Geometric Constant Of Banach Space And Application

From the beginning Clarkson introduced the modulus of uniformly convexity to study the characters of unit ball in Banach spaces and Kirk proved that normal structure implies fixed point property, the geometrical theory of Banach spaces and fixed point problems are one of pop research subjects all the while. Recently, many authors characterize the geometrical structure of Banach spaces by geometrical constants and get some geometrical conditions which imply normal structure by using the relations of constants and coefficients. These methods give us some ideas.The moduli and constants of Banach spaces and their applications in fixed point theory are studied in this thesis. Firstly, by learning the former experiences, we generalize the modulus of W~*-convexity which was introduced by Ji Gao in 2004 and define generalized modulus of W~*-convexity. Some properties of the new constant are discussed thoroughly, such as some equivalent definitions and its ultrapower form. We also characterize the uniformly nonsquare by generalized modulus of W~*-convexity and obtain some geometrical conditions which imply fixed point property by using the relations between the new constant and other classical coefficients. These results are new and extend some known conclusions.Secondly, to proceed from uniformly convexity, we introduce a new concept of convexity,r-uniformly convexity. We also characterize r-uniformly convexity by some known constants and prove that r -uniformly convexity is a super-property: a Banach space X is r -uniformly convexity if and only if its ultropower X% is r -uniformly convexity. Using the definition of r -uniformly convexity, we get some results of fixed point property.