛼0 ) > 𝐹∗ (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦)𝑇 ∇𝐹∗ (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼0 ) < 0;

Then, 𝐹 is a fuzzy pseudo-invex mapping 𝜂 on 𝐾. Proof. Set 𝑥, 𝑦 ∈ 𝐾, 𝑥 ≠ 𝑦, and 𝜂(𝑥, 𝑦)𝑇 ∇𝐹(𝑦) ⪰ 0 holds. Thus, we need to show that 𝐹(𝑥) ⪰ 𝐹(𝑦); that is, 𝐹∗ (𝑥)(𝛼) ≥ 𝐹∗ (𝑦)(𝛼) and 𝐹∗ (𝑥)(𝛼) ≥ 𝐹∗ (𝑦)(𝛼), for all 𝛼 ∈ [0, 1]. Assume the contrary, that is, 𝐹(𝑥)𝐹(𝑦). Thus, there exists some 𝛼0 ∈ [0, 1], such that 𝐹∗ (𝑥) (𝛼0 ) < 𝐹∗ (𝑦) (𝛼0 )

(95)

𝐹∗ (𝑥) (𝛼0 ) < 𝐹∗ (𝑦) (𝛼0 ) .

(96)

or

(iv) ∇𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy strictly pseudo-invex monotone with respect to 𝜂 on 𝑘. Then, 𝐹 is a fuzzy strictly pseudoinvex mapping 𝜂 on 𝑘. Proof. Let 𝑥, 𝑦 ∈ 𝐾, 𝑥 ≠ 𝑦, such that 𝜂(𝑥, 𝑦)𝑇 ∇𝐹(𝑦) ⪰ 0. Thus, we need to show that 𝐹(𝑥) ≻ 𝐹(𝑦). ̃ 𝐹(𝑦); then, there By contradiction, suppose that 𝐹(𝑥) ≻ exists some 𝛼0 ∈ [0, 1], such that

Without loss of generality, suppose that 𝐹∗ (𝑥) (𝛼0 ) < 𝐹∗ (𝑦) (𝛼0 ) .

(97)

By hypothesis (iii), 𝑇 ̃ (𝑥, 𝑦)) (𝛼 ) < 0, 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦 + 𝜆𝜂 0

(98)

̃ (𝑥, 𝑦)) (𝛼 ) < 0, × ∇𝐹 (𝑦 + 𝜆𝜂 0

(99)

̃ ∈ (0, 1). for some 𝜆 Since ∇𝐹 is a pseudo-invex monotone with respect to 𝜂, thus ̃ (𝑥, 𝑦) , 𝑦)𝑇 ∇𝐹∗ (𝑦) (𝛼 ) < 0. 𝜂 (𝑦 + 𝜆𝜂 0

𝐹∗ (𝑥) (𝛼0 ) ≤ 𝐹∗ (𝑦) (𝛼0 ) .

(103)

or

Without loss of generality, suppose that (104)

By hypothesis (iii),

𝑇

∗

(102)

𝐹∗ (𝑥) (𝛼0 ) ≤ 𝐹∗ (𝑦) (𝛼0 ) .

̃ ∈ (0, 1). for some 𝜆 It follows from (98) and (i) that ̃ (𝑥, 𝑦) , 𝑦) 𝜂 (𝑦 + 𝜆𝜂

𝐹∗ (𝑥) (𝛼0 ) ≤ 𝐹∗ (𝑦) (𝛼0 )

(100)

𝑇 ̃ (𝑥, 𝑦)) (𝛼 ) ≤ 0, 𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦 + 𝜆𝜂 0

(105)

̃ ∈ (0, 1). for some 𝜆 It follows from (i) and above inequity (105) that ̃ (𝑥, 𝑦))𝑇 𝜂 (𝑦, 𝑦 + 𝜆𝜂 ̃ (𝑥, 𝑦)) (𝛼 ) ≥ 0. × ∇𝐹 (𝑦 + 𝜆𝜂 0 ∗

(106)

10

Abstract and Applied Analysis

Since ∇𝐹 is a strictly pseudo-invex monotone with respect to 𝜂, thus ̃ (𝑥, 𝑦))𝑇 ∇𝐹∗ (𝑦) (𝛼 ) > 0. 𝜂 (𝑦, 𝑦 + 𝜆𝜂 0

𝑇

𝜂 (𝑥, 𝑦) 𝐹 (𝑦) ⪰ 0

(107)

̃ ̃ ̃ ∈ (0, 1), it follows From 𝜂(𝑦, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = −𝜆𝜂(𝑥, 𝑦) and 𝜆 that 𝑇

𝜂 (𝑥, 𝑦) ∇𝐹∗ (𝑦) (𝛼0 ) < 0. 𝑇

Since 𝐹 : 𝐾 → (𝐸)𝑛 is a fuzzy pseudo-invex monotone with respect to 𝜂, thus

𝑇

That is, there are

(108)

𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼) ≥ 0

∗

This contradicts 𝜂(𝑥, 𝑦) ∇𝐹 (𝑦)(𝛼) ≥ 0.

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑦) 𝐹∗ (𝑥) (𝛼) ≥ 0,

4. The Existence of a Solution to the Fuzzy Variational-Like Inequality

𝑇

𝑇

∀𝑥 ∈ 𝐾.

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑦) 𝐹∗ (𝑥) (𝛼) ≥ 0, for all 𝛼 ∈ [0, 1]. Therefore, 𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑥) (𝛼) < 0

(FVLI)

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼) < 0,

Definition 20. Let 𝐾 be an invex set with respect to 𝜂(𝑥, 𝑥∗ ) : 𝐾 × 𝐾 → 𝑅𝑛 . A fuzzy mapping 𝐹 : 𝐾 → (𝐸)𝑛 is called 𝜂-hemicontinuous, if for 𝑥, 𝑦 ∈ 𝐾, for all 𝛼 ∈ [0, 1], the mappings 𝑡 → 𝐹∗ (𝑦 + 𝑡𝜂(𝑥, 𝑦))(𝛼) and 𝑡 → 𝐹∗ (𝑦 + 𝑡𝜂(𝑥, 𝑦))(𝛼) are continuous at 0+ , with 𝑡 ∈ [0, 1]. Lemma 21. Let 𝐾 be a nonempty convex set in 𝑅𝑛 , and suppose that

𝑇

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼) < 0. In particular, for 𝛼0 ∈ [0, 1], 𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑥) (𝛼0 ) < 0

(i) 𝐹 : 𝐾 → (𝐸) is a fuzzy pseudo-invex monotone with respect to 𝜂 and 𝜂-hemicontinuous on 𝐾; (ii) 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 satisfies (a) 𝜂(𝑦, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = −𝜆𝜂(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝐾, 𝜆 ∈ [0, 1], (b) 𝜂(𝑥, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = (1 − 𝜆)𝜂(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝐾, 𝜆 ∈ [0, 1]; (iii) for any fixed 𝑦 ∈ 𝐾, 𝑥 → 𝜂(𝑥, 𝑦) is linear; that is, for 𝑥(𝑖) ∈ 𝐾, 𝑖 = 1, 2, . . . , 𝑛, 𝑡𝑖 ∈ [0, 1], ∑𝑛𝑖=1 𝑡𝑖 = 1, with 𝜂(∑𝑛𝑖=1 𝑡𝑖 𝑥(𝑖) , 𝑦) = ∑𝑛𝑖=1 𝑡𝑖 𝜂(𝑥(𝑖) , 𝑦). 𝑇

