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Abstract and Applied Analysis
Volume 2013 (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

1College of Mathematics, Chengdu University of Information Technology, Chengdu 610225, China
2Uncertainty Decision-Making Laboratory, School of Business and Administration, Sichuan University, Chengdu 610064, China

Received 11 May 2013; Accepted 1 September 2013

Academic Editor: Kanishka Perera

Copyright © 2013 Zezhong Wu and Jiuping 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.

Abstract

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 [79]. 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., [1121] 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 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 [2325]. For further research of variational inequalities of fuzzy mappings, one can refer to [2630]. 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 variational-like 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 variational-like 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, 3133]. 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 pseudo-invex 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 variational-like 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.

2. Preliminaries

A fuzzy set of is a mapping . For each such fuzzy set , its -cut set is denoted by for all . The support of is denoted by ; that is, . The closure of is defined as .

Definition 1 (see Wu and Xu [6]). A fuzzy number is a fuzzy set with the following properties:(1) is normal; that is, there exists such that ;(2) is convex fuzzy set; that is, , , ;(3) is compact.
Let denote the family of fuzzy numbers; that is, denotes the family of compact and convex fuzzy set on . Obviously, is a nonempty compact convex subset of (denoted by ) for all and for all .
A precise number is a special case of fuzzy number encoded as
However, a precise number will be denoted as usual, in particular, number 0. The fuzzy numbers are represented by and , respectively. For each real number , the addition and scalar multiplication are defined as follows:
It is well known that for all and
For , , if and only if    , and if and only if and .

Definition 2. For , , if and only if for every , and .   If , , then .    if and only if and , such that or .
For , if either or , then and are comparable; otherwise, they are noncomparable.
If , there exists such that , then we say the Hukuhara difference of and exists, call the H-difference of and , and denote .
It is obvious that if the H-difference exists, then , .

Definition 3. A mapping is said to be a fuzzy mapping. Denote , for all .

Definition 4 (see Buckley and Feuring [34]). Let be a fuzzy mapping from the set of real numbers to the set of all fuzzy numbers, and let . Assume that the partial derivatives of , with respect to for each exist and are denoted by , , respectively. Let for , . If defines the -cut of a fuzzy number for each , then is said to be differentiable and is written as , for all , .

Definition 5 (see Panigrahi et al. [4]). Let be a fuzzy mapping, where is an open set. Let . Let ,   stand for the “partial differentiation” with respect to the th variable . Assume that, for all , , have continuous partial derivatives so that , are continuous. Define
If each , defines the -cut of a fuzzy number, then is called differentiable at , and it can be represented as is said to be the gradient of the fuzzy mapping at .
is said to be an -dimensional fuzzy vector if and only if the components of are composed by fuzzy numbers, denoted by . The set of all -dimensional fuzzy vectors is denoted by .
A level vector of fuzzy vector is defined as The addition and the scalar multiplication of fuzzy vectors and are defined as

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 is said to be(a) fuzzy invex with respect to , if and only if (b) fuzzy incave with respect to , if and only if (c) fuzzy strictly invex with respect to , if and only if (d) fuzzy strictly incave with respect to , if and only if (e) fuzzy pseudo-invex with respect to , if and only if (f) fuzzy strictly pseudo-invex with respect to , if and only if

Example 7 (see Wu and Xu [6]). Let represent the reproduction rate of some germ:
So, , . Then there is , such that is a fuzzy invex mapping, where , , represents the predicted quantity, and represents the actual reproduction quantity.

Example 8. Consider the fuzzy mapping , . Then, there is an such that is a fuzzy pseudoinvex mapping, where , .

Remark 9. For an invex fuzzy mapping, there must exist , such that holds.

Proof. Since is a comparable fuzzy mapping, then for all , there is or Without loss of generality, suppose that Thus, for all ,
If , for any , the result holds.
If , for given .
(1) When .
(i) If , then take Thus, that is, From , there is so, Hence, That is, From (22) and (26), it follows that
(ii) If , then take Thus, From and , there is so, Hence, That is, From (29) and (33), it follows that
(iii) If is indefinite, there is a vector , such that holds.
Take Thus, That is, From and , there is so, Hence, That is, From (38) and (42), it follows that
(2) When .
(i) If , then take Thus, On the other hand, That is, From (45) and (47), it follows that
(ii) If , then take Thus, That is, On the other hand, that is, From (51) and (53), it follows that
(iii) If is indefinite, there is a vector , such that holds.
Take Thus, That is, On the other hand, that is, From (58) and (60), it follows that This completes the proof.

Definition 10. A comparable fuzzy mapping is said to be(a) fuzzy invex monotone on , if such that for any , (b) fuzzy pseudo-invex monotone on , if such that for any , (c) fuzzy strictly invex monotone on , if such that for any , , (d) fuzzy strictly pseudo-invex monotone if and only if such that for any , ,

Definition 11. The function is said to be a skew function if

Definition 12. Let , is said to be invex at with respect to if, for each , , is said to be an invex set with respect to if is invex at each .

