Some Geometric Properties In Musielak-Orlicz Spaces

As an important part of Banach space—Musielak-Orlicz space.Its theory provides intuitive background for general functional analysis and has been applied directly in many fields.For example,the research has played a great role in mathematics and physics.Therefore,throughing the scholars constantly dig into the characteristics of the space,Musielak-Orlicz spaces theory has been greatly developed and laid a foundation for future research.This paper mainly discusses the Kadec-Klee property of Musielak-Orlicz sequence space and Musielak-Orlicz function space.The paper is divided into three parts,which are as follows:Firstly,the research aim and meaning of this paper.On the one hand,the development status and research results of Orlicz space theory is reviewed.On the other hand,Musielak-Orlicz spaces theory research results are summarized.Secondly,the property of Musielak-Orlicz sequence space are discussed.Defining the p-Amemiya norm,and the expression of p-Amemiya norm are considered when k(x)=? and k(x)??.Then we give the sufficient criterion for x?S(l?,p)to be Kadec-Klee point and the necessary and sufficient conditions for l?,p to have the Kadec-Klee property.Finally,in the case of N function,we give the condition that l?,p to have uniform Kadec-Klee property and to be nearly uniformly convex are obtained.Finally,the Kadec-Klee property and fixed point problems in the function space are generalized.In this paper,the case that the Orlicz norm of {kn} is bounded or unbounded is discussed.A new method adopted to make {kn} always bounded in this chapter.Based on the above conclusions,three equivalent conditions of Kadec-Klee property are proved.It is also known that the Kadec-Klee property closed connect with the approximation compact and fixed point property of non-expansive mappings Therefore,we obtain that a class of non-reflexive Musielak-Orlicz function spaces have fixed point property,and an example.