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Bifurcation from simple eigenvalues. (English) Zbl 0219.46015

Let \(G\) map a subset of a Banach space \(W\) into a Banach space \(Y\) and let \(C\) be a curve in \(W\) such that \(G(C) = \{0\}\). A point \(p\in C\) is called a bifurcation point for \(G\) with respect to \(C\) if every neighborhood of \(C\) contains zeros of \(G\) not on \(C\). The main result of the paper gives sufficient conditions for \(c\) to be a bifurcation point and for this case precisely characterizes \(G^{-1}(0)\) near \(c\). A stronger and useful specialization treats the case where \(C\) is a line segment which arises often in practice. Several illustrative applications are given, in particular to ordinary and partial differential equations.

MSC:

46B99 Normed linear spaces and Banach spaces; Banach lattices
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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