Theorem 13. If a differentiable fuzzy mapping is invex on with respect to and is a skew function. Then, is fuzzy invex monotone with respect to the same .

Proof. Let be invex on , then there exists , such that That is, there are for all .
By changing for , That is, there are for all .
From (69) and (71), it follows that As , then Therefore, from (73), there is

Corollary 14. If a differentiable fuzzy mapping is invex on with respect to and is a skew function. Then, is fuzzy pseudo-invex monotone with respect to the same .

Proof. From Theorem 13, it follows that Thus, for all .
If , for all .
Thus, from (76) and (77), holds. Therefore, .

Theorem 15. If a differentiable fuzzy mapping is strictly invex on with respect to and is a skew function. Then, is fuzzy strictly invex monotone on with respect to the same .

Proof. Assume that is strictly invex on , then there exists , such that, for any , , Thus, there exists some , such that or For , without loss of generality, suppose that By changing for , By (82), (83) and since is a skew function, we have On the other hand, for other , Therefore,

Theorem 16. If a differentiable fuzzy mapping is strictly pseudo-invex on with respect to and is a skew function. Then, is fuzzy strictly pseudo-invex monotone on with respect to the same .

Proof. Let be a fuzzy strictly pseudo-invex, then there exists , such that for any , , We need to show that there exists , such that for all , .
By contradiction, suppose that , then there exists some , such that or Without loss of generality, assume that As , then Since is strictly pseudo-invex on , then for , there is which is a contradiction.

Theorem 17. Let be a fuzzy strictly pseudo monotone mapping on with respect to ; then, is a fuzzy pseudo monotone with respect to on .

Proof. As is a strictly pseudo monotone with respect to on , then for any , , is strictly pseudo monotone with respect to on . Thus, , also is pseudo monotone with respect to on ; that is, is pseudo monotone with respect to on .

Theorem 18. Let be a differentiable mapping, and suppose that (i) satisfies the following conditions: (a) , (b) ; (ii) is an invex set with respect to ;(iii) for each , some ,(a) implies ,or(b) implies ; (iv) is fuzzy pseudo-invex monotone with respect to on . Then, is a fuzzy pseudo-invex mapping on .

Proof. Set , , and holds. Thus, we need to show that ; that is, and , for all .
Assume the contrary, that is, . Thus, there exists some , such that or Without loss of generality, suppose that By hypothesis (iii), for some .
It follows from (98) and (i) that for some .
Since is a pseudo-invex monotone with respect to , thus From and , (100) becomes This contradicts .

Theorem 19. Let be a differentiable mapping, and suppose that (i) satisfies the following conditions:(a) , (b) ;(a) , (b) ;(ii) is an invex set with respect to ; (iii)for each , some , (a) implies ,or(b) implies ; (iv) is fuzzy strictly pseudo-invex monotone with respect to on .Then, is a fuzzy strictly pseudoinvex mapping on .

Proof. Let , , such that . Thus, we need to show that .
By contradiction, suppose that ; then, there exists some , such that or Without loss of generality, suppose that By hypothesis (iii), for some .
It follows from (i) and above inequity (105) that Since is a strictly pseudo-invex monotone with respect to , thus From and , it follows that This contradicts .

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

Let , , , the fuzzy variational-like inequality problem be: find , (denoted by ), such that

Definition 20. Let be an invex set with respect to . A fuzzy mapping is called -hemicontinuous, if for , for all , the mappings and are continuous at , with .

Lemma 21. Let be a nonempty convex set in , and suppose that(i) is a fuzzy pseudo-invex monotone with respect to and -hemicontinuous on ;(ii) satisfies(a) , for all , , (b) , for all , ;(iii) for any fixed , is linear; that is, for , , , , with . Then, for , , for all if and only if , for all .

Proof. By contradiction, suppose that there exists a , such that . Thus, there exists some , or Without loss of generality, assume that Since is a fuzzy pseudo-invex monotone with respect to , thus That is, there are for all .
Therefore, In particular, for , This contradicts .
By contradiction, suppose that there exists a , such that . Thus, there exists some , or Without loss of generality, assume that Since is an invex set and by condition (a), we knowthat When satisfies (iii), there is Since is -hemicontinuous on , it follows that Therefore, Let , then This contradicts , for all .
By the hypothesis of the pseudo-invex monotonicity of and the linearity of , the existence theorem can be obtained.

Theorem 22. Let be a nonempty convex set in , suppose that (i) is fuzzy pseudo-invex monotone with respect to and -hemicontinuous on ; (ii) satisfies (a) , for all , , (b) , for all , ;(iii) for any fixed , is linear; that is, for , , , , for , with . Then, there exists , such that , for all .

Proof. Let That is, for all , there are