Adv. Math. Econ. 8, 155-184 (2006)

On differentiability and bifurcation

Gilles Evéquoz^{1} and Charles Alexander Stuart^{2}

1 Ecole Polytechnique Fédérale de Lausanne,Institut d’Analyse et de Calcul Scientfique,CH-1015Lausanne,Switzerland

2 Ecole Polytechnique Fédérale de Lausanne,Institut d’Analyse et de Calcul Scientfique,CH-1015Lausanne,Switzerland

Received: April 7, 2005 Revised: September 9, 2005

JEL classification: C65

Mathematical Subject Classification (2000): 47J15, 58E07, 35P30

Abstract. For a function acting between Banach spaces, we recall the notions of Hadamard and w-Hadamard differentiability and their relation to the common notions of Gâteaux and Fréchet differentiability. We observe that even for a function F: H → H that is both Hadamard and w-Hadamard differentiable but not Fréchet differentiable at 0 on a real Hilbert space H, there may be bifurcation for the equation F (u) = λu at points λ which do not belong to the spectrum of F′(0). We establish some necessary conditions for λ to be a bifurcation point in such cases and we show how this result can be used in the context of partial differential equations such as

−Δu(x) + q(x)u(x) = λe^{−|x|} tanh(e^{|x|}u(x)) for u ∈ H^{2}(ℝ^{N})

where this situation occurs.