Bifurcation at isolated singular points of the Hadamard derivative

Abstract

For Banach spaces X and Y, we consider bifurcation from the line of trivial solutions for the equation F (λ, u) = 0, where F : ℝ × X → Y with F (λ, 0) = 0 for all λ ∈ ℝ. The focus is on the situation where F (λ, ·) is only Hadamard differentiable at 0 and Lipschitz continuous on some open neighbourhood of 0, without requiring any Fréchet differentiability. Applications of the results obtained here to some problems involving nonlinear elliptic equations on ℝN, where Fréchet differentiability is not available, are presented in some related papers, which shed light on the relevance of our hypotheses.