Complex Rotundity And Some Pointwise Properties Of Orlicz Spaces

The thesis mainly consists of three parts, the first is the fixed point property of mean non-expansive mapping; the second is the criteria for compactly locally uni-formly rotund points of Orlicz spaces; the last is the complex convexity of Musielak-Orlicz spaces.We first discuss the fixed point property of mean non-expansive mapping. There are lots of conclusions related to fixed point properties of non-expansive mapping and many people are interested in this problem. In this paper, the mean non-expansive mapping is introduced. By using geometric coefficients of Banach space X, the fixed point properties of non-expansive mapping are extended. A general sufficient condition of fixed point property of mean non-expansive mapping is obtained.Second, compactly locally uniformly rotund points of Orlicz spaces are studied. It is well known that compactly locally uniformly rotund points are very important in Banach spaces. A point x of the unit sphere of Banach space X is locally uniform rotund if and only if x is both a compactly locally uniformly rotund point and a strongly U point of the unit ball of Banach space X. We give the criteria for compactly locally uniformly rotund points in Odicz spaces with Luxemburg norm and Orlicz norm.At last, the complex midpoint locally uniformly rotundity of Musielak-Orlicz function spaces and Musielak-Orlicz sequence spaces are investigated. In these years, much work in the area of the geometry of Banach spaces was directed at the complex geometry of complex Banach spaces. It is well known that the complex geometric properties of complex Banach spaces have applications in various branches of knowledge, among others in Harmonic Analysis Theory, Operator Theory, Banach Algebras, C~*-Algebras, Differential Equation Theory, Quantum Mechanics Theory and Hydrodynamics Theory. We first introduce the concepts of complex strongly extreme points and complex midpoint locally uniform rotundity for complex Banach spaces, and show that a complex locally uniformly rotund point is a complex strongly extreme point and a complex strongly extreme point is a complex extreme point which im-ply the introduced concepts are reasonable. Moreover, criteria for them in Musielak- Orlicz function spaces and Musielak-Orlicz sequence spaces are given.