Auxiliary principle and fuzzy variational-like inequalities
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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. 1. Introduction 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]. Equally important is variational inequality theory, which constitutes a significant and important extension of the variational principle. A useful and important generalization of variational inequalities is generalized mixed variational-like inequality. These kinds of variational inequalities have potential and significant applications in optimization theory [16, 17], structural analysis [14] and economics [5, 16]. 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. It is useful to remark that these methods are not constructive. Thus, the development of an efficient and implementable technique for solving variational-like inequalities is one of the most interesting and important problems in variational inequality theory. Although there exist many numerical methods (e.g., the projection method and its variant forms, linear approximation, descent and Newton’s methods) for variational inequalities, there are very few methods for general variational-like inequalities. One method used in the literature is to develop an auxiliary technique for solving various mixed variational-like inequalities.
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 479–494 DOI: 10.1155/JIA.2005.479
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Auxiliary principle and fuzzy variational-like inequalities
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 auxilia
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