Nonlinear analysis in geodesic spaces

This thesis deals with nonlinear analysis in geodesic metric spaces, particularly in CAT(0) spaces. A major aim is to investigate the convex feasibility problem associated with the nonempty closed convex sets A and B. This problem is normally investigated in a Hilbert space setting, but in this work we have placed it into a CAT(0) setting. In chapter 2 we begin with metric spaces and move into geodesic metric spaces in order to obtain the structure which we will need in later chapters. In chapter 3 we introduce CAT(0) spaces and investigate their properties, especially those that we will need in chapters 5, 6 and 7. We also introduce new work - "Polarization in CAT(0) spaces". More new work occurs in Chapter 4 where we develop hyperplanes and half-spaces in a CAT(0) space setting and then prove a separation theorem in CAT(0) spaces. Chapter 5 covers fixed point theory in CAT(0) spaces. We use an appropriate notion of weak sequential convergence (introduced in chapter 3) to develop a fixed point theory for nonexpansive type mapping. In chapter 6 we prove that the project project algorithm works in CAT(0) spaces. In chapter 7 we set up a prototype for a CAT(0) space of non-constant curvature, develop its geometry and then investigate the Douglas-Rachford algorithm in this space.