Then, for 𝑦 ∈ 𝐾, 𝜂(𝑥, 𝑦) 𝐹(𝑦) ≺ 0, for all 𝑥 ∈ 𝐾 if and only if 𝜂(𝑥, 𝑦)𝑇 𝐹(𝑥) ≺ 0, for all 𝑥 ∈ 𝐾. Proof. ⇒ By contradiction, suppose that there exists a 𝑥 ∈ 𝐾, such that 𝜂(𝑥, 𝑦)𝑇 𝐹(𝑥) ≺ 0. Thus, there exists some 𝛼0 ∈ [0, 1], 𝑇

(109)

(114)

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑥) (𝛼) < 0

𝑛

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑥) (𝛼0 ) < 0,

(113)

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼) ≥ 0

Let 𝐾 ⊂ 𝑅𝑛 , 𝜂(𝑥, 𝑥∗ ) : 𝐾 × 𝐾 → 𝑅𝑛 , 𝐹 : 𝐾 → (𝐸)𝑛 , the fuzzy variational-like inequality problem be: find 𝑥∗ ∈ 𝐾, 𝐹(𝑥∗ ) ∈ (𝐸)𝑛 (denoted by 𝐹(𝑥∗ ) = (𝐹1 (𝑥∗ ), 𝐹2 (𝑥∗ ), . . . , 𝐹𝑛 (𝑥∗ ))𝑛 ), such that 𝜂 (𝑥, 𝑥∗ ) 𝐹 (𝑥∗ ) ≺ 0,

(112)

󳨐⇒ 𝜂 (𝑥, 𝑦) 𝐹 (𝑥) ⪰ 0.

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼0 ) < 0.

(115)

This contradicts 𝜂(𝑥, 𝑦)𝑇 𝐹(𝑦) ≺ 0. ⇐ By contradiction, suppose that there exists a 𝑥 ∈ 𝐾, such that 𝜂(𝑥, 𝑦)𝑇 𝐹(𝑦) ≺ 0. Thus, there exists some 𝛼0 ∈ [0, 1], 𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼0 ) < 0,

(116)

or 𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼0 ) < 0.

(117)

Without loss of generality, assume that 𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼0 ) < 0.

(118)

Since 𝐾 is an invex set and by condition (a), we knowthat 𝑦 + 𝑡 (𝑥 − 𝑦) ∈ 𝐾,

𝑡 ∈ [0, 1] , 𝜂 (𝑦, 𝑦) = 0.

(119)

When 𝜂 satisfies (iii), there is

or 𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑥) (𝛼0 ) < 0.

(110)

= 𝑡𝜂 (𝑥, 𝑦) + (1 − 𝑡) 𝜂 (𝑦, 𝑦)

Without loss of generality, assume that 𝑇 ∗

𝜂 (𝑥, 𝑦) 𝐹 (𝑥) (𝛼0 ) < 0.

𝜂 (𝑦 + 𝑡 (𝑥 − 𝑦) , 𝑦)

(111)

= 𝑡𝜂 (𝑥, 𝑦) ,

𝑡 ∈ [0, 1] .

(120)

Abstract and Applied Analysis

11

Since 𝐹 is 𝜂-hemicontinuous on 𝐾, it follows that

Then, there exists some 𝛼0 ∈ [0, 1], such that

+

𝑡 󳨀→ 0 ,

𝑇

𝐹∗ (𝑦 + 𝑡 (𝑥 − 𝑦)) (𝛼0 ) 󳨀→ 𝐹∗ (𝑦) (𝛼0 ) .

𝜂 (𝑥(𝑖) , 𝑥) 𝐹∗ (𝑥) (𝛼) < 0,

(121)

(127)

𝑖 = 1, 2, . . . , 𝑛.

Therefore, Therefore,

𝑇

𝜂 (𝑥, 𝑦) 𝐹∗ (𝑦) (𝛼0 ) < 0

𝑛

𝑇

󳨐⇒ 𝜂 (𝑦 + 𝑡 (𝑥 − 𝑦) , 𝑦)

× 𝐹∗ (𝑦 + 𝑡 (𝑥 − 𝑦)) (𝛼0 ) < 0.

(123)

This contradicts 𝜂(𝑥, 𝑦)𝑇 𝐹(𝑥) ≺ 0, for all 𝑥 ∈ 𝐾. By the hypothesis of the pseudo-invex monotonicity of 𝐹 and the linearity of 𝜂, the existence theorem can be obtained. Theorem 22. Let 𝐾 be a nonempty convex set in 𝑅𝑛 , suppose that (i) 𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy pseudo-invex monotone with respect to 𝜂 and 𝜂-hemicontinuous on 𝐾; (ii) 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 satisfies (a) 𝜂(𝑦, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = −𝜆𝜂(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝐾, 𝜆 ∈ [0, 1], (b) 𝜂(𝑥, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = (1 − 𝜆)𝜂(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝐾, 𝜆 ∈ [0, 1]; (iii) for any fixed 𝑦 ∈ 𝐾, 𝑥 → 𝜂(𝑥, 𝑦) is linear; that is, for 𝑥(𝑖) ∈ 𝐾, 𝑖 = 1, 2, . . . , 𝑛, 𝑡𝑖 ∈ [0, 1], 𝑖 = 1, 2, . . . , 𝑛, for ∑𝑛𝑖=1 𝑡𝑖 = 1, with 𝜂(∑𝑛𝑖=1 𝑡𝑖 𝑥(𝑖) , 𝑦) = ∑𝑛𝑖=1 𝑡𝑖 𝜂(𝑥(𝑖) , 𝑦). Then, there exists 𝑥∗ ∈ 𝐾, such that 𝜂(𝑥, 𝑥∗ )𝑇 𝐹(𝑥∗ ) ≺ 0, for all 𝑥 ∈ 𝐾.

𝜂 (∑𝑡𝑖 𝑥(𝑖) , 𝑥) 𝐹∗ (𝑥) (𝛼) < 0.

(129)

𝜂 (𝑥, 𝑥)𝑇 𝐹∗ (𝑥) (𝛼) < 0.

(130)

𝑖=1

That is,

From 𝜂(𝑥, 𝑥) = 0, it follows that 0 < 0, which is absurd. So 𝑥 = ∑𝑛𝑖=1 𝑡𝑖 𝑥(𝑖) ∈ ⋃ 𝑌∗ (𝑥(𝑖) )(𝛼). Similarly, 𝑥 = ∑𝑛𝑖=1 𝑡𝑖 𝑥(𝑖) ∈ ⋃ 𝑌∗ (𝑥(𝑖) )(𝛼) can be proofed. Thus, conv ({𝑥(1) , 𝑥(2) , . . . , 𝑥(𝑛) }) ⊂ ⋃ 𝑌∗ (𝑥(𝑖) ) (𝛼) ,

can be obtained. Therefore, 𝑌∗ (𝑦)(𝛼), 𝑌∗ (𝑦)(𝛼) are KKM mappings. Let 𝑇

𝑌 (𝑦) = {𝑥 ∈ 𝐾 | 𝜂 (𝑦, 𝑥) 𝐹 (𝑦) ≺ 0} ,

𝑌 (𝑦) = {𝑥 ∈ 𝐾 | 𝜂 (𝑦, 𝑥) 𝐹 (𝑥) ≺ 0} , ∀𝑦 ∈ 𝐾.

