Computation For Fixed Point Index And Properties Of Cones In Banach Spaces

Some consequences about computation of fixed point index for completely continuous operator are proved in this thesis by retract and retraction where the conditions about cone or the whole space in previous conclusions are replaced with convex closed sets. They unify and extend several results about computation of fixed point index and topological degree in the references, and are explained and compared. Meanwhile, the dimension reduction property of Leray-Schauder degree and the obtuse angle principle in infinitely dimensional Hilbert space are given which are the supplements and analogies of the dimension reduction property of Brouwer degree in finitely dimensional space and the acute angle principle in Hilbert space.On the other hand, various characteristics of nonnegative cones and sign-changing cones are discussed in some concrete Banach spaces through giving the proofs or constructing counterexamples. The contents are whether or not these cones are solid and generating, whether or not those cones are extensible, fully regular, regular and normal, and whether or not those cones are minihedral and strongly minihedral. These Banach spaces include usual sequence spaces and function spaces.Finally, a complete summary is provided about the properties of cones in these concrete Banach spaces discussed above and known results in the references. Some open problems are also posed.