Cubic Stability via Two-Uniform Smoothness

Abstract

Let a fixed point of a family of maps carry a leading cubic correction, Ts (x) − fs =(x − fs ) + Φs (x − fs ). If Φs is a restoring force — negative in the appropriate sense on theunit sphere, with controlled remainder — is the fixed point stable? On Hilbert space this iselementary: test Φs against the inner product. On a general Banach space there is no innerproduct to test against, and the natural substitute, a Hahn–Banach norming functional, is ingeneral not even unique. We show the argument survives regardless: the norming pairing isalways a subgradient of 12 ∥ · ∥2 , a purely convex-analytic fact needing no uniqueness, and oncethe space is two-uniformly smooth — a classical, purely metric property of the norm alone— the entire Hilbert-space mechanism goes through verbatim, with the smoothness constantappearing exactly where the parallelogram identity used to sit. The result specializes toLp (µ), p ≥ 2, via a classical smoothness estimate for the p-norm.