Nonsimple Nonlinear Eigenvalues and Isolated Spectra

Definitions and mathematical statement

The verified clean statement contains two questions:

1. Describe the nonlinear spectral behavior near an eigenvalue $\lambda_0$ that is not simple for the operator considered in the report.
2. Characterize a class of homogeneous maps on ${\mathbb{R}}^N$ for which every eigenvalue is isolated.

The original short extraction retained only the first question; the supplied verification says the second was restored from the same open-problem paragraph.

For clarity, in the usual degree-one interpretation a homogeneous map is $F:{\mathbb{R}}^N\to{\mathbb{R}}^N$ satisfying $F(tx)=tF(x)$ for the prescribed positive scalars $t>0$. Its real eigenvalue set is \[ \sigma_{\mathrm{eig}}(F)=\{\lambda\in{\mathbb{R}}:\exists x\in{\mathbb{R}}^N\setminus\{0\},\ F(x)=\lambda x\}. \] An eigenvalue $\lambda$ is isolated if some $\varepsilon>0$ satisfies \[ \sigma_{\mathrm{eig}}(F)\cap(\lambda-\varepsilon,\lambda+\varepsilon)=\{\lambda\}. \] Thus the second target seeks structural hypotheses on a class $\mathcal H$ implying \[ \forall F\in\mathcal H\ \forall\lambda\in\sigma_{\mathrm{eig}}(F), \qquad\lambda\text{ is isolated in }\sigma_{\mathrm{eig}}(F). \] This concerns isolation of spectral values, not uniqueness or isolation of eigenvectors. The degree-one positive-homogeneity convention is made explicit for interpretation; the extract does not specify the degree, negative-scalar behavior, regularity, or other intended restrictions on the class.

For the first question, the relevant operator, its domain, a parameter family, and the meaning of nonlinear eigenvalue multiplicity are absent. Consequently no particular Fučik operator, perturbation equation or branching theorem can be supplied without inventing the missing setup. "Not simple" must be matched to the source's multiplicity notion rather than defined by a linear-eigenspace formula for an arbitrary nonlinear map.

The literature assessment checked on 21 August 2026 reports partial results for multiple-eigenvalue cases, including special Fučik-branch analyses, while the requested general nonlinear spectral description remains unresolved in that assessment. It supplies no general class solving the second question. These are dated literature-triage claims requiring independent verification.

Short English statement

Describe what happens near a multiple nonlinear eigenvalue, and find structural conditions on homogeneous finite-dimensional maps that make every eigenvalue isolated. The source restores both questions but omits the operator and class hypotheses needed for a sharper theorem.

Sources

• Mini-Workshop: Nonlinear Spectral and Eigenvalue Theory (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/08, pp. 407--438, https://doi.org/10.4171/owr/2004/08.
• Literature assessment checked 21 August 2026: "Oberwolfach report," https://oa.tib.eu/renate/server/api/core/bitstreams/aaddbeeb-c549-49a0-995d-8691ab5156bd/content; "Fuccik-spectrum paper," https://arxiv.org/abs/1407.0190; "Recent discrete Fuccik paper," https://arxiv.org/abs/2412.11709.
• Local UnsolvedMath registry, permanent ID $30000032$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. The clean statement is labelled corrected and verified; supplied verification explicitly restores both questions without inventing the operator.