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We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool { T λ } of nonexpansive mappings. The difficulty of handling a possibly uncountable family is settled down to dealing with a countable family of auxiliary mappings { S n } associated to { T λ } , in the sense that the approximate fixed points of { S n } provide the common fixed points of { T λ }. Algorithms with strong convergence for solving the associated variational inequality problems are presented. Applications to convex minimization problems and convex feasibility problems are provided, together with numerical examples for comparisons of our algorithms with the existing ones. [ABSTRACT FROM AUTHOR]