Exceptional Family of Elements and the Solvability of Variational Inequalities for Unbounded Sets in Infinite Dimensional Hilbert Spaces

It is well known that the theory of ̈ariational inequalities is now very well developed. The development of this theory has been stimulated by the diversity of applications in physics, mechanics, elasticity, fluid mechanics, w x economics, engineering, etc. 19 . The number of papers published on this subject is impressive. The solvability of variational inequalities has been studied by several methods using, for example, coercï ity conditions, compactness, fixed-point theory, KKM-mappings, min-max theory, and many other topological methods. The relations between variational inequalities and complementarity problems are well known. A new method is now used in the study of solvability of complementarity problems. This new method is based on the concept of exceptional family of elements and it is related to the topological degree to the concept of zero-epi mapping and to the w x Leray]Schauder alternatï e 4, 5, 8, 10]18, 23, 26, 27]29 . w x In 20 , T. E. Smith used the notion of an exceptional sequence of elements, which is more restrictive than our notion and is not related to

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