Generalized Modulus Of Smoothness Of Banach Space And Application In Fixed Point

The study of the geometry of Banach space has been the focus of many researchers' attentions,especially the geometric structure and various geometric constants of Banach space,these different geometric constants can not be only used of the geometric structure of Banach space,can also be combined with the fixed point theory,it is an important clue for the research of the fixed point property.This paper focuses on the geometric constant in Banach space and its applications in fixed point theory,the study of the research at the same time also introduces new geometric constants,the geometric constants and functions that we are familiar with together,trying to find the relationship of the new constant and normal structure.The whole article includes three aspects of the research contents.Firstly,the geometric constants of Banach space and their developments are briefly introduced,in which the modulus of smoothness and the generalized modulus of smoothness are introduced.Secondly,by means of the relations between modulus of convexity and the modulus of smoothness,based on the analysis of the properties of convex modulus,respectively discuss some geometric properties in Banach space of the modulus of smoothness and its relationship with normal structure,at the same time,make a detailed analysis of the generalized modulus of smoothness,such as the conditions of continuity,monotonicity,convexity and smoothness in the characterizations of Banach spaces with normal structure,with some important conclusions obtained combined with the fixed point theory,the launch of its fixed point property conditions.Finally,on the basis of the promotion of the modulus of smoothness,and generalize a new constant,with new constants and generalized modulus of smoothness to characterize the smoothness of Banach space at the same time,get a new sufficient conditions can satisfy the uniformed normal structure.