Generalised Topological Degree and Bifurcation Theory

[摘要] The objective of this thesis is to give sufficient conditions for global bifurcation of solutions to the nonlinear eigenvalue problem:F(X,lambda) = 0, where F : X x IR→Y, with X x IR, Y Banach spaces and (x,lambda) ∈ X x IR. F(.,lambda) is assumed to belong to the class of A-proper maps and to be of the non-standard form, an A-proper, linear operator A - lambdaB : X → Y plus a nonlinear mapping R(.,lambda) : X → Y. R(X,lambda) is taken to satisfy a smallness condition in x at the origin in X. Our analysis is based on an extension of known methods, for obtaining global bifurcation results, which have been used successfully when the mappings involved are compact or k-set contractive. Chapter One is an introduction to the concepts used throughout the thesis, including Fredholm maps of index zero, A-proper maps and generalised topological degree. In Chapter Two we state and prove our main global bifurcation theorem in terms of the generalised degree; this result forms the basis for the proofs of all the main theorems in the thesis. Chapters Three and Four contain various global bifurcation theorems, for different sets of hypotheses imposed on the mapping F and the underlying spaces X x IR and Y. Finally, in Chapter Five we apply our results to certain classes of ordinary differential equations and obtain existence results, for periodic solutions in one case and not necessarily periodic solutions in another. The main results are: Theorem 2.10; Theorems 3.3 and 3.13; Theorems 4.7, 4.12, 4.15 and 4.18.

[发布机构] University:University of Glasgow

[关键词] Theoretical mathematics