Some The Fixed Point Properties In Banach Spaces

Using geometric properties to deal with the fixed point property of nonexpansive mapping in Banach spaces have being highly develpmented since Kirk proved that a Banach space with normal structure have weak fixed point property in 1965. The paper will divided five parts,the contents is following.(1)Discussing the relationships between property (L) and Opial property, uniformly Opial property and Garcia-Falset's constant R(X). We have the following results:The equality , Opial modulus hold. The dual space of X with uniformly Opial property and reflexive has weak fixed point property. Meanwhile, we give a necessary and sufficient condition for that Orlicz sequence spaces equipped with the Luxemburg norm has property (L).(2)Relationship between Garcia-Falset geometric coefficient R(X) and the fixed point property. In this chapter, we get that if Garcia-Falset geometric coefficient then the mapping T: has weak fixed point property in Banach space X..(3) -property and the fixed point property. To study the dual space of a Banach space X with uniformly Kadec-Klee property,we introduce a new property, namely -property and prove that if X is separable andhas -property then X has UKK property. We also get that if the space X has W-property then X have weak Banach-Saks property and the weak fixed point.(4)Calculating Garcia-Falset coefficient R(X) in classical Banach spacesIn this Chapter, we get the equalities hold. Meanwhile, we give calculating program of R(X) in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm.(5) Calculating convex modulus . Define . We get the equality convex coefficient forholds..