Fixed Points for Mean Non-expansive Mappings

Abstract For a Banach Space X Garcia-Falset introduced the coefficient R(X) and showed that if R(X) < 2 then X has a fixed point. In this paper, we define a mean non-expansive mapping T on X in the sense that ∥Tx − Ty∥ ≤ a∥x − y∥ + b∥x − Ty∥ for any x, y ∈ X, where a, b ≥ 0, a + b ≤ 1. We show that if then T has a fixed point in X. Supported by the National Natural Science Foundation of China (No. 10461006), the Natural Science Foundation of Shandong Province (Y002A10) and the Younger Foundation of Yantai University (SX05Z9).