## 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.

### MSC:

 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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