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Bifurcation points of variational inequalities. (English) Zbl 0621.49006

Let K be a closed convex cone in a Hilbert space H with its vertex at the origin. Let \(A: H\to H\) be a linear (in general nonsymmetric) completely continuous operator, \(N: {\mathbb{R}}\times H\to H\) a nonlinear completely continuous operator satisfying the condition \(\lim_{\| v\| \to 0}(N(\mu,v)/\| v\|)=0\) uniformly on bounded \(\mu\)-intervals. An arbitrary couple of simple characteristic values \(\mu^{(0)}\), \(\mu^{(1)}\) \((0<\mu^{(0)}<\mu^{(1)})\) of the operator A having eigenvectors \(u^{(0)},u^{(1)}\), respectively, in the interior of K (with \(-u^{(0)},-u^{(1)}\not\in K)\) is considered. Under certain assumptions it is proved that there exists a bifurcation point \([\mu_{\infty},0]\) of the variational inequality (I) \(v\in K\), (II) \(<v-\mu Av+N(\mu,v)\), \(w-v>\geq 0\) for all \(w\in K\) with \(\mu_{\infty}\in (\mu^{(0)},\mu^{(1)})\). In some cases this ensures the existence of an infinite sequence of bifurcation points of (I), (II). The proof is based on a known global bifurcation result, which is applied to the corresponding equation with penalty.

MSC:

49J40 Variational inequalities
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
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References:

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