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Remarks on Krasnosel’skij bifurcation theorem. (English) Zbl 0681.47031

A classical bifurcation theorem of M. A. Krasnosel’skij [see, e.g., “Topological methods in the theory of nonlinear integral equations”, London (1974; Zbl 0070.330)] states that any eigenvalue \(\mu_ 0\) of odd multiplicity of a linear operator T is a bifurcation point for the problem \(Tu+F(u)=\mu u,\) provided that the nonlinearity F satisfies \(F(0)=0\) and \(F'(0)=0\). In the present paper, the author generalizes this result to the case of non-differentiable F. For example, the following is shown: let \[ k(F)=\lim_{r\to 0}\inf \{k:\quad \| Fu\| \leq k\| u\| \quad for\quad \| u\| \leq r\}, \] and suppose that there exists \(\lambda <\mu_ 0<\nu\) such that \[ [\lambda,\nu]\cap \sigma (T)=\{\mu_ 0\}\quad and\quad \min \{\| (T-\lambda I)^{- 1}\|^{-1},\quad \| (T-\nu I)^{-1}\|^{-1}\}>k(F) \] (which trivially holds in Krasnosel’skij’s theorem, since \(k(F)=0\) in case \(F'(0)=0)\); then \(\mu_ 0\) is a bifurcation point. An illuminating application to integro-differential equations is also given.
Reviewer: J.Appell

MSC:

47J05 Equations involving nonlinear operators (general)
34B15 Nonlinear boundary value problems for ordinary differential equations
35B32 Bifurcations in context of PDEs
45K05 Integro-partial differential equations

Citations:

Zbl 0070.330
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