 Fixed point indices and existence theorems for semilinear equations in cones

Cremins, Casey Timothy (1997) Fixed point indices and existence theorems for semilinear equations in cones. PhD thesis, University of Glasgow.

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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.

Item Type: Thesis (PhD) Doctoral Q Science > QA Mathematics College of Science and Engineering > School of Mathematics and Statistics > Mathematics Webb, Prof. J.R.L. 1997 Angi Shields glathesis:1997-3520 Copyright of this thesis is held by the author. 17 Jul 2012 10 Dec 2012 14:08 http://theses.gla.ac.uk/id/eprint/3520 View Item