∗

(134)

𝑌∗ (𝑦) (𝛼) ⊂ 𝑌∗ (𝑦) (𝛼) .

(125)

∗

As 𝑌∗ (𝑦)(𝛼), 𝑌∗ (𝑦)(𝛼) are KKM mappings, thus 𝑌 (𝑦)(𝛼), 𝑌∗ (𝑦)(𝛼) are also KKM mappings. By Lemma 21, we have

Let 𝑥(𝑖) ∈ 𝐾, 𝑖 = 1, 2, . . . , 𝑛, 𝑡𝑖 ∈ [0, 1], ∑𝑛𝑖=1 𝑡𝑖 = 1. Suppose that

𝑖=1

∀𝑦 ∈ 𝐾. (133)

∗

∀𝑦 ∈ 𝐾.

𝑥 = ∑𝑡𝑖 𝑥 ∈ ⋃ 𝑌 (𝑥 ) (𝛼) .

𝑇

𝑌∗ (𝑦) (𝛼) ⊂ 𝑌 (𝑦) (𝛼) ,

𝑌∗ (𝑦) (𝛼) = {𝑥 ∈ 𝐾 | 𝜂 (𝑦, 𝑥) 𝐹∗ (𝑥) (𝛼) < 0} ,

(𝑖)

∀𝑦 ∈ 𝐾,

Let 𝑥 ∈ 𝑌(𝑦); that is, 𝜂(𝑦, 𝑥)𝑇 𝐹(𝑥) ≺ 0. By Lemma 21, 𝜂(𝑦, 𝑥)𝑇 𝐹(𝑦) ≺ 0 holds, that is, 𝑥 ∈ 𝑌(𝑦). Thus,

𝑇

∗

𝑇

𝑌 (𝑦) = {𝑥 ∈ 𝐾 | 𝜂 (𝑦, 𝑥) 𝐹∗ (𝑦) (𝛼) < 0} ,

𝑌∗ (𝑦) (𝛼) = {𝑥 ∈ 𝐾 | 𝜂 (𝑦, 𝑥) 𝐹∗ (𝑥) (𝛼) < 0} ,

𝑇

(132)

That is, for all 𝛼 ∈ [0, 1],

(124)

That is, for all 𝛼 ∈ [0, 1], there are

∀𝑦 ∈ 𝐾,

∀𝑦 ∈ 𝐾.

𝑌∗ (𝑦) = {𝑥 ∈ 𝐾 | 𝜂 (𝑦, 𝑥) 𝐹∗ (𝑦) (𝛼) < 0} ,

𝑇

(131)

conv ({𝑥(1) , 𝑥(2) , . . . , 𝑥(𝑛) }) ⊂ ⋃ 𝑌∗ (𝑥(𝑖) ) (𝛼)

Proof. Let

(𝑖)

𝑇

𝑛

𝑇

̂ 𝑦) 𝐹∗ (𝑥) ̂ (𝛼0 ) < 0. 𝜂 (𝑥,

(128)

𝑖=1

From (iii), for fixed 𝑥 ∈ 𝐾,

Let 𝑥̂ = 𝑦 + 𝑡(𝑥 − 𝑦), then

𝑛

𝑇

∑𝑡𝑖 𝜂 (𝑥(𝑖) , 𝑥) 𝐹∗ (𝑥) (𝛼) < 0.

(122)

(126)

∗

⋂ 𝑌∗ (𝑦) (𝛼) = ⋂ 𝑌 (𝑦) (𝛼) , 𝑦∈𝐾

𝑦∈𝐾

⋂ 𝑌∗ (𝑦) (𝛼) = ⋂ 𝑌∗ (𝑦) (𝛼) .

𝑦∈𝐾

𝑦∈𝐾

(135)

12

Abstract and Applied Analysis ∗

As 𝑌∗ (𝑦)(𝛼), 𝑌 (𝑦)(𝛼) are closed for every 𝑦 ∈ 𝐾 and 𝐾 is ∗ a bounded set, then 𝑌∗ (𝑦)(𝛼), 𝑌 (𝑦)(𝛼) are bounded; hence, ∗ 𝑌∗ (𝑦)(𝛼), 𝑌 (𝑦)(𝛼) are compact. Therefore,

That is, there are 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) ≥ 0 𝑇

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼) > 0,

∗

⋂ 𝑌∗ (𝑦) (𝛼) = ⋂ 𝑌 (𝑦) (𝛼) ≠ 0, 𝑦∈𝐾

𝑦∈𝐾

⋂ 𝑌∗ (𝑦) (𝛼) = ⋂ 𝑌∗ (𝑦) (𝛼) ≠ 0.

𝑦∈𝐾

𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) ≥ 0

(136)

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼) > 0,

𝑦∈𝐾

Hence, there exists 𝑥∗ ∈ 𝐾, such that 𝑇

∀𝑦 ∈ 𝐾,

𝑇

∀𝑦 ∈ 𝐾.

𝜂 (𝑦, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) < 0, 𝜂 (𝑦, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) < 0,

(141)

(137)

for all 𝛼 ∈ [0, 1]. Since 𝜂 is a skew function, there is 𝜂(𝑥, 𝑥∗ ) + 𝜂(𝑥∗ , 𝑥) = 0. Thus 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) ≥ 0 𝑇

󳨐⇒ 𝜂 (𝑥∗ , 𝑥) 𝐹∗ (𝑥) (𝛼) < 0,

That is, 𝑇

𝜂 (𝑦, 𝑥∗ ) 𝐹 (𝑥∗ ) ≺ 0,

∀𝑦 ∈ 𝐾.

(138)

The pseudo-invex monotonicity of 𝐹 assures us of the existence of a solution to (FVLI), but not the uniqueness of such a solution. To achieve this, we assume the strictly pseudo-invex monotonicity of 𝐹. Theorem 23. Let 𝐾 be a nonempty convex set in 𝑅𝑛 , and suppose that (i) 𝐹 : 𝐾 → (𝐸)𝑛 is strictly fuzzy pseudo-invex monotone with respect to 𝜂 and 𝜂-hemicontinuous on 𝐾; 𝑛

(ii) 𝜂 : 𝐾 × 𝐾 → 𝑅 is a skew function and satisfies (a) 𝜂(𝑦, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = −𝜆𝜂(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝐾, 𝜆 ∈ [0, 1], (b) 𝜂(𝑥, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = (1 − 𝜆)𝜂(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝐾, 𝜆 ∈ [0, 1]; (iii) for any fixed 𝑦 ∈ 𝐾, 𝑥 → 𝜂(𝑥, 𝑦) is linear; that is, for 𝑥(𝑖) ∈ 𝐾, 𝑖 = 1, 2, . . . , 𝑛, 𝑡𝑖 ∈ [0, 1], ∑𝑛𝑖=1 𝑡𝑖 = 1, with 𝜂(∑𝑛𝑖=1 𝑡𝑖 𝑥(𝑖) , 𝑦) = ∑𝑛𝑖=1 𝑡𝑖 𝜂(𝑥(𝑖) , 𝑦). ∗ 𝑇

∗

∗

Then, there exists unique 𝑥 ∈ 𝐾, such that 𝜂(𝑥,𝑥 ) 𝐹(𝑥 ) ≺ 0, for all 𝑥 ∈ 𝐾. Proof. From Theorem 17, 𝐹 is a fuzzy pseudo-invex monotone. Also from Theorem 22, there exists a solution for problem (FVLI). Suppose that (FVLI) has two distinct solutions 𝑥∗ , 𝑥. Then, 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹 (𝑥∗ ) ≺ 0, 𝑇

𝜂 (𝑥∗ , 𝑥) 𝐹 (𝑥) ≺ 0.

