On The Homotopy Invariance Of Topological Degree For Maximal Monotone Mappings

In this paper, the main work is the homotopy invariance of the degree for maximal monotone mappings. The homotopy invariance of the degree for a cluster of maximal monotone mappings is studied. Moreover, the homotopy invariance of the degree for the subdifferentials of convex functions is discussed. The whole paper consists of three chapters.In chapter1, the background of topological degree theory and the history of the homotopy invariance of the degree for maximal monotone mappings are introduced. In addition, the background of the subdifferentials of convex functions is also reviewed.In chapter2, the main topic is the homotopy invariance of the degree for a cluster of maximal monotone mappings. The first step is to construct Hausdorff continuity for the cluster of maximal monotone mappings and their domains. Then by transforming the multi-valued mapping to single-valued mapping with its Yosida approximation, approach-ing its topological degree with the degree of its Yosida approximation, the homotopy invariance of the degree for the cluster of maximal monotone mappings is deduced. Meanwhile, some basic properties of the topological degree are obtained. Similarly, under some additional assumptions, a theorem about the homotopy invariance of the degree for the sum of two maximal monotone mappings is proved.In chapter3, the homotopy invariance of the degree for the subdifferentials of convex functions is considered. In real reflexive space, the subdifferential of lower semicontinuous proper convex function is multi-valued maximal monotone mapping. Thus the homotopy invariance of the degree of the subdifferential is studied.