Consider F∈C(R×X,Y) such that F(λ, 0) = 0 for all λ∈R, where X and Y are Banach spaces. Bifurcation from the line R×{0} of trivial solutions is investigated in cases where F(λ, · ) need not be Fréchet differentiable at 0. The main results provide sufficient conditions for μ to be a bifurcation point and yield global information about the connected component of {(λ,u):F(λ,u)=0 and u≠0}∪{(μ,0)} containing (μ, 0). Some necessary conditions for bifurcation are also formulated. The abstract results are used to treat several singular boundary value problems for which Fréchet differentiability is not available. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

中文翻译：

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Hadamard 导数孤立奇异点处的全局分岔

考虑 F∈C(R×X,Y) 使得 F(λ, 0) = 0 对于所有 λ∈R，其中 X 和 Y 是 Banach 空间。在 F(λ, · ) 不需要在 0 处 Fréchet 可微的情况下，研究了来自平凡解的线 R×{0} 的分岔。主要结果为 μ 作为分岔点提供了充分条件，并产生关于{(λ,u):F(λ,u)=0 和 u≠0}∪{(μ,0)} 的连通分量包含 (μ, 0)。还制定了分岔的一些必要条件。抽象结果用于处理 Fréchet 可微性不可用的几个奇异边值问题。本文是主题问题“微分方程和差分方程中的拓扑度和不动点理论”的一部分。