Another means for a Banach space, (locally) uniformly convex and uniformly smooth, is to be presented.

Weak convergence result of the Mann iteration method to a fixed point of strict pseudo-contractive mapping in a real 2-uniformly smooth Banach space which is uniformly convex is proved.

Let X be a uniformly smooth Banach space,T:X→X be a continuous strongly accretive operator and Lipschitzian strongly accretive operator,respectively;then we will discuss the Mann sequence iteration and Ishikawa sequence iteration for the fixed point of nonlinear equation Sx=f-Tx+x.

The relations among these uniform convexity and uniform smoothness, very smoothness, uniform extreme smoothness and extreme smoothness are studed.

This paper studies the relations among the average uniform convexity, average local uniform convexity, average weak local uniform convexity, uniform smoothness, very smoothness, uniform extreme smoothness and extreme smoothness.

This paper extends the strong convergence of the CQ method for Ishikawa iteration in Hilbert spaces to smooth and uniformly convex Banach spaces by using the projection operator PK which is introduced by XU Hong-kun.

Describe the boundedness of the operator-valued martingale transforms,and as an application the p-uniform smoothness and q-uniform convexity of Banach spaces are characterized by the boundedness of martingale transform operators.

By using the property of supporting functionals,the paper obtain a characterization of uniform smoothness.

We define the TC modulus of convexity and smoothness of Banach spaces and characterize uniform convexity and uniform smoothness.