A fixed-point index and existence theorems for semilinear equations in cones

Abstract

The purpose of this thesis is to develop fixed point indices for A-proper 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 A-proper 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, A-proper maps, Fredholm operators of index zero, and the definition and properties of the generalised degree for A-proper maps. In Chapter 2, we define a fixed point index for A-proper 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 A-proper 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 Lan-Webb fixed point index for weakly inward A-proper at 0 maps to semilinear operators. This index is also extended to include unbounded sets. Existence theorems of positive and non-negative 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 non-negative solution to a periodic boundary value problem; and, a non-negative solution to a Volterra integral equation.