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Multivalued    $$*$$  ∗ -nonexpansive mappings are studied in Banach spaces. The demiclosedness principle is established. Here we focus on the problem of solving a variational inequality which is defined on the set of fixed points of a multivalued    $$*$$  ∗ -nonexpansive mapping. For this purpose, we introduce two algorithms approximating the unique solution of the variational inequality.

Some Results on the Approximation of Solutions of Variational Inequalities for Multivalued Maps on Banach Spaces