Generalized Modulus Of Smoothness Of Banach Space And Application

The study of Banach space geometry theory has always been focused on by many researchers,especially in the study of geometry structure of Banach spaces and various of geometric constants.To apply these different geometric constants can not only through the study of geometric structure of Banach spaces,but also need to combine with the fixed point theory.In this paper,we mainly study Generalized Modulus of Smoothness of Banach Space and Application.This paper mainly studies from three aspects.At the first,we introduce the geometric constants of Banach space and developments of Banach space theorem briefly.In addition,the modulus of smoothness and the generalized modulus of smoothness are introduced.Secondly,introduced the definition of the generalized modulus of convexity and the dual relationship between the generalized modulus of convexity and the generalized modulus of Smoothness.Given an inequality for the generalized modulus of convexity when Vx,y(?)X and ||x||2+||y||2=2.Proved the equivalence conditions of strict convexity of the generalized modulus of convexity in Banach spaces and researched the relationship between the generalized modulus of convexity and the uniform normal structure.Finally,proved that ? and t satisfy some conditions,the inequality relation between px(t?)and rp(?),that(?)a is monotonic in ?(?)(0,2],and that Banach space has a uniform normal structure.Studied the relationship between the generalized modulus of Smoothness and t in Banach space.proved three equivalent conditions of uniformly non-square and four equivalent propositions about the generalized modulus of smoothness.In addition,it is proved that Banach spaces and super-reflexive banach spaces satisfy the conditions of(?)and pX?(t)<?+3/2t?(x)-1 have a uniform normal structure and researched about generalized deformable modules.