Some Geometric Constants In Banach Spaces And Their Applications

The geometric theory of Banach space is an important branch of functional analysis,which is widely used in many fields.Among them,the fixed point theory is the most important application.It provides theoretical support for the study of many practical problems,such as the study of many practical problems in mathematics and physical engineering.An important tool to study the geometric properties of Banach spaces is the geometric constants of Banach spaces.Therefore,the study of geometric constants related to fixed point properties in Banach spaces has always been a hot topic.Orlicz space is a concrete Banach space.It has a broad prospect to find the specific description and characterization of some geometric properties and geometric constants of Banach space in Orlicz space.In this paper,the geometric properties and fixed points of Banach spaces are studied from the geometric constants of Banach spaces.Firstly,The purpose and significance of this research,and describe the geometric theory and fixed point theory of Banach space,as well as the development process,development status and important achievements of Orlicz space theory at home and abroad are introduced in this paper.Secondly,a new geometric constant,namely U convex coefficient,is introduced in Banach space.The relationship between U convex coefficient and some geometric properties such as uniform non-squareness and normal structure are studied,and it is proved that Banach space satisfying U0(X)<1/2 has normal structure.Then,by studying the relationship between U convex coefficient and constant R(X),it is found that Banach space X has fixed point property.Then,when?(u)u?A(u??)the criterion of the smooth points of Orlicz sequence space is given,and the equivalent conditions of smooth points,very smooth points and strong smooth points are given,and the criterion of the smoothness of Orlicz sequence space in this case is obtained.It is proved that when?(u)/u?A(u??)the Orlicz sequence spaces don't have the U property,and the U convex module in Lebesgue sequence space is also calculeted.Finally,the result of Brailey Sims are generalized and we prove that weakly orthogonal Banach lattices with M constants ?n(X)<n has the weak fixed point property.In addition,the expression formula and estimation formula of M constant ?n(X)in Orlicz sequence space for the Luxemburg norm are obtained and provied that the sufficient and necessary conditions for ?n(l?)in Orlicz spaces.