Some Geometric Properties And Application In Banach Space

Since early last century, Banach contraction mapping principle is presented by Banach, it has been widely applications in many fields. In 1965, W. A. Kirk had proved the property that Banach space with the normal structure has the weak fixed point property. In 1965, KK properties was introduced by Kadec-Klee and the definition of nearly uniformly convex and Uniform Kadec-Klee property were introduced by Huff in 1980, they had got that a Banach space is NUC if and only if it is reflexive and it has (UKK ),and a Banach space with (UKK )has ( KK )property. In 1992, the definition of WUKK and WUKK ' had been introduced by D. KUTZAROVA, E. MALUTA & S. PRUS, and WUKK implies FPP had been proved by D. Van Dulst and B. Sims, and the new definition of UKK * and WUKK * had been introduced in 1983.In this paper, three new geometric properties, namely W^*U KK ' properties and W^*U KK^(α) and W^*U KK^(α′) in the dual space X * of Banach space X , are introduced. Moreover, the relationships of three new geometric properties and fixed point property are studied.The thesis is devided into three parts. First of all, some basic concepts and basic properties of Banach space are introduced. And the relationship of some geometric properties is shown.Second, according to the discussion of KK properties and UKK properties and WUKK properties and WUKK ' properties, three new geometric properties that W^*U KK ' properties and W^*U KK^(α) and W^*U KK^(α′) in the dual space X * of Banach space X are introduced.Finally, it is presented that if the dual space X * of Banach space X has W^*U KK ' properties and W^*U KK^(α) and W^*U KK^(α′) in the dual space X *, Banach space X has the weak fixed point property.