Title:

Fixed point indices and existence theorems for semilinear equations in cones

The purpose of this thesis is to develop fixed point indices for Aproper semilinear operators defined on cones in Banach spaces and use the results to obtain existence theorems to semilinear equations. We consider semilinear equations of the form Lx = Nx where L is a linear Fredholm operation of index zero, N a nonlinear operator such that L  N is Aproper at zero relative to a projection scheme L. Chapter 1 is an introduction to basic concepts used throughout the thesis, including; Banach spaces, linear operators, Aproper maps, Fredholm operators of index zero, and the definition and properties of the generalised degree for Aproper maps. In Chapter 2, we define a fixed point index for Aproper maps on cones in terms of the generalised degree and derive the basic properties of this index. We then extend the definition to include unbounded sets. A more general fixed point index than that of Chapter 2 is developed in Chapter 3 for Aproper maps based on limits of a finite dimensionally defined index. Properties of the index are given and a definition for unbounded sets is provided. Chapter 4 extends the LanWebb fixed point index for weakly inward Aproper at 0 maps to semilinear operators. This index is also extended to include unbounded sets. Existence theorems of positive and nonnegative solutions to semilinear equations on cones are established in Chapter 5 using the fixed point indices of Chapters 2, 3, and 4. Finally, in Chapter 6, we apply some of the existence theorems of Chapter 5 to several differential and integral equations. We prove the existence of: a positive solution to a Picard boundary value problem; a nonnegative solution to a periodic boundary value problem; and, a nonnegative solution to a Volterra integral equation.
