A new method for obtaining eigenvalues of variational inequalities: operators with multiple eigenvalue. (English) Zbl 0621.49005

Kučera, Milan

Czech. Math. J. 32(107), 197-207 (1982).

Let K be a closed convex cone in a Hilbert space H, \(A: H\to H\) a linear symmetric completely continuous operator. The paper deals with an eigenvalue problem for the variational inequality (I) \(u\in K\), (II) (\(\lambda\) u-Au, v-u)\(\geq 0\) for all \(v\in K\). It is proved that under certain assumptions there exists a connected and unbounded in \(\epsilon\) branch of solutions [\(\lambda\),u,\(\epsilon\) ]\(\in {\mathbb{R}}\times H\times {\mathbb{R}}\) of the corresponding equation with the penalty \(\lambda u- Au+\epsilon \beta u=0\), starting with \(\epsilon =0\) at a given eigenvalue \(\lambda_ 0\) of the operator A and converging to a new eigenvalue \(\lambda_{\infty}\) and eigenvector \(u_{\infty}\) of the variational inequality (I), (II) if \(\epsilon \to +\infty\). In some cases, this method yields the existence of infinitely many eigenvalues of (I), (II) converging to zero with the corresponding eigenvectors on the boundary of K. Unlike in the author’s previous papers concerning this topic, no assumption about the multiplicity of the initial eigenvalue \(\lambda_ 0\) is necessary.

MSC:

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

Keywords:

eigenvalue problem; variational inequality; branch of solutions

References:

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