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For a non expansive mapping, T, acting on a Banach space we determine fixed points of T via converges of sequence. Main result is established at chapter three where the arbitrary sequences are within the open interval (0,1). In chapter four we establish strong convergence for a family of non-expansive self mappings defined in a closed subset of a Banach space X. The result is achieved by developing and proving systems and necessary and sufficient conditions for strong convergence in X.