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

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... al-like inequalities are established. The results presented in this paper are new and they generalize, improve, and unify a number of recent results. 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. Recently, 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. However, the proof of the uniqueness part in [12, Theorem 3.1] is not quite right. Also the proof of the existence part is based on the assumption that the auxiliary problem has a solution. Unfortunately he did not show the existence of the solution for this auxiliary problem. Subsequently, Huang and Deng [10] introduced and studied a class of generalized set-valued strongly nonlinear mixed variational-like inequalities which includes the known class of mixed variational-like inequalities introduced by Noor [12] as a special case. On the other hand, in 2001, Ansari and Yao [1] considered and studied a class of mixed variational-like inequality problems for single-valued mappings by using the auxiliary principle technique. 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. 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 in [1, 10, 19] .