| Geometric parameters or moduli of a Banach space are mainly introduced in this thesis. Their properties and the relationship among uniform nonsquareness, normal structure and uniform normal structure are studied, and the contact with fixed points is also established.TheUβ- modulus of convexity uβ(ε)andUβ- space are defined. We prove that if there existsδ> 0, such that uβ(1 -δ)> 0, then R ( X ) < 2; We also obtain the results that if uβ(1) > 0, then X is uniformly nonsquare, and that if there isδ> 0, such that uβ(1 / 2 -δ)> 0, then X has normal structure, and therefore X has the fixed point property.Next, the coefficient E ( X ), introduced by J.Gao recently, is also studied. Furthermore, a sufficient and necessary condition, which imply a Banach space X is super reflexive is equivalent to (E|-)(X) < 8, is given.Generalizing all the previous results, and three sufficient conditions for Banach space X to have fixed point property are obtained.Followed that, the generalization of Clarkson's moduli of convexity is investigated. Those are some geometric properties ofδαX(ε). The functional properties of this modulus, such as monotonity and continuity, are also obtained. Moreover, some sufficient conditions for X to have normal structure are obtained, that is, if there exists anε, 0≤ε≤1, such thatδαX (1 +ε) > (1 -α)ε, then X has a normal structure; If there exists anε∈[0,1] and anα∈[0,1], such thatδαX(1 +ε) > f(ε), then X has a normal structure. These results generalize the conclution of J.Gao.Finally, a sufficient condition is achieved that nonexpansive set-valued mappings have fixed points on Banach space by means of moduli of smoothness nonsquare constants.
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