β Property Of Musielak-Orlicz Spaces

According to the requirement of various subjects and applications, there are many generalizations of Orlicz spaces. Musielak-Orlicz spaces are common generalization. Uniform rotundity, βproperty and weak βproberty are important geometric properties in Banach spaces. There are many important applications in approximate theory, control theory and variation inequalities etc.. In this paper, we mainly investigate some properties of Musielak-Orlicz spaces. It consists of three 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, and it evaluated and summarized the pioneers'main research results and showing the background and significance of the content of each part in this paper. Secondly the criteria for βproperty, local uniform convexity, local weak uniform convexity and local βproperty in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm are given in the chapter 2. It is well known that βproperty,weak βproperty and weak fixed point property are all important concepts in Banach Theory. People have paid much attention on it. There are many discussions about the criteria for weak βproperty and weak fixed point property in the chassical Orlicz spaces. However, because of the complication of Musielak-Orlicz spaces, at present, there are only a few discussions about them. Last we give Garcia-Falset coefficient R (l_M)and the relation between βproperty and weak fixed point property in the chapter 3. We give that a Banach space is CLUR if and only if the space has CLkR and properties. Furthermore, the sufficient and necessary condition for CLkR property in Orlicz sequence spaces equipped with the Orlicz norm is given in the chapter 4. WM1...