Abstract
The purpose of this paper is to introduce the concept of fuzzy variational-like inequalities and to study the existence problem and the iterative approximation problem for solutions of certain kinds of fuzzy variational-like inequalities in Hilbert spaces. By using the general auxiliary principle technique, Ky Fan's KKM theorem, Nadler's fixed point theorem, and some new analytic techniques, some existence theorems and some iterative approximation schemes for solving this kind of fuzzy variational-like inequalities are established. The results presented in this paper are new and they generalize, improve, and unify a number of recent results.
Highlights
In recent years, the fuzzy set theory introduced by Zadeh [18] in 1965 has emerged as an interesting and fascinating branch of pure and applied sciences
The applications of fuzzy set theory can be found in many branches of physical, mathematical and engineering sciences, see [2, 6, 20]
Some special cases of mixed variational-like inequalities have been studied by Tian [16] and Parida and Sen [15] by using Berge maximum theorem in finite and infinite dimensional spaces
Summary
The fuzzy set theory introduced by Zadeh [18] in 1965 has emerged as an interesting and fascinating branch of pure and applied sciences. The auxiliary principle technique was suggested by Glowinski et al [9] in 1981 These days it is a useful and powerful tool for solving various mixed variational-like inequalities. Noor [12] extended the auxiliary principle technique to study the existence and uniqueness of a solution for a class of generalized mixed variational-like inequalities for set-valued mappings with compact values. Inspired and motivated by recent research in this interesting field, the purpose of this paper is to introduce the concept of fuzzy generalized set-valued mixed variational-like inequalities and to study the existence problem and the iterative approximation problem for solutions of certain kinds of fuzzy generalized set-valued mixed variational-like inequalities in Hilbert spaces. The results presented in this paper are new and they generalize, improve and unify a number of recent results in [1, 10, 19]
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