The Calculation And Application Of Some Geometric Constants In Banach Space

The geometric theory of Banach spaces is an important research direction of functional analysis,which has been widely applied in many fields of modern mathematics,such as fixed point theory,cybernetics,approximation theory,theory of martingale,harmonic analysis and so on.In particularly,Kirk proved that the Banach space with normal structure has fixed point property.Since then,the research on the existence of fixed point for nonexpansive mappings has been developed rapidly.However,it is very difficult to give the characterization of geometry structure of Banach space.In recent years,many geometrical constants have been introduced by many scholars,which can describe the geometry structure of Banach space.In this thesis,we not only study the calculation method of the constant for some concrete spaces,but also give some application in the existence of fixed point for nonexpansive mappings.Firstly,we present some sufficient conditions for which the Banach space X has uniform normal structure in terms of the Banas type modulus of convexity.Moreover,some inequalities concerning the weakly convergent sequence coefficient,James type constant,the coefficient of weak orthogonality and the Benavides coefficient were established,which enable us to obtain some sufficient conditions for normal structure.Meanwhile,the parameterized Jordan von-Neumann type constant was introduced,some sufficient conditions for the normal structure on this constant are obtained.The results obtained in this paper represent an extension as well as refinement of previous known results.Secondly,we give some sufficient conditions for the(DL)condition in terms of the James type constant,parameterized James constant,parameterized Jordan von-Neumann type constant,which imply the existence of fixed points for multivalued nonexpansive mappings.Finally,we give a method to compute some geometrical constants under the absolute normalized norms,which give some practical example in the application of geometric theory of Banach spaces.