Generalized Topological Degree Theory For VMO Functions And Continuous Bounded Operators

This paper mainly deals with Generalized Topological Degree theory for VMO functions and continuous bounded Operators. We depart this paper into three Chapters. In the first Chapter, we recall the background and recent results of Topological Degree theory. The Chapter two and three deal with Topological Degree theory for the sum of VMO functions and Multi-Valued Upper Semi-Continuous Mappings with boundaries and Generalized Topological Degree theory for continuous bounded operator respectively. In the second Chapter, we define the Topological Degree theory for the sum of VMO functions and Multi-Valued Upper Semi-Continuous Mappings with boundaries by means of the existent results of VMO functions degree and Multi-Value Upper Semi Continuous Mappings degree. Then we study the property of the newly defined topological degree. We find it have basic property of the generic topological degree. Meanwhile, it has new property, we call it property of essR.In Chapter three, we consider constructing a new generalized degree function for continuous bounded operator by means of degree theory of Multi-Value Upper Semi Continuous Mappings. We will prove this generalized degree function has the crucial properties of the general topological degree, such as normality, sovlability,homotopy and additively.