Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E * , C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E , and T : C → E a non-expansive nonself-mapping with F ( T ) ≠ ∅ . In this paper, we study the strong convergence of two sequences generated by x n + 1 = α n x + ( 1 − α n ) ( 1 / n + 1 ) ∑ j = 0 n ( P T ) j x n and y n + 1 = ( 1 / n + 1 ) ∑ j = 0 n P ( α n y + ( 1 − α n ) ( T P ) j y n ) for all n ≥ 0 , where x , x 0 , y , y 0 ∈ C , { α n } is a real sequence in an interval [ 0 , 1 ] , and P is a sunny non-expansive retraction of E onto C . We prove that { x n } and { y n } converge strongly to Q x and Q y , respectively, as n → ∞ , where Q is a sunny non-expansive retraction of C onto F ( T ) . The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.