Invariant curves for mappings. (English) Zbl 0606.47056

The main result concerns a smooth map T of a Banach space X into itself which has an unstable fixed point \(x_ 0\). We prove that if the spectral radius \(\lambda_ 0\) of the Fréchet derivative of T at \(x_ 0\) is an eigenvalue which exceeds one and appropriate additional assumptions hold, then there is a smooth invariant curve emanating from \(x_ 0\) which might be called the ”most unstable manifold” of \(x_ 0\). The curve is parametrized by a smooth function satisfying a functional equation involving T and \(\lambda_ 0\). This result is shown to be especially useful when the map T possesses certain monotonicity conditions. In this case, the curve can be shown to be monotone and to terminate on a stable fixed point of T.


47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
46G05 Derivatives of functions in infinite-dimensional spaces
47J05 Equations involving nonlinear operators (general)
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