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

• Published in: Journal of inequalities and applications Vol. 2022; no. 1; pp. 1 - 16
• Main Authors: Xie, Ting, Li, Dapeng
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 08-04-2022; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Decision making; Dynamic stability; Estimates; Fuzzy set theory; Fuzzy sets; Fuzzy systems; Inequalities; Intuitionistic fuzzy relation; Iterative algorithms; Iterative methods; Level set; Linguistics; Lyapunov stability; Mathematics; Mathematics and Statistics; Neural networks; Projection method; Variational inequality; Projection method; Level set; Variational inequality; 26E50; 34D20; Lyapunov stability; Intuitionistic fuzzy relation; 26E25
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-022-02777-1

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.