(139)

Since 𝐹 is a fuzzy strictly pseudo-invex monotone on 𝐾, then 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹 (𝑥∗ ) ⪰ 0 𝑇

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹 (𝑥) ≻ 0.

(140)

𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) ≥ 0

(142)

𝑇

󳨐⇒ 𝜂 (𝑥∗ , 𝑥) 𝐹∗ (𝑥) (𝛼) < 0. That is, 𝑇

𝜂 (𝑥∗ , 𝑥) 𝐹 (𝑥) ≺ 0,

(143)

which contradicts 𝜂(𝑥∗ , 𝑥)𝑇 𝐹(𝑥) ≺ 0.

5. Qualities of the Fuzzy Variational-Like Inequality A well-known fact in mathematical programming is that the variational inequality problem has a close relationship with the optimization problem. Similarly, the fuzzy variational inequality problem also has a close relationship with the fuzzy optimization problem. Consider the unconstrained fuzzy vector optimization problem: min 𝐹 (𝑥) , 𝑥∈𝐾

(FP)

where 𝐾 is a subset of n dimension Euclidean space 𝑅𝑛 , 𝐹 : 𝐾 → 𝐸 is a fuzzy mapping. A point 𝑥 ∈ 𝐾 is called a feasible point. If 𝑥∗ ∈ 𝐾 and no 𝑥 ∈ 𝐾, 𝐹(𝑥) ≺ 𝐹(𝑥∗ ), then 𝑥∗ is called an optimal solution, a global optimal solution, or simply a solution to the problem (FP). If 𝑥∗ ∈ 𝐾 and there exists an 𝛿-neighborhood 𝑁𝛿 (𝑥∗ ) around 𝑥∗ , such that for no 𝑥( ≠ 𝑥∗ ) ∈ 𝐾 ∩ 𝑁𝛿 (𝑥∗ ), 𝐹(𝑥) ≺ 𝐹(𝑥∗ ), then 𝑥∗ is called a local optimal solution. Similarly, if 𝑥∗ ∈ 𝐾 and there exists an 𝛿-neighborhood 𝑁𝛿 (𝑥∗ ) around 𝑥∗ , such that for no 𝑥 ∈ 𝐾 ∩ 𝑁𝛿 (𝑥∗ ), 𝐹(𝑥) ⪯ 𝐹(𝑥∗ ), then 𝑥∗ is called a strict local optimal solution. The following lemmas and theorems discuss the properties of fuzzy variational inequality. Lemma 24 (see Wu and Xu [6]). Let 𝐹 : 𝐾 → 𝐸 be a fuzzy differentiable pseudoinvex mapping. If (𝑥∗ , ∇𝐹(𝑥∗ )) is a solution of (FVLI), then 𝑥∗ is a local optimal solution of (FP).

Abstract and Applied Analysis

13

Lemma 25 (see Wu and Xu [6]). Let 𝐹 : 𝐾 → 𝐸 be a fuzzy differentiable strictly pseudoinvex mapping. If (𝑥∗ , ∇𝐹(𝑥∗ )) is a solution of (FVLI), then 𝑥∗ is a strictly local optimal solution of (FP).

Theorem 28 gives the equivalent relationship of two fuzzy variational-like inequalities.

Theorem 26. Let 𝐹 : 𝐾 → 𝐸 be a differentiable fuzzy mapping, and suppose that

(i) 𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy pseudo-invex monotone with respect to 𝜂 and 𝜂-hemicontinuous on 𝐾; (ii) 𝜂 satisfies the following conditions:

Theorem 28. Let 𝐾 be an invex set with respect to 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 . Suppose that

(i) 𝜂 satisfies the following conditions: (a) 𝜂(𝑦, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = −𝑡𝜂(𝑥, 𝑦), (b) 𝜂(𝑥, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = (1 − 𝑡)𝜂(𝑥, 𝑦);

(a) 𝜂(𝑦, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = −𝜆𝜂(𝑥, 𝑦), (b) 𝜂(𝑥, 𝑦 + 𝜆𝜂(𝑥, 𝑦)) = (1 − 𝜆)𝜂(𝑥, 𝑦). Then, 𝑥∗ ∈ 𝐾 satisfies 𝜂(𝑥, 𝑥∗ )𝑇 𝐹(𝑥∗ ) ≺ 0, for all 𝑥 ∈ 𝐾 if and only if it satisfies 𝜂(𝑥, 𝑥∗ )𝑇 𝐹(𝑥) ≺ 0, for all 𝑥 ∈ 𝐾.

(ii) 𝐾 is an invex set with respect to 𝜂; (iii) for each 𝑥 ≠ 𝑦, some 𝛼0 ∈ [0, 1], (a) 𝐹∗ (𝑦)(𝛼0 ) > 𝐹∗ (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦)𝑇 ∇𝐹∗ (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼 ) < 0, 0

Proof. ⇒ By contradiction, suppose that there exists an 𝑥 ∈ 𝐾, such that 𝜂(𝑥, 𝑥∗ )𝑇 𝐹(𝑥) ≺ 0. Thus, there exists some 𝛼0 ∈ [0, 1], 𝑇

or

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼0 ) < 0, ∗

∗

𝑇

∗

(b) 𝐹 (𝑦)(𝛼0 ) > 𝐹 (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦) ∇𝐹 (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼 ) < 0;

or

0

𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼0 ) < 0.

𝑛

(iv) ∇𝐹 : 𝐾 → (𝐸) is fuzzy pseudo-invex monotone with respect to 𝜂 on 𝑘. If (𝑥∗ , ∇𝐹(𝑥∗ )) is a solution of (FVLI), then 𝑥∗ is a local optimal solution of (FP). Proof. From Theorem 18, we know that 𝐹 is a fuzzy pseudo invex. By Lemma 24, we can show it. Theorem 27. Let 𝐹 : 𝐾 → 𝐸 be a differentiable fuzzy mapping, and suppose that (i) 𝜂 satisfies the following conditions:

(144)

(145)

Without loss of generality, assume that 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼0 ) < 0.

(146)

Since 𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy pseudo-invex monotone with respect to 𝜂, thus 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹 (𝑥∗ ) ⪰ 0 𝑇

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹 (𝑥) ⪰ 0.

