One of the most important tools in nonlinear functional analysis is the Leray-Schauder degree for compact vector fields defined on the closure of open subsets of some Banach space. The Leray-Schauder theory can be used to prove the existence of solutions of differential equations. In some cases in differential equations, we need to find the positive solutions. For example, if our equation is a partial differential equation for the concentration of a chemical, the only relevant solutions are the non-negative ones. In some cases, after some transformations, the problem is to find the fixed points in cones of positive mappings (mapping the cone itself). The fixed point index in cones has been used a great deal to study the fixed points of positive mappings in cones. Amann obtained the index relative to a cone at zero under an assumption of non-degeneracy. Dancer obtained a formula for the index of an isolated fixed point of positive mappings under a non-degeneracy assumption, which was extended by Du to strict set contractions. Also, Dancer and Du obtained an index formula for the boundary points of product cones. In this thesis, we extend these results for cones or products of cones to cover more general situations. We introduce new methods for index calculations not covered by the earlier theory. My results are not complete, but it seems very hard to obtain a complete calculation.
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