| This dissertation deals with fixed point theory in gauge space with its application. On the basis of discussing the topological structure of gauge space and its completeness, we establish several fixed point theorems for the generalized contraction mappings on complete gauge spaces and give their applications. The main contents are as follows:In chapter 1, we introduce the basal concepts of gauge space and some examples, consider its topological structure, and prove the nested theorem of closed setes in gauge space.In chapter 2, we first give a generalized type of the Prigon's fixed point theorem and a straightforward proof of it. At the same time, we give a positive answer to an open question about convergency of iterative sequence for generalized contraction which is bring forwarded by Frigon in [3]. Secondly, by using the introduced function φ, we establish some fixed point theorems for the set-valued φ-contraction mappings in gauge spaces. These results unify and generalize the corresponding results of Cain and Nashed [1], Figon [3], Espinola and Kirk[4], Tarafder [5] et al.In chapter 3, as a application of fixed point theorems in gauge spaces, by using the results in chapter 2 we study the existence and uniqueness of solution for integral equation of Voletrra type in C(R). |