(147)

That is, for any 𝛼 ∈ [0, 1], 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) ≥ 0

(a) 𝜂(𝑦, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = −𝑡𝜂(𝑥, 𝑦), (b) 𝜂(𝑥, 𝑦 + 𝑡𝜂(𝑥, 𝑦)) = (1 − 𝑡)𝜂(𝑥, 𝑦);

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼) ≥ 0, 𝑇

(ii) 𝐾 is an invex set with respect to 𝜂;

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) ≥ 0

(iii) for each 𝑥 ≠ 𝑦, some 𝛼0 ∈ [0, 1],

(148)

𝑇

(a) 𝐹∗ (𝑦)(𝛼0 ) > 𝐹∗ (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦)𝑇 ∇𝐹∗ (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼 ) < 0, 0

or

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼) ≥ 0. Therefore, 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼) < 0 𝑇

∗

∗

𝑇

∗

(b) 𝐹 (𝑦)(𝛼0 ) > 𝐹 (𝑥)(𝛼0 ) implies 𝜂(𝑥, 𝑦) ∇𝐹 (𝑦 + ̃ 𝜆𝜂(𝑥, 𝑦))(𝛼0 ) < 0; (iv) ∇𝐹 : 𝐾 → (𝐸)𝑛 is fuzzy strictly pseudo-invex monotone with respect to 𝜂 on 𝑘. If (𝑥∗ , ∇𝐹(𝑥∗ )) is a solution of (FVLI), then 𝑥∗ is a strictly local optimal solution of (FP). Proof. From Theorem 19, we know that 𝐹 is a fuzzy strictly pseudo-invex. By Lemma 25, we can show it.

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) < 0, 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼) < 0

(149)

𝑇

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼) < 0. In particularly, for 𝛼0 ∈ [0, 1], there is 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥) (𝛼0 ) < 0 𝑇

󳨐⇒ 𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼0 ) < 0. This contradicts 𝜂(𝑥, 𝑥∗ )𝑇 𝐹(𝑥∗ ) ≺ 0.

(150)

14

Abstract and Applied Analysis Find 𝑥 ∈ 𝐾 ⊂ 𝑅𝑛 , 𝐹(𝑥) ∈ (𝐸)𝑛 , such that

⇐ By contradiction, suppose that there exists an 𝑥 ∈ 𝐾, such that 𝜂(𝑥, 𝑥∗ )𝑇 𝐹(𝑥∗ ) ≺ 0. Thus, there exists some 𝛼0 ∈ [0, 1], 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼0 ) < 0,

𝐹(𝑥)𝑇 𝑥 = 0, 𝐹 (𝑥) ∈ 𝐾∗ ,

(151)

(FCP)

𝐾∗ = {𝑧 ∈ (𝐸)𝑛 | 𝑧𝑇 𝑦 ≺ 0, ∀𝑦 ∈ 𝐾} .

or 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼0 ) < 0.

(152)

Without loss of generality, assume that 𝑇

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼0 ) < 0.

(153)

Since 𝐾 is invex set with respect to 𝜂 and assumption (ii), it follows that ∗

∗

𝑥 + 𝜆𝜂 (𝑥, 𝑥 ) ∈ 𝐾, 𝜂 (𝑥∗ + 𝜆𝜂 (𝑥, 𝑥∗ ) , 𝑥∗ ) = 𝜆𝜂 (𝑥, 𝑥∗ ) ,

The following theorem shows the equivalence between the fuzzy variational-like inequality and the fuzzy generalized complementarity problem. Theorem 29. The fuzzy variational-like inequality (FVLI) is equivalent to the fuzzy generalized complementarity problem (FCP), when 𝐾 is a convex cone. Proof. At first, we show (FCP)⊂(FVLI). Suppose that (𝑥∗ , 𝐹(𝑥∗ )) is a solution of (FCP), then 𝑇

𝐹(𝑥∗ ) 𝑥∗ = 0,

(154)

𝑇

𝜆 ∈ [0, 1] .

𝐹(𝑥∗ ) 𝑦 ≺ 0,

Again, since 𝐹 is 𝜂-hemicontinuous on 𝐾, there is 𝜆 󳨀→ 0+ , ∗

∗

∗

∗

𝐹 (𝑥 + 𝜆𝜂 (𝑥, 𝑥 )) (𝛼0 ) 󳨀→ 𝐹 (𝑥 ) (𝛼0 ) .

𝑇

̃ 𝐹(𝑥∗ ) (𝑦 − 𝑥∗ ) ⪰ 𝜇𝐶−1𝑦 (𝛼) ⪰ −𝑝,

(156)

× 𝐹∗ (𝑥∗ + 𝜆𝜂 (𝑥, 𝑥∗ )) (𝛼0 ) < 0. Set 𝑥̃ = 𝑥∗ + 𝜆𝜂(𝑥, 𝑥∗ ), then 𝑥̃ ∈ 𝐾.

(157)

∀𝑦 ∈ 𝐾,

𝑇

∗

[𝐹(𝑥∗ ) 𝑥∗ ] (𝛼0 ) > 𝑞̃1∗ (𝛼0 )

(161)

(162)

or

6. Solution to the Fuzzy Variational-Like Inequality

𝑇

[𝐹(𝑥∗ ) 𝑥∗ ] (𝛼0 ) > 𝑞̃1∗ (𝛼0 ) . ∗

In order to solve the fuzzy variational-like inequality, it is important find an equivalent fuzzy problem. Next, we discuss the equivalent fuzzy generalized complementarity problem of (FVLI), where 𝜂(𝑦, 𝑥) = 𝑦 − 𝑥. Denote (FVLI): find 𝑥 ∈ 𝐾 ⊂ 𝑅𝑛 , 𝐹(𝑥) ∈ (𝐸)𝑛 , such that ∀𝑦 ∈ 𝐾,

(160)

where 𝑝̃ ⪰ 0 is the tolerance level which a decision maker can tolerate in the accomplishment of the fuzzy variation-like inequality 𝐹(𝑥)𝑇 (𝑦 − 𝑥) ≺ 0. By contradiction, suppose that 𝐹(𝑥∗ )𝑇 𝑥∗ ≠ 0, then there exists some 𝛼0 ∈ [0, 1], 𝑞̃1 ⪰ 0, such that

This contradicts 𝜂(𝑥, 𝑥∗ )𝑇 𝐹(𝑥) ≺ 0, for all 𝑥 ∈ 𝐾.

𝐹(𝑥)𝑇 (𝑦 − 𝑥) ≺ 0,

∀𝑦 ∈ 𝐾.

𝑇

𝑇

󳨐⇒ 𝜂 (𝑥∗ + 𝜆𝜂 (𝑥, 𝑥∗ ))

(159)

Therefore, (𝑥∗ , 𝐹(𝑥∗ )) is also a solution of (FVLI). Next, we show (FVLI)⊂(FCP). Let (𝑥∗ , 𝐹(𝑥∗ )) be a solution of (FVLI) which the degree of membership 𝛼 ∈ [0, 1], then

𝜂 (𝑥, 𝑥∗ ) 𝐹∗ (𝑥∗ ) (𝛼0 ) < 0

𝑇

𝐹(𝑥∗ ) (𝑦 − 𝑥∗ ) ≺ 0,

(155)

Therefore,

̃ 𝑥∗ ) 𝐹∗ (𝑥) ̃ (𝛼0 ) < 0, 𝜂 (𝑥,

∀𝑦 ∈ 𝐾.

Combining (158) and (159), we have 𝑇

∗

(158)

(FVLI)

where 𝐹 : 𝐾 → (𝐸)𝑛 . It is well known that variational inequalities are equivalent to the generalized complementary problem over a convex cone [35]. Similarly, we consider solving the fuzzy variationallike inequality in view of the following fuzzy generalized complementarity problem.

