The Modulus Of Near Smoothness In Orlicz Sequence Spaces

Geometry theory of Banach space is an important research content of functional analysis.Geometry theory is an important tool for studying geometrical structure and the fixed point in Banach space.Some geometric constants in Banach space and Orlicz spaces are studied in this thesis.It is obtained that calculation formula of the geometric as well as its relationship with other geometric properties in Banach space and Orlicz spaces The thesis is divided into four chapters,the main research is as follows:In Chapter one,we introduce the development history and the background of the theories of Banach spaces and Orlicz spaces,decribe the singnificance of some geometric properties in Banach spaces and Orlicz spaces,and give the main contents.In Chapter two,a new geometric constant C_z^p?X?for a Banach space X is introduced,called the generalized Zbaganu constant.Next,it is shown that the upper and the lower bounds of the constant estimation for any Banach space X.Moreover,it gives the equivalent conditions of X to be the uniformly non-square and that discusses the relationship between the James constant and C_z^p?X?Finally,the relationship between C_z^p?X?and the fixed point property are found.In Chapter three,we prove that the modulus of nearly uniform smoothness in Kothe sepaces without absolutely continuous norm is ?X?t?= t.Meanwhile,the formular of the modulus of nearly uniform smoothness in Orlicz sequence spaces equipped with the Luxemburg norm is given.As corollary,we get criterion that Orlicz sequence spaces equipped with Luxemburg norm is nearly uniform smoothness.Finally,the equivalent conditions of R(a,l_?)<1+a and RW(a,l_?)<1+a are given.In Chapter four,some new results on the modulus of nearly uniform smoothness in Orlicz sequence spaces equipped with the Orlicz norm and some its application to the fixed point property are presented.Some useful formulas for this modulus are presented and their usefulness on some examples are illustrated.