Generalized variational inequalities for linguistic interpretations using intuitionistic fuzzy relations and projected dynamical systems

The fuzzy set theory enables us to represent our knowledge under multiple interpretations and axiomatic foundations from linguistic to computational representations. While the intuitionistic fuzzy set, as a generalization of the fuzzy set, cannot only represent the tolerance levels, but also the intolerance levels which a decision maker can tolerate and cannot tolerate in the accomplishment of a linguistic interpretation, in this paper, we introduce the generalized variational inequalities for linguistic interpretations using intuitionistic fuzzy relations. It is shown that such problems can be transformed into the classical (nonfuzzy) generalized variational inequalities by means of level sets of the intuitionistic fuzzy relation. Furthermore, the equivalence between the generalized variational inequalities with intuitionistic fuzzy relations and the fuzzy fixed point problems is established. Finally, based on the projection method, we propose an iterative algorithm and a projected neural network model for the generalized variational inequalities with intuitionistic fuzzy relations, and the stability of the proposed projected dynamical system is also investigated.

and y ∈ f (x) such that y T xx ≥ 0, ∀x ∈ M, (1.1) where M ⊆ R n , and f : M → 2 R n is a set-valued function. There is an equivalence between set-valued mappings and binary relations, thus the more convenient discussion framework system can be chosen between the two according to the actual needs. Indeed, for a given set-valued mapping f : M → R n , we can induce a relation by = (x, y) ∈ R n × R n : x ∈ M and y ∈ f (x) .
Conversely, for a given relation ∈ R n × R n , its domain is dom( ) = x ∈ R n : (x, y) ∈ , for some y ∈ R n , and for every x ∈ dom( ), its image set is That is, f is a set-valued mapping from dom( ) to the family of subsets of R n . Thus, for a given set-valued mapping f on M, the generalized variational inequality (1.1) can also be represented as follows: the generalized variational inequality, denoted by GVI(M, ), is to find all solutions (x, y) such that x ∈ M, y, xx ≥ 0, ∀x ∈ M, (x, y) ∈ , (1.2) where M ⊆ R n , ⊆ R n × R n . The most basic result on the existence of solutions to the variational inequality VI(M, F) requires the set M to be compact and convex, and the mapping f to be continuous [7], which is given by Brouwer's fixed point theorem. To proceed from this result, extended conclusions are derived by replacing the compactness of the set M (closed which is possibly unbounded) with additional conditions on F (e.g., pseudomonotone, strongly monotone, coercive with respect to M) [7,8,15]. Similarly, the most fundamental existence theorem for GVI(M, F) can also be proved by Kakutani fixed point theorem which is for a set-valued function. In the classical set, the nature of the element is required to be explicit, that is, it can be explicitly indicated that any element has or does not have this property. However, in the objective world, many phenomena, which are based on numerous fuzzy phenomena and multi-valued logic, have fuzziness. For example, the linguistic interpretations such as "young" and "old", "long" and "short" are fuzzy concepts in people's concepts. Such vague concepts with unclear denotations cannot be expressed by the usual binary logic and therefore cannot be described by the classical set. In 1965, Zadeh [26] introduced the fuzzy set theory, which offers a wide variety of techniques for analyzing imprecise data and enables us to represent our knowledge under varied interpretations and axiomatic foundations from linguistic to computational representations. A fuzzy set u on R is a mapping u : R → [0, 1], and u(x) is the degree of membership of the element x in the fuzzy set u. The fuzzy set is a generalization of the classical set whose characteristic function is valued in {0, 1}. By fuzziness, we mean a type of imprecision which is associated with fuzzy sets, that is, classes in which there is no sharp transition from membership to nonmembership. In fact, in sharp contrast to the notion of a class or a set in mathematics, most of the classes in the real world do not have crisp boundaries which separate those objects which belong to a class from those which do not. For notational