(163)

Or there exists some 𝛼1 ∈ [0, 1], 𝑞̃2 ≻ 0, such that 𝑇

∗

[𝐹(𝑥∗ ) 𝑥∗ ] (𝛼1 ) < 𝑞̃2∗ (𝛼1 )

(164)

or 𝑇

[𝐹(𝑥∗ ) 𝑥∗ ] (𝛼1 ) > 𝑞̃2∗ (𝛼1 ) . ∗

(165)

Without loss of generality, suppose that 𝑇

∗

𝑇

∗

[𝐹(𝑥∗ ) 𝑥∗ ] (𝛼0 ) > 𝑞̃1∗ (𝛼0 ) , [𝐹(𝑥∗ ) 𝑥∗ ] (𝛼1 ) < 𝑞̃2∗ (𝛼1 ) .

(166)

Abstract and Applied Analysis

15

Since 𝐾 is a convex cone, then when 𝑦 = 𝜆𝑥∗ with 𝜆 > 1, there is −𝑝̃ , 𝜆−1

𝑇

𝐹(𝑥∗ ) 𝑥∗ ⪰

𝑝̃ ⪰ 0.

(167)

1, if 𝑑𝑗 𝑥 ⪰ 0, { { 𝜇𝑗 (𝑥) = {𝜇𝑗 (𝑑𝑗 𝑥) , if − 𝑡𝑗 ≺ 𝑑𝑗 𝑥 ⪯ 0, ∀𝑗 = 1, 2, . . . , 𝑚, { if 𝑑𝑗 𝑥 ⪯ −𝑡𝑗 , {0,

(168)

1, if 𝐹(𝑥)𝑇 𝑥 ⪰ 0, { { 𝜇𝑠𝑚+1 (𝑥) = {𝜇𝑚+1 (𝐹(𝑥)𝑇 𝑥) , if − 𝑡𝑚+1 ≺ 𝐹(𝑥)𝑇 𝑥 ⪯ 0, { if 𝐹(𝑥)𝑇 𝑥 ⪯ −𝑡𝑚+1 , {0,

When 𝑦 = 0, 𝑇

̃ 𝐹(𝑥∗ ) 𝑥∗ ⪯ 𝑝. 𝑇

∗

If [𝐹(𝑥∗ ) 𝑥∗ ] (𝛼0 ) > 𝑞̃1∗ (𝛼0 ), then ∗

[

−𝑝̃ ] (𝛼0 ) > 𝑝̃1∗ (𝛼0 ) . 𝜆−1

(169)

This leads to a contradiction. 𝑇 If [𝐹(𝑥∗ ) 𝑥∗ ]∗ (𝛼1 ) < 𝑞̃2∗ (𝛼1 ), then 𝑝̃∗ (𝛼1 ) < 𝑞̃2∗ (𝛼1 ) .

(170)

This also leads to a contradiction. Therefore, 𝐹(𝑥∗ )𝑇 𝑥∗ = 0. Furthermore, from (161), it follows that 𝑇

𝐹(𝑥∗ ) (𝑦 − 𝑥∗ ) ⪰ −𝑝̃ 𝑇

̃ 󳨐⇒ 𝐹(𝑥∗ ) 𝑦 ⪰ −𝑝,

(171) ∀𝑦 ∈ 𝐾.

Therefore, 𝑇

𝐹(𝑥∗ ) 𝑦 ≺ 0.

In (174), each fuzzy inequality can be represented by a fuzzy set 𝑆𝑗 with the corresponding membership function 𝜇𝑠𝑗 (𝑥), for 𝑗 = 1, 2, . . . , 2𝑚 + 2; that is

(172)

This shows that (𝑥∗ , 𝐹(𝑥∗ )) is a solution of (FCP). To solve (FVLI) with 𝐾 = {𝑥 ∈ 𝑅𝑛 | 𝐷𝑥 ⪰ 𝑜, 𝐷 ∈ 𝑅 , and 𝐷 = [𝑑𝑖 ], for all 𝑖 = 1, 2, . . . , 𝑚} being a convex cone, we consider the problem (FCP) with 𝑑𝑖 𝑥 ⪰ 0, for all 𝑖 = 1, 2, . . . , 𝑚. That is,

1, if − 𝐹(𝑥)𝑇 𝑥 ⪰ 0, { { 𝑇 𝜇𝑠𝑚+2 (𝑥) = {𝜇𝑚+2 (−𝐹(𝑥) 𝑥) , if − 𝑡𝑚+1 ≺ −𝐹(𝑥)𝑇 𝑥 ⪯ 0, { if − 𝐹(𝑥)𝑇 𝑥 ⪯ −𝑡𝑚+2 , {0,

where 𝑡𝑗 ⪰ 0, for all 𝑗 = 1, 2, . . . , 𝑚 are the tolerance level of 𝑑𝑗 𝑥 ⪰ 0, 𝑗 = 1, 2, . . . , 𝑚. 𝑡𝑗 ⪰ 0, for all 𝑗 = 𝑚 + 1, 𝑚 + 2, . . . , 2𝑚 + 2 other fuzzy inequalities in which 𝑡𝑗 are fuzzy numbers. To find a solution to problem (175), we define a fuzzy ̃ of (175) as the fuzzy set resulting from the decision 𝐷 intersection of fuzzy set 𝑠𝑗 , 𝑗 = 1, 2, . . . , 2𝑚 + 2. By choosing the commonly used “minimum operator” for the fuzzy set ̃ intersections, we can define the membership function for 𝐷 as 𝜇𝑗 = min {𝜇𝑠𝑗 (𝑥)} .

𝑚×𝑛

𝑑𝑖 𝑥 ⪰ 0, ∗

𝐹 (𝑥) ∈ 𝐾 ,

𝐹(𝑥)𝑇 𝑥 = 0, ∀𝑖 = 1, 2, . . . , 𝑚,

(173)

where 𝐾∗ = {𝑧 ∈ (𝐸)𝑛 | 𝑧𝑇𝑦 ≺ 0, for all 𝑦 ∈ 𝐾}. Similarly [26], it can be shown that (173) can be rewritten as follows: find 𝑥 ∈ 𝐾, 𝐹(𝑥) ∈ (𝐸)𝑛 such that 𝑑𝑖 𝑥 ⪰ 0,

𝐹(𝑥)𝑇 𝑥 = 0,

𝑑𝑖󸀠 𝐹 (𝑥) ⪰ 0,

∀𝑖 = 1, 2, . . . , 𝑚,

(174)

where 𝑑𝑖󸀠 is a normal to 𝑑𝑖 . Of course, (174) can be also rewritten as follows: find 𝑥 ∈ 𝐾, 𝐹(𝑥) ∈ (𝐸)𝑛 such that 𝐹(𝑥)𝑇 𝑥 ⪰ 0,

𝑑𝑖 𝑥 ⪰ 0, 𝑇

−𝐹(𝑥) 𝑥 ⪰ 0, 𝑑𝑖󸀠 𝐹 (𝑥) ⪰ 0,

∀𝑖 = 1, 2, . . . , 𝑚.

