In this paper we proves the convergence of a convex sequence $x_n = \lambda x_{n−1} + (1 −\lambda )f(x_{n−1})$, $\lambda \in (0, 1)$, to a fixed point of the nonexpansive completely continuous operator in the normed $f_{\lambda}$-orbitally complete spaces with $\lambda$-uniformly convex sphere. Further we shall prove some fixed point theorems of the star-shaped sets.