Extensions To Nonlinear Maps
For nonlinear operators the operator norm and logarithmic norm are defined in terms of the inequalities
where is the least upper bound Lipschitz constant of, and is the greatest lower bound Lipschitz constant; and
where and are in the domain of . Here is the least upper bound logarithmic Lipschitz constant of, and is the greatest lower bound logarithmic Lipschitz constant. It holds that (compare above) and, analogously, where is defined on the image of .
For nonlinear operators that are Lipschitz continuous, it further holds that
If is differentiable and its domain is convex, then
- and
Here is the Jacobian matrix of, linking the nonlinear extension to the matrix norm and logarithmic norm.
An operator having either or is called uniformly monotone. An operator satisfying is called contractive. This extension offers many connections to fixed point theory, and critical point theory.
The theory becomes analogous to that of the logarithmic norm for matrices, but is more complicated as the domains of the operators need to be given close attention, as in the case with unbounded operators. Property 8 of the logarithmic norm above carries over, independently of the choice of vector norm, and it holds that
which quantifies the Uniform Monotonicity Theorem due to Browder & Minty (1963).
Read more about this topic: Logarithmic Norm
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