Degree theory for multivalued (S) type mappings and fixed point theorems

The main purpose of this paper is devoted to generalizing the results of Browder ^1,2]. This paper consists of four parts. In the first part, we introduce the concepts of multivalued (S) and (S) type mappings and the concepts of the limits of multivalued (S) and (S) type mappings. These kinds of mappings contain many monotone type mappings, such as maximal monotone mapping, bounded pseudo-monotone mapping and bounded generalized pseudo-monotone mapping, as its special cases. In the second part we define the pseudo-degree for (S) type mapping and the degree for (S) ₊ type mapping. These two kinds of degrees are all the generalizations of the degree defined by Browder ^1,2].As applications, we utilize the degree theory presented in part 2 to study the existence of solutions for the multivalued operator equations (see part 3) and to obtain some new fixed point theorems in part 4.Research Supported by the National Natural Science Founation of China.