󸀠 𝑥 ⪰ 0, if 𝑑𝑗−𝑚−2 󸀠 if − 𝑡𝑗 ≺ 𝑑𝑗−𝑚−2 ⪯ 0, ∀𝑗 = 𝑚 + 3, 𝑚 + 4, . . . , 2𝑚 + 2, 󸀠 if 𝑑𝑗−𝑚−2 𝑥 ⪯ −𝑡𝑗 , (176)

1, { { 󸀠 { {𝜇𝑗 (𝑑𝑗−𝑚−2 𝑥) , 𝜇𝑗 (𝑥) = { { { { {0,

𝑗

(177)

Therefore, a solution to problem (175) can be taken as the solution with the highest membership in the fuzzy decision ̃ and obtained by solving the following problem: set 𝐷 max 𝜇𝑗 = max 𝑥

𝑥

min

𝑗=1,2,...,2𝑚+2

{𝜇𝑠𝑗 (𝑥)} .

(178)

Problem (178) is an unconstrained optimization, but its objective function is not continuous and derivable. It cannot be solved by traditional optimization methods, but it may be solved by genetic algorithms [36, 37]. Next, we discuss a special genetic algorithm with mutation along the weighted gradient direction developed by Wang and Tang [37–39]. For individual 𝑥, let 𝜇min (𝑥) = min{𝜇𝑠1 (𝑥), 𝜇𝑠2 (𝑥), . . . , 𝜇𝑠2𝑚+2 (𝑥)}. If 𝜇min (𝑥) ≤ 𝜇𝑠𝑗 (𝑥) < 1, then move along the gradient direction of 𝜇𝑠𝑗 (𝑥), and the value of 𝜇𝑠𝑗 (𝑥) may be improved. The smaller 𝜇𝑠𝑗 (𝑥) is, the greater the improvement in 𝜇𝑠𝑗 (𝑥) that may be achieved. Based on the above idea, construct the direction as follows: 2𝑚+2

(175)

𝐺 (𝑥) = ∑ 𝑤𝑗 ∇𝜇𝑠𝑗 (𝑥) , 𝑗=1

where 0 < 𝜇𝑠𝑗 (𝑥) ≤ 1.

(179)

16

Abstract and Applied Analysis

𝐺(𝑥) is called the weighted gradient direction of 𝜇𝐷̃, and 𝑤𝑗 is the gradient direction weight defined as follows: if 𝜇𝑠𝑗 (𝑥) = 1, { {0, 1 𝑤𝑗 = { , if 𝜇min ≤ 𝜇𝑠𝑗 < 1, { 𝜇 − 𝜇min + 𝜖 { 𝑠𝑗

(180)

Step 4. Set iteration index 𝑘 = 1. Step 5. Calculate the fitness function 𝐹(𝑗) and selection probabilities 𝑃(𝑗) by means of (183). Step 6. Produce new individual 𝑥(𝑗) with the parent 𝑥(𝑖) as 𝑥(𝑗)𝑘 = 𝑥(𝑖)(𝑘−1) + 𝛽𝑘 𝐺(𝑥(𝑖)(𝑘−1) ). 𝐺(𝑥) is defined by (179).

where 𝜖 is a sufficiently small positive number and 1/𝜖 is the largest weight. The child 𝑥𝑗𝑘+1 generated from 𝑥𝑖𝑘 by mutation along the weighted gradient direction 𝐺(𝑥) can be described as follows:

Step 7. For individual 𝑗, calculate the membership function 𝜇𝑆𝑗 (𝑥(𝑗) ) with (182), and update optimal degree of membership 𝜇max and the upper and the lower values of criteria 𝑟.

𝑥𝑗𝑘+1 = 𝑥𝑖𝑘 + 𝛽𝑘 𝐺 (𝑥𝑖𝑘 ) ,

(181)

Step 8. Set 𝑘 + 1 = 𝑘; if 𝑘 ≤ max-gen, go to Step 5; otherwise, go to Step 9.

where 𝛽𝑘 is a random step length of the Erlang distribution generated by a random number generator with declining means. The degree of membership is calculated as follows:

Step 9. Output the optimal membership degree 𝜇max and the upper and the lower values of criteria preferred by the decision maker, then stop.

𝜇𝑠𝑗 (𝑥(𝑗) ) = {

𝜇min (𝑥(𝑗) ) , if 𝜇min > 𝛼0 , 𝜖𝜇min (𝑥(𝑗) ) , else,

(182)

where 𝛼0 is the acceptable satisfaction degree preferred by the decision maker and 𝜖 ∈ 𝑈(0, 1). For individual 𝑗 (𝑗 = 1, 2, . . . , pop-size), calculate its fitness function 𝐹(𝑗) and selected probability 𝑃(𝑗) by 𝐹 (𝑗) = 𝜇𝑠𝑗 (𝑥(𝑗) ) + 𝜖̃, 𝑃 (𝑗) =

𝐹 (𝑗) pop-size

∑𝑗=1

𝐹 (𝑗)

,

(183)

where 𝜖̃ is a small positive number. It is used to guarantee a nonzero denominator. The procedure can be written as follows. Step 1. Transforming (FVLI) into problem (175). Step 2. Initialize. (i) Input an acceptable satisfaction degree 𝛼0 and the largest max-gen and pop-size. (ii) Input the criteria set: 𝐶𝐼 = 1, 2, . . . , 𝑛, 𝑛 + 1, . . . , 𝑛 + 2𝑚 + 2, 𝑛 + 2𝑚 + 3, where 𝑗 = 1, 2, . . . , 𝑛 stand for the variables, 𝑗 = 𝑛 + 1, . . . , 𝑛 + 2𝑚 + 2 stand for inequality constraints to (175), and 𝑗 = 𝑛 + 2𝑚 + 3 stands for the sum of weighted satisfaction degree, respectively. Give the initial values and the upper and the lower values of criteria 𝑟, 𝑟𝑖 ∈ 𝐶𝐼 . (iii) Input the types of membership function that describes the fuzzy constraints. (iv) Input the weights 𝑤𝑗 . Step 3. Randomly produce the initial population and calcuup late their membership degrees by 𝑥𝑖 (𝑗) = 𝜉𝑥𝑖 , where 𝜉 ∈ up 𝑈(0, 1), 𝑖 = 1, 2, . . . , 𝑛; 𝑗 = 1, 2, . . . , pop-size and 𝑥𝑖 is the upper bound of the 𝑖th element of variable 𝑥. The membership function 𝜇𝑆𝑗 (𝑥(𝑗) ) is calculated with (182).

7. Conclusion In this paper, we have introduced the concepts of invex monotonicity and pseudoinvex monotonicity of fuzzy mapping and discussed the relationship between invex monotonicity(pseudoinvex monotonicity) and invexity (pseudoinvexity) of fuzzy mapping. We have put forward the fuzzy variational-like inequality and also discussed extended qualities of it. Finally, we discussed the existence of a theorem for a solution and solving method to the fuzzy variationallike inequality. We got some useful results and gave a method of solving the fuzzy variational-like inequality of a genetic algorithm.

Acknowledgments This research was supported by the Key Program of NSFC (Grant no. 70831005); the National Science Foundation for Distinguished Young Scholars, China (Grant no. 70425005); The Young and Middle-Aged Leader Scientific Research Foundation of Chengdu University of Information Technology (J201218), China; Bringing in Qualified Personnel Projects of Chengdu University of Information Technology (KYTZ201203), China.

