Banach Space Geometry Constant And Its Nature

Banach space geometric structure and the fixed point property are mainly throughthe geometric constants and moduli to carry out research, so the geometric structure andgeometric constants of the intrinsic link between the research became a hot issue. Recently,many research work mainly by using di?erent geometric constant, to explore the Banachspace has normal structure (uniform normal structure) of the suffcient condition.Moduli or geometric constants of Banach space are mainly introduced in the the-sis.Their properties and the relationship among uniform nonsquareness,normal structureand uniform normal structure are studied. The paper is organized as follows:First, U^(β)(ε), the generalization of the modulus of U-convexity, is investigated. Thefunctional properties of this modulus, such as monotonity and continuity, are also obtained.Moreover, we get some suffcient conditions for X to have normal structure, that is, thereexist aε, 0≤ε≤2, if U_X^(β)(ε) > max {0, (1-β)(ε-1)} then X has uniform normalstructure;Also, if U_X^(β)(ε) > max{0, (2-2β)(ε-1)},then X and X* has uniformly normalstructure. Moreover,some suffcient conditions for which a Banach space has normal struc-ture by the modulus and Benavides coe?cient are obtained,which generalize some resultsof Gao and Zuo. which generalized a known result of J. Gao.Next, lower bounds for the weakly convergent sequence coeffcient in term of param-eterized James cnstant and generalized James constant is established. Some suffcientconditions which imply normal structure are obtained by the bounds, and therefore aBanach space X has fixed point property.Finally, A new geometric constant E_p(t, X) is introduced,it is a generalization of Gaoconstant E(t, X). Monotonicity, continuity and convexity of this new constant are dis-cussed. Some suffcient conditions for which a Banach space has normal structure bymeans of the constant and weak orthogonal coe?cient are obtained.