Neumann, Jan An abstract differential equation and the potential bifurcation theorems by Krasnosel’skij. (English) Zbl 0633.34053 Commentat. Math. Univ. Carol. 28, 261-276 (1987). 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 Keywords:potential bifurcation theorems; first order differential equation; Hilbert space; potential operator Citations:Zbl 0050.343 PDFBibTeX XMLCite \textit{J. Neumann}, Commentat. Math. Univ. Carol. 28, 261--276 (1987; Zbl 0633.34053) Full Text: EuDML