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Topological identification of multiple solutions to parametrized nonlinear equations. (English) Zbl 0615.47042

Let L:E\(\to F\) be an isomorphism of Banach spaces, let \(H:E\times R^ n\to F\) be a completely continuous mapping, and let \(B:E\to R^ n\) be a bounded linear mapping onto a euclidean space. The solutions (y,\(\lambda)\) to the problem: \(Ly=H(y,\lambda)\), \(By=0\) can be represented as the fixed points of a mapping \(T:E\times R^ n\to E\times R^ n\). Nielsen fixed point theory may be extended to produce lower bounds for the number of fixed points of such maps. A typical problem to which the theory applies is a ”three-point boundary value problem” \(y=y(t):[0,1]\to R^ n\) and \(\lambda \in R^ n\) which solves \(y''=h(t,y,y',\lambda)\) such that \(y(0)=y(1/2)=y(1)=0\).

MSC:

47H10 Fixed-point theorems
55M20 Fixed points and coincidences in algebraic topology
34B15 Nonlinear boundary value problems for ordinary differential equations
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