Remarks on non-linear spectral theory. (English) Zbl 0539.47042

The authors investigate spectral properties [in the sense of M. Furi, M. Martelli and A. Vignoli, Ann. Mat. Pura Appl., IV. Ser. 118, 229–294 (1978; Zbl 0409.47043)] of asymptotically linear mappings in Banach spaces. For example, they derive the following result.
Let \(X\) be a real Banach space, \(f: X\to X\) an asymptotically linear mapping with asymptotic derivative \(T\) and assume that \(f-T\) is compact. Let \(\lambda_ 0\) be an eigenvalue of \(T\) of odd algebraic multiplicity and with \(| \lambda_ 0|>r_ e(T)\) (= the essential spectral radius of the complexification of \(T\)). Then \(\lambda_ 0\) is an asymptotic bifurcation point for \(f\), i.e. there is a sequence \((\lambda_ n,x_ n)\) in \(R\times X\) such that \(fx_ n=\lambda_ nx_ n\), \(\| x_ n\| \to \infty\) and \(\lambda_ n\to \lambda_ 0\).


47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47J05 Equations involving nonlinear operators (general)


Zbl 0409.47043