purposes, it is convenient to have a device for indicating that a fuzzy set is obtained from a nonfuzzy set by fuzzifying the boundaries of the latter set. In 1970, Bellman and Zadeh [3] employed a wavy bar under a symbol which defines the nonfuzzy set. On the other hand, the classical binary relations were also extended to the fuzzy binary relations on two ordinary sets [20]. For two given ordinary sets A and B, a fuzzy relation is a fuzzy subset of the set A × B. The uncertainty environment for a variational inequality leads to certain degrees of fuzziness in the classical relation. In 2001, Hu [17] introduced the fuzzy variational inequality over a compact set by using the tolerance approach. Subsequently, Hu [18] investigated the generalized variational inequality with fuzzy relation and showed that such problems can be transformed into regular optimization problems. In 2009, Hu and Liu [19] discussed mathematical programs with fuzzy parametric variational inequalities. In 2019, Xie and Gong [25] investigated the generalized variational-like inequalities for fuzzy-vector-valued functions.
With the research of fuzzy sets, Atanassov [1] presented the intuitionistic fuzzy set (IFS, for short) which is more powerful and sensitive than the fuzzy set in dealing with imperfect information and imprecise information. Actually, the intuitionistic fuzzy set is a straightforward generalization of Zadeh's fuzzy set: a fuzzy set gives a degree to which an element belongs to the set, while an intuitionistic fuzzy set gives both a membership degree and a nonmembership degree. The membership and nonmembership values induce the hesitancy degree, which models the hesitancy of deciding the degree to which an object satisfies a particular property. For instance, this situation can be found in group decision making problems. Consider a voting situation in which human voters can be divided into three groups: vote for, vote against, and abstain. If we take x 1 , 0.6, 0.3 as an element of intuitionistic fuzzy set A of voting, then we can interpret it as "the voting for the candidate x 1 is 0.6 for 0.3 against and 0.1 abstentions". IFS theory has been widely applied in various fields such as decision analysis, pattern recognition, machine learning, image processing, and so on [12][13][14]. In this paper, we further discuss the generalized variational inequalities with intuitionistic fuzzy relations. In addition, real-time solutions to theses problems are always needed in engineering applications, and thus they have to be solved in real time to optimize the performance of dynamical systems. As parallel computational models, recurrent neural networks possess many desirable properties for real-time information processing. In 2003, M.A. Noor [24] investigated some implicit projected dynamical systems associated with quasi-variational inequalities by using the techniques of the projection and the Wiener-Hopf equations. Indeed, by means of level sets of the intuitionistic fuzzy relations, the generalized variational inequalities for linguistic interpretations using intuitionistic fuzzy relations can be transformed into the classical (nonfuzzy) generalized variational inequalities, and we further propose a projection neural network for solving the generalized variational inequalities with intuitionistic fuzzy relations. The stability of the projected dynamical system is also discussed.
The aim of this paper is to investigate the generalized variational inequality for linguistic interpretations using intuitionistic fuzzy relations and the stability of the associated dynamical system. The rest of the paper is structured as follows. In Sect. 2, we recall some preliminaries with respect to intuitionistic fuzzy sets. In Sect. 3, we introduce the generalized variational inequality for linguistic interpretations using intuitionistic fuzzy relations. In Sect. 4, we prove the existence theorem of solutions to the generalized variational inequalities with intuitionistic fuzzy relations. Furthermore, we propose an iterative algorithm. In Sect. 5, based on the projection method, we propose a projection neural network for solving the proposed problems, which is a dynamical system, and the stability of the projected dynamical system associated with generalized variational inequalities with intuitionistic fuzzy relations is discussed. Section 6 concludes this paper.

