Some The Fixed Point Properties In Banach Spaces

Uniform rotundity is an important geometric property in Banach spaces. There are many important applications in approximate theory, control theory and variation inequalities etc. According to the requirement of various subjects and applications,Orlicz spaces is extended from the classical spaces Lp. There are also manygeneralizations of Orlicz spaces, one of them is Musielak-Orlicz space. In the paper, we mainly investigate pointwise geometric properties in Orlicz spaces or Musielak-Orlicz spaces. For convenient to studying the geometric properties, we introduce a new definition, namely uniformly locally convex point in measure( -URP, for short). Let X be a Banach space. An element x S(X) is called a uniformly locally convex point in measure if xn B(X) and ||xn + x|| - 2= xn -- x. X is called to be uniformly local convex in measure if each point inthe unit sphere of X is uniformly local convex point in measure. It is well known that uniformly locally convex point (URP, for short) is precise and point wise property of uniformly local convex. Studying works from macroproperty into microproperty is a qualitative leap in development of Orlicz spaces geometric theory. The paper is consist of four parts. The main results of this paper are summarized as following:First it reviewed that a survey of the theory of Orlicz and Musielak-Orlicz spaces. We show the background and significance of the content of each part in this paper.Secondly the criteria for that uniformly locally convex point in measure of Orlicz spaces equipped with the Orlicz norm and Luxemberg norm are given in the chapter 2. Meanwhile we get necessity and sufficient condition for that Orlicz spaces equipped with both norms are uniformly locally covex in measure as corollaries.Last we give criteria of uniform rotundity point and weak uniform rotundity point in Musielak-Orlicz sequence spaces in chapter 3. For Musielak-Orlicz function spaces case, we only give necessity and sufficient condition of the spaces are uniformly local convex since the case is very complex.