A fold bifurcation theorem of degenerate solutions in a perturbed nonlinear equation. (English) Zbl 1230.37060

Consider a nonlinear equation \(F(\varepsilon,\lambda,u)=0\), where \(\varepsilon\) is a perturbation parameter, \(\lambda\) is a real number, \(u\) belongs to a Banach space, and \(F\) is a differentiable mapping into a Banach space. The authors use the generalized saddle-node bifurcation theorem to obtain a general bifurcation result.


37G10 Bifurcations of singular points in dynamical systems
37H20 Bifurcation theory for random and stochastic dynamical systems
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[1] M. G. Crandall and P. H. Rabinowitz, “Bifurcation from simple eigenvalues,” Journal of Functional Analysis, vol. 8, pp. 321-340, 1971. · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2
[2] M. G. Crandall and P. H. Rabinowitz, “Bifurcation, perturbation of simple eigenvalues and linearized stability,” Archive for Rational Mechanics and Analysis, vol. 52, pp. 161-180, 1973. · Zbl 0275.47044 · doi:10.1007/BF00282325
[3] P. Liu and Y.-W. Wang, “The generalized saddle-node bifurcation of degenerate solution,” Polskiego Towarzystwa Matematycznego, vol. 45, no. 2, pp. 145-150, 2005. · Zbl 1124.47045
[4] P. Liu, J. Shi, and Y. Wang, “Imperfect transcritical and pitchfork bifurcations,” Journal of Functional Analysis, vol. 251, no. 2, pp. 573-600, 2007. · Zbl 1139.47042 · doi:10.1016/j.jfa.2007.06.015
[5] K.-C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer, Berlin, Germany, 2005. · Zbl 1206.76017 · doi:10.1080/03091920500291805
[6] H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, vol. 156 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2004. · Zbl 1032.35001
[7] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, vol. 426 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2001. · Zbl 0978.47048 · doi:10.1201/9781420035506
[8] C. Pötzsche, “Nonautonomous bifurcation of bounded solutions I: a Lyapunov-Schmidt approach,” Discrete and Continuous Dynamical Systems B, vol. 14, no. 2, pp. 739-776, 2010. · Zbl 1215.37017 · doi:10.3934/dcdsb.2010.14.739
[9] C. Pötzsche, “Persistence and imperfection of nonautonomous bifurcation patterns,” Journal of Differential Equations, vol. 250, no. 10, pp. 3874-3906, 2011. · Zbl 1218.37035 · doi:10.1016/j.jde.2010.12.019
[10] J. Jang, W.-M. Ni, and M. Tang, “Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model,” Journal of Dynamics and Differential Equations, vol. 16, no. 2, pp. 297-320, 2004. · Zbl 1072.35091 · doi:10.1007/s10884-004-2782-x
[11] P. Gray and S. K. Scott, Chemical Oscillations and Instabilities: Nonlinear Chemical Kinetics, Proceedings in Nonlinear Science, Clarendon Press, Oxford, UK, 1990.
[12] J. Shi, “Persistence and bifurcation of degenerate solutions,” Journal of Functional Analysis, vol. 169, no. 2, pp. 494-531, 1999. · Zbl 0949.47050 · doi:10.1006/jfan.1999.3483
[13] M. Z. Nashed, Ed., Generalized Inverse and Applications, Academic Press, New York, NY, USA, 1976.
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