Preliminaries
For convenience of the reader, the basic properties of intuitionistic fuzzy sets is provided in this section. Let U be a nonempty set called the universe of discourse.

Definition 2.1 ([1]) Let U be a given set. An intuitionistic fuzzy set in U is an expression A given by
The numbers μ A (x) and ν A (x) denote, respectively, the degree of membership and the degree of nonmembership of the element x in the set A. We call π(x) = 1μ A (x)ν A (x) the intuitionistic index or the hesitancy degree of the element x in the set A. We will denote by IFSs(U) the set of all the intuitionistic fuzzy sets in U.
The operations of IFS are defined as follows [1,2]: for every A, B ∈ IFSs(U),

Definition 2.2 ([11]) An intuitionistic fuzzy set
Note that a fuzzy set u is called a fuzzy number if u is normal, convex, and upper semicontinuous, and

Definition 2.3 ([5]) An intuitionistic fuzzy relation is an intuitionistic fuzzy subset of
for all x ∈ U. We will denote by IFR(U × V ) the set of all the intuitionistic fuzzy subsets in U × V . If U = V , then we say that R is an intuitionistic fuzzy relation on U.

The complementary relation of R is
The (α, β)-level set of an intuitionistic fuzzy relation R is defined as Example 2.1 Assume that Mr. X wants to buy a car. Let U = {u 1 , u 2 , u 3 , u 4 , u 5 } be a set of five candidate cars. Suppose that the set of candidate cars U can be characterized by a set of parameters where v j (j = 1, 2, 3, 4) stands for "being cheap", "being beautiful", "being safe", and "being comfortable", respectively. Mr. X thinks u 1 is very expensive and this fuzzy information cannot be expressed only by the two crisp numbers 0 and 1, a membership degree can be used instead, which is associated with each element and represented by a real number valued in the interval [0, 1]. Furthermore, Mr. X thinks u 1 is 0.8 in being cheap, 0.1 against, and the hesitancy degree is 0.1. In that case, the characteristics of five candidate choices under four parameters are represented by an intuitionistic fuzzy relation matrix R(u i , v j ) 5×4 , which describes the attractiveness of the cars which Mr. X is going to buy, as follows:

The generalized variational inequality with intuitionistic fuzzy relation
be a set-valued mapping and is an intuitionistic fuzzy relation on M × R n with its membership function μ and nonmembership function ν . Then the generalized variational inequality with intuitionistic fuzzy relation, denoted by GVI(M, ), is defined as , here the wavy bar under a symbol plays the role of a fuzzifier, that is, a transformation which takes a nonfuzzy set into a fuzzy set which is approximately equal to it. In other words, y = ∼ f (x) is a fuzzy equality and "= ∼ " denotes the fuzzified version of "=" with the linguistic interpretation "approximately equal to".
. . , n, which actually determines an intuitionistic fuzzy set, whose membership function and nonmembership function are denoted by μ j , ν j , j = 1, 2, . . . , n, respectively. The membership grade μ j (x, y) can be interpreted as the degree to which the regular equality y j = f j (x), j = 1, 2, . . . , n, is satisfied. To specify the membership functions μ j , it is commonly assumed that μ j (x, y) should be 0 if the regular equality y j = f j (x) is strongly violated and 1 if it is satisfied, which is analogous to the nonmembership functions. In this sense, we can obtain a membership function and a nonmembership function in the following forms, respectively: where c j , d j ≥ 0, are the tolerance levels which a decision maker can tolerate in the accomplishment of the fuzzy equality y j = ∼ f j (x), and s j , t j ≥ 0 are the intolerance levels. We usually assume that μ L j , ν R j ∈ [0, 1] are continuous and nondecreasing, respectively, on [-d j , 0], [0, t j ], and μ R j , ν L j ∈ [0, 1] are continuous and nonincreasing, respectively, on [0, d j ], [-s j , 0].
As shown in Fig. 1, μ j (x, y) and ν j (x, y) is the membership function and the nonmembership function of y j = ∼ f j (x), respectively, where μ L j (-c j ) = 0, μ R j (-d j ) = 0, ν L j (s j ) = 0, and

The existence theorem and iterative algorithm of solutions to the generalized variational inequalities with intuitionistic fuzzy relations
where · and · denote norm and inner product on R n , respectively.
(2) Lipschitz continuous, if there exists a constant L ∈ (0, 1) such that where D is the Hausdorff metric on R n .
For simplicity, if the image of R is strongly monotone and Lipschitz continuous, we also say R is strongly monotone and Lipschitz continuous, respectively.

Definition 4.2 ([4])
The distance of a point x 0 ∈ R n to a closed set C ⊆ R n , in the norm · , is defined as The infimum here is always achieved. We refer to any point z ∈ C which is closest to x 0 , i.e., satisfies zx 0 = dist(x 0 , C), as a projection of x 0 on C, denoted by P C (x 0 ). In other words, P C : R n → C and P C (x 0 ) = argmin{ x 0x : x ∈ C}, we refer to P C as a projection on C.
Remark 4.1 In general, there can exist more than one projection of x 0 on C, that is, there are several points in C which are closest to x 0 . However, we can construct the projection of a point on a set which is unique. For instance, if C is closed and convex and the norm is strictly convex such as the Euclidean norm, then for any x 0 , there is always exactly one z ∈ C which is closest to x 0 . Conversely, if for any x 0 there is a unique Euclidean projection of x 0 on C, then C is closed and convex.
In what follows, we suppose that the norm of x ∈ R n is the Euclidean norm, i.e., x = x T x 1/2 = x 2 1 + · · · + x 2 n 1/2 .

