A product topology convergence scheme for finding a function that it's values are common fixed points of a family of representations

In this paper, using a family of representations of nonexpansive mappings, we introduce an algorithm in a product space $E^I$ consisting of all functions from a nonempty set $I$ to a Banach space $E$. Then we prove the product topology convergence of the proposed algorithm to an element of $E^I$ such that it’s values are the common fixed points of the representations of the family.