Invariant curves for mappings. (English) Zbl 0606.47056

Smith, Hal L.

SIAM J. Math. Anal. 17, 1053-1067 (1986).

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)

Keywords:

smooth linearization; unstable fixed point; Fréchet derivative; most unstable manifold