| Multivalued analysis is a part of modern mathematic, which is booming at 1940s.It has been an important part of nonlinear analysis ,as a mathematic theory and powerful tool in building nonlinear problem mathematic models and solving nonlinear problems. It is widely applied in various fields such as control theory and differential game theory, mathematic economics and decision theory, nonlinear optimal theory, biology mathematic, physics and topology, functional analysis, convex analysis and nonsmooth analysis, differential equation and differential inclusions and so on. Its spirit has also penetrated in the research of many social science, natural science and technology fields. Nowadays, the research of multivalued analysis theory and application is still booming.Decomposable set is an important concept of multivalued analysis. the definition of a decomposable set formally looks like that of a convex set .In this paper ,we have mainly discussed the property of decomposable sets in Lp (Ω,Χ)(1≤p≤∞).Firstly , we have obtained some basic characterizes of decomposable sets ,which are similar to those of convex sets. Nevertheless, though the further research, we discovered that it must be an unbounded set and be dense in the whole space when it has an inner point in Lp (Ω,Χ)(1≤p<∞). If p =∞,the conclusion is not real .Using the knowledge related to multivalued analysis, we get the sufficient and necessary conditions for a decomposable set having inner points in L∞(Ω,Χ).Finally ,this paper primarily study the strong and weak convergence property of decomposable sets, that means we can find a decomposable composition which is convergent in the strong topology to every weak convergence sequence .There is no doubt that this result will play an important part in multivalued analysis and differential inclusions. |
