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An abstract differential equation and the potential bifurcation theorems by Krasnosel’skij. (English) Zbl 0633.34053
Some bifurcation theorems for the problem \(\lambda x-F'(x)=0\) in a Hilbert space, where F’ is a potential operator, that are originally due to M. A. Krasnosel’skij [Mat. Sb. Nov. Ser. 33(75), 199-214 (1953; Zbl 0050.343)], are reproved under slightly weakened assumptions by studying the asymptotic behavior of the ordinary differential equation \[ u'(t)=F'(u(t))-<F'(u(t)),u(t)>\cdot u(t)\cdot | u(t)|^{-2}, \] a method that goes back to I. V. Skrypnik.
Reviewer: H.Engler
MSC:
34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
35B32 Bifurcations in context of PDEs
47J05 Equations involving nonlinear operators (general)
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
Citations:
Zbl 0050.343
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