Some Geometric Properties And Application In Banach Spaces

With the establishment of Banach space theory,its related results have been widely concerned in algebra,control theory,calculus and other fields,and have brought more common applications to science and technology in other fields.Therefore,it is important to study the relationship between many different properties and fixed points of Banach space.This paper is divided into four parts,as follows:Firstly,the source,research purpose and significance of the subject are introduced,and briefly expounds the development status of Banach space theory at home and abroad.Secondly,we give a new geometric property of Banach space — kUKK,and prove that Banach space with this property has weak Banach-saks property.The sufficient and necessary condition that Banach space is kNUC is reflexive and has kUKK property;Finally,in view of the important role that geometric constant plays in the geometric properties of Banach space,a new geometric constant is given by the definition of kUKK.When the constant is less than k,Banach space has the property of weak fixed point,and the specific value of the constant in Cesaro sequence space is calculated.Thirdly,the definition of the ? properties is given in w-hyperbolic space.The definition of ? modulus in ?w-hyperbolic space proves its approximation compactness and thus has normal structure.Secondly,the nonempty intersection property in ?w-hyperbolic space is proved.Finally,the existence of fixed points for asymptotically averaged nonexpansive maps in ?w-hyperbolic space is proved.Finally,based on the properties of k UKK,the definition of k UKK operator is introduced,and the relationship between kNUC operator and kUKK operator is proved.The sufficient and necessary conditions for the operator in Banach space to be kNUC are reflexive and has kUKK property;The properties of kUKK operator are discussed,and the problem of minimum norm control for continuous distributed parameter system in Banach space with k UKK operator is studied.