Fixed point theorems of nonexpansive mappings on weakly compact sets

In this paper, we prove that if K is a nonempty weakly compact set in a Banach space X, \(T: K \rightarrow K\) is a nonexpansive map satisfying \(\dfrac{x+Tx}{2} \in K\) for all \(x \in K\), then T has a fixed point whenever X is uniformly rotund with respect to every k-dimensional subspace or X satisfies the property (P). These results improve the results of Veeramani and Radhakrishnan et al.