Lemma 4.1 ([21])
Let M ⊆ R n be a closed and convex set. Then where L ∈ (0, 1). Since R and Then we can obtain a sequence {x n } of points of M satisfying x n+1 ∈ [ R(x n )] β α and x n+1x n ≤ L x nx n-1 ≤ L n x 1x 0 for all n ≥ 1. Therefore, we have for all n, m ≥ 1, thus, the sequence {x n+1 } is a Cauchy sequence, which implies that x n → x ∈ R n . Therefore, the sequence [ R(x n )] β α converges to x n+1 ∈ [ R(x)] β α weakly, and since Thus, for a constant ρ > 0, we have ρy, Therefore, x = min x∈M Theorem 4.2 indicates that GVI(M, ) is equivalent to the following fuzzy fixed point problem: where y ∈ [ (x)] β α . Accordingly, we can give the following iterative algorithm.
Step 2. Since . Let Step 3. Find x n and y n by the following iterative methods:   (x n , y n ) and (x, y) be the solutions to (4.7) and (3.1), respectively. If is strongly monotone and Lipschitz continuous, then x n → x strongly, and y n → y strongly.

Stability of the dynamical system for generalized variational inequalities with intuitionistic fuzzy relations
Let M ⊆ R n be a closed and convex set, be an intuitionistic fuzzy relation on M × R n . Consider the following projected neural network associated with the generalized variational inequality with intuitionistic fuzzy relation (3.1): where ρ > 0, λ are constants, y ∈ [ (x)] β α . x(t) = (x 1 (t), x 2 (t), . . . , x m (t)) T denotes the state vector of neurons, m is the number of neurons, and the initial value x 0 is given randomly. It is a dynamical system.
Without loss of generality, consider the following nonlinear dynamical system [6]: where t ∈ R, x ∈ M ⊆ R n , x 0 is the initial state. If there exists a state x in the state space satisfying f t, x = 0, ∀t ≥ t 0 , then we say x is an equilibrium state or an equilibrium point of system (5.2). The equilibrium point x is said to be stable in the sense of Lyapunov if, for any ε > 0, there exists δ > 0, when x is said to be asymptotically stable if x is stable and satisfies x(t) → x (t → ∞); x is said to be globally asymptotically stable if, for any initial point, x is asymptotically stable; x is said to be globally exponentially stable if, for any solution of the system x(t), there exist k > 0, η > 0 such that System (5.2) is said to globally converge to the set M ⊆ R n if, for any initial point, the solution of the system x(t) satisfies Lemma 5.1 (LaSalle's invariance principle [23]) Let f (t, x) be continuous in system (5.2). If there exists a continuously differentiable function V : R n → R 1 satisfying the following conditions: (i) There exists a constant r > 0 such that the set M r = {x ∈ R n : V (x) ≤ r} is bounded; (ii) for all x ∈ M r , dV (x) dt ≤ 0, then for all x 0 ∈ M r , when t → ∞, x(t) converges to the largest invariant subset of the set {x ∈ R n : dV (x) dt ≤ 0}.

Theorem 5.2
Let M ⊆ R n be a closed and convex set, be an intuitionistic fuzzy relation on M ×R n . If is Lipschitz continuous, then for any x 0 ∈ R n there exists a unique continuous solution x(t) of dynamical system Proof Let Then, for any x 1 , x 2 ∈ R n , since is Lipschitz continuous, and by (4.4), we have is Lipschitz continuous. Then, by the existence and uniqueness theorem of solutions for an ordinary differential equation, for any x 0 ∈ R n , there exists a unique continuous solution x(t) of dynamical system (5.1) with x(t 0 ) = x 0 over [t 0 , T].
On the other hand, since for any x ∈ R n where k 1 = λ{ x + P M [x ] }, k 2 = λ(2 + ρL). Therefore, by Lemma 5.2, we have It implies that x(t) is bounded on [t 0 , T), then by the extension theorem of solutions for an ordinary differential equation, we have T = ∞.

Conclusions
In this paper, we have investigated the generalized variational inequalities with intuitionistic fuzzy relations. We have obtained the existence theorem of solutions to the generalized variational inequalities with intuitionistic fuzzy relations. Furthermore, we have analyzed an iterative algorithm and a projected neural network model for this type variational inequality by using the projection method, and the proposed projected dynamical system is shown to be stable in the sense of Lyapunov, globally convergent and globally exponentially convergent under various conditions.