Local tangential cone condition

Let $X,Y$ be real Hilbert spaces, let $U\subseteq X$ be open, and let $F:U\to Y$ be Fréchet differentiable at $x\in U$. For $\delta >0$ such that the closed ball

${\bar{B}}_{\delta }(x):=\{x+h:h\in X,\mathrm{\parallel }h{\mathrm{\parallel }}_X\le \delta \}$

is contained in $U$, the local tangential cone condition at $x$ on radius $\delta$ with constant $\eta$ holds if $0\le \eta <1$ and

$\mathrm{\parallel }F(x+h)-F(x)-F^{\mathrm{\prime }}(x)h{\mathrm{\parallel }}_Y\le \eta \mathrm{\parallel }F(x+h)-F(x){\mathrm{\parallel }}_Y(\mathrm{\parallel }h{\mathrm{\parallel }}_X\le \delta )\mathrm{.}$

Nonlinear forward maps between real Hilbert spaces

Fréchet differentiability on an open real Banach domain

The norm triangle inequality for real inner-product spaces

Christian Clason, Regularization of Inverse Problems, equation (9.6)

Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, §2.3

This is the one-basepoint local condition: the derivative is fixed at $x$ while $h$ varies over the ball. Some treatments impose a stronger two-point condition uniformly for every pair of points in a set. The constant restriction $0\le \eta <1$ is part of this definition. Fréchet differentiability alone does not imply this condition.