References [1] S. S. L. Chang and L. A. Zadeh, “On fuzzy mapping and control,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 2, pp. 30–34, 1972. [2] S. Nanda and K. Kar, “Convex fuzzy mappings,” Fuzzy Sets and Systems., vol. 48, no. 1, pp. 129–132, 1992. [3] H. Yan and J. Xu, “A class of convex fuzzy mappings,” Fuzzy Sets and Systems, vol. 129, no. 1, pp. 47–56, 2002. [4] M. Panigrahi, G. Panda, and S. Nanda, “Convex fuzzy mapping with differentiability and its application in fuzzy optimization,” European Journal of Operational Research, vol. 185, no. 1, pp. 47– 62, 2008. [5] Z. Wu and J. Xu, “Nonconvex fuzzy mappings and the fuzzy prevariational inequality,” Fuzzy Sets and Systems. An International

Abstract and Applied Analysis

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16] [17]

[18]

[19]

[20]

[21]

[22]

[23]

Journal in Information Science and Engineering, vol. 159, no. 16, pp. 2090–2103, 2008. Z. Wu and J. Xu, “Generalized convex fuzzy mappings and fuzzy variational-like inequality,” Fuzzy Sets and Systems, vol. 160, no. 11, pp. 1590–1619, 2009. G. Ruiz-Garz´on, R. Osuna-G´omez, and A. Rufi´an-Lizana, “Generalized invex monotonicity,” European Journal of Operational Research, vol. 144, no. 3, pp. 501–512, 2003. J.-P. Crouzeix and J. A. Ferland, “Criteria for differentiable generalized monotone maps,” Mathematical Programming, vol. 75, no. 3, pp. 399–406, 1996. A. Daniilidis and N. Hadjisavvas, “On the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity,” Journal of Mathematical Analysis and Applications, vol. 237, no. 1, pp. 30–42, 1999. R. W. Cottle, F. Giannessi, and J.-L. Lions, Eds., Theorems of Alternative, Quadratic Programs and Complementary Problems, Variational Inequalities and Complementary Problems, John Wiley & Sons, New York, NY, USA, 1980. G. Ruiz-Garz´on, R. Osuna-G´omez, and A. Rufi´an-Lizana, “Relationships between vector variational-like inequality and optimization problems,” European Journal of Operational Research, vol. 157, no. 1, pp. 113–119, 2004. C. Guang-Ya and Y. Xiao-Qi, “The vector complementary problem and its equivalences with the weak minimal element in ordered spaces,” Journal of Mathematical Analysis and Applications, vol. 153, no. 1, pp. 136–158, 1990. G. Y. Chen, “Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem,” Journal of Optimization Theory and Applications, vol. 74, no. 3, pp. 445–456, 1992. G.-Y. Chen and B. D. Craven, “Approximate dual and approximate vector variational inequality for multiobjective optimization,” Australian Mathematical Society A, vol. 47, no. 3, pp. 418– 423, 1989. G. Chen and G. Cheng:, Vector Variational Inequality and Vector Optimization, vol. 285 of Lecture Notes in Economics and Mathematics Systems, Springer, New York, NY, USA, 1986. F. Giannessi, Variational Inequalities and Vector Equilibrium, Kluwer Academic Publishers, London, UK, 2000. D. E. Ward and G. M. Lee, “On relations between vector optimization problems and vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 113, no. 3, pp. 583–596, 2002. X. Q. Yang, “On the gap functions of prevariational inequalities,” Journal of Optimization Theory and Applications, vol. 116, no. 2, pp. 437–452, 2003. X. Q. Yang, “Generalized convex functions and vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 79, no. 3, pp. 563–580, 1993. X. Q. Yang, “Vector variational inequality and vector pseudolinear optimization,” Journal of Optimization Theory and Applications, vol. 95, no. 3, pp. 729–734, 1997. N. J. Huang, X. Q. Yang, and W. K. Chan, “Vector complementarity problems with a variable ordering relation,” European Journal of Operational Research, vol. 176, no. 1, pp. 15–26, 2007. S.-S. Chang and Y.-G. Zhu, “On variational inequalities for fuzzy mappings,” Fuzzy Sets and Systems, vol. 32, no. 3, pp. 359–367, 1989. M. A. Noor, “Variational inequalities for fuzzy mappings. (I),” Fuzzy Sets and Systems, vol. 55, no. 3, pp. 309–312, 1993.

17 [24] M. A. Noor, “Variational inequalities for fuzzy mapping (II),” Fuzzy Sets and Systems, vol. 97, no. 1, pp. 101–107, 1998. [25] M. A. Noor, “Variational inequalities for fuzzy mappings. (III),” Fuzzy Sets and Systems, vol. 110, no. 1, pp. 101–108, 2000. [26] C.-F. Hu, “Solving variational inequalities in a fuzzy environment,” Journal of Mathematical Analysis and Applications, vol. 249, no. 2, pp. 527–538, 2000. [27] C.-F. Hu, “Generalized variational inequalities with fuzzy relation,” Journal of Computational and Applied Mathematics, vol. 146, no. 1, pp. 47–56, 2002. [28] J. Y. Park and J. U. Jeong, “Generalized strongly quasivariational inequalities for fuzzy mappings,” Fuzzy Sets and Systems, vol. 99, no. 1, pp. 115–120, 1998. [29] J. Y. Park, S. Y. Lee, and J. U. Jeong, “Completely generalized strongly quasivariational inequalities for fuzzy mappings,” Fuzzy Sets and Systems, vol. 110, no. 1, pp. 91–99, 2000. [30] J. Zhu, C.-K. Zhong, and G.-P. Wang, “Vector-valued variational principle in fuzzy metric space and its applications,” Fuzzy Sets and Systems, vol. 119, no. 2, pp. 343–354, 2001. [31] G. Chen, C. Goh, and X. Yang, “On gap function for vector variational inequalities,” European Journal of Operational Research, vol. 82, pp. 55–72, 2000. [32] G. Y. Chen and S. J. Li, “Existence of solutions for a generalized vector quasivariational inequality,” Journal of Optimization Theory and Applications, vol. 90, no. 2, pp. 321–334, 1996. [33] X. Q. Yang and J. C. Yao, “Gap functions and existence of solutions to set-valued vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 115, no. 2, pp. 407– 417, 2002. [34] J. J. Buckley and T. Feuring, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 110, no. 1, pp. 43–54, 2000. [35] H.-F. Wang and H.-L. Liao, “Variational inequality with fuzzy convex cone,” Journal of Global Optimization, vol. 14, no. 4, pp. 395–414, 1999. [36] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Mich, USA, 1975. [37] J. Tang and D. Wang, “An interactive approach based on a genetic algorithm for a type of quadratic programming problems with fuzzy objective and resources,” Computers & Operations Research, vol. 24, no. 5, pp. 413–422, 1997. [38] J. Tang and D. Wang, “Modelling and optimization for a type of fuzzy nonlinear programming problems in manufacturing systems,” in Proceedings of the 35th IEEE Conference on Decision and Control, pp. 4401–4405, December 1996. [39] D. Wang, “An inexact approach for linear programming with a GA for a type of quardratic programming problem with fuzzy objective and resources,” Fuzzy Sets and Systems, vol. 24, pp. 261–281, 1995.

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