Abstract: We introduce a new hybrid generalized f-projection method for finding a common element of the fixed point set of a countable family of totally quasi--asymptotically nonexpansive mappings.We prove the strong convergence theorems of new iterative algorithms under some suitable conditions in a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property.Results extend and unify the previously known results announced by many other authors.