A remark on the global structure of the solution set for a generic class of non-linear eigenvalue problems
Fordyce A. Davidson · Discovery Research Portal (University of Dundee) · 2002
We discuss the structure of the solution set for a generic class of non-linear eigenvalue problems in a Banach space. In particular, we introduce an extension of a classical result from global bifurcation theory which has applications to the study of physical systems. 1 Introduction In this note we consider the following non-linear eigenvalue problem, u = G(; u); (1) where, 2 IR, u 2 E, a real Banach space with norm k \\Delta k and G : E j IR \\Theta E ! E is compact and continuous. The norm in E is defined to be k(; u)k = (jj 2 + kuk 2 ) 1=2 . We assume that (1) has the trivial solution (; u) = (; 0) for all values of and are interested in bifurcation from this trivial solution. Let C denote the closure of the set of non-trivial solutions to (1). In his classic paper [10], Rabinowitz considers the eigenvalue problem (1) under the assumptions G(; u) = Lu +H(;u); where H(; u) is o(kuk) for u near zero uniformly on bounded intervals of and L is a compact linear map on E. ...