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An extension of the Rabinowitz bifurcation
theorem to Lipschitz potential operators
in Hilbert spaces


Authors: Alexander Ioffe and Efim Schwartzman
Journal: Proc. Amer. Math. Soc. 125 (1997), 2725-2732
MSC (1991): Primary 58E05; Secondary 49K99
DOI: https://doi.org/10.1090/S0002-9939-97-04061-6
MathSciNet review: 1415327
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Abstract | References | Similar Articles | Additional Information

Abstract: The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations $Ax + {\varphi _{\lambda }(x)\ }= \lambda x$ with $\varphi $ continuous jointly in $(\lambda ,x)$ and $ {\varphi _{\lambda }(\cdot )\ }$ of class $C^{1,1}$. We also prove a bifurcation theorem for critical points of the function $g_{\lambda }(x)$ which is just continuous and changes at $x=0$ an isolated minimum (in $x$) to isolated maximum when $\lambda $ passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.


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  • 1. P.H. Rabinowitz, A bifurcation theorem for potential operators, J. Functional Analysis 25 (1977), 412-424. MR 57:3928
  • 2. M.A. Krasnosel' ski, Topological Methods in the Theory of Nonlinear Integral Equations, GITTL, Moscow 1956 (Russian); English translation: Pergamon Press, Oxford-London-New York 1964. MR 28:2414
  • 3. R. Böhme, Die Lösung der Verzweigungsgleichungen für Eigenwertprobleme, Math. Z. 127 (1972), 105-126. MR 47:910
  • 4. A. Marino, La biforcazione nel caso variazionalle, Conf. Sem. Mat. Univ. Bari, no. 132 (1973). MR 50:1068
  • 5. J.B. McLeod and R.E.L. Turner, Bifurcation of nondifferentiable operators with an application to elasticity, Arch. Rat. Mech. Anal. 63 (1976/77), 1-45. MR 57:13611
  • 6. H. Kielhöffer, A bifurcation theorem for potential operators, J. Functional Analysis 77 (1988), 1-8. MR 89f:58032
  • 7. M. Degiovanni and M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Math. Pura. Appl. 167 (1994), 73-100. MR 96a:58043
  • 8. J.M. Corvellec, M. Degiovanni and M. Marzocchi, Deformation properties of continuous functionals and critical point theory, Topol. Methods in Nonlinear Analysis 1 (1993),151-171. MR 94c:58026
  • 9. A. Ioffe and E. Schwartzman, Metric critical point theory 1. Morse regularity and homotopic stability of a minimum, J. Math. Pures et Appl., 75 (1996), 125-153. CMP 96:10
  • 10. A. Ioffe and E. Schwartzman, Metric critical point theory 2.Deformation techniques and characterization of critical levels of continuous functions, to appear in OT series volume dedicated to the memory of I. M. Glasman, Birkhäuser.
  • 11. G. Katriel, Mountain pass theorem and a global homeomorphism theorem, Ann. Inst. Henri Poincaré - Analyse Non-linéaire 11 (1994), 189-211. MR 95f:58022
  • 12. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley 1983. MR 85m:49002
  • 13. J.P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley Interscience 1984. MR 87a:58002
  • 14. I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474. MR 80h:49007
  • 15. Shui-Nee Chow and J.K. Hale, Methods of Bifurcation Theory, Springer-Verlag, 1982. MR 84e:58019
  • 16. E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol II/B, Springer-Verlag, 1990. MR 91b:47002

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Additional Information

Alexander Ioffe
Affiliation: Department of Mathematics The Technion Haifa 32000, Israel
Email: ioffe@math.technion.ac.il

Efim Schwartzman
Affiliation: Department of Mathematics The Technion Haifa 32000, Israel

DOI: https://doi.org/10.1090/S0002-9939-97-04061-6
Keywords: Critical point, $C^{1, 1}$-function, bifurcation, modulus of regularity, Palais-Smail condition
Received by editor(s): February 5, 1996
Additional Notes: The research was supported by the US-Israel BSF grant 90-00455 and by the Israel Ministry of Science and Technology grant 3501–1–91
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society

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