Probably the answer depends on the context, where you need the generalized concept.

Unfortunately, usually the linearized operator is not bounded anymore, because it contains differential operators.

As pointed out by Mikael, the [Lipschitz constant][1] may be one possibility. For example, in the [Crandall][2]-[Liggett][3] theory of nonlinear semigroups the Hille-Yosida generation theorem is generalized to contractions. Or in the [Banach fixed point theorem][4] the convergence of the geometric series is generalized to iterations of nonlinear maps.

But there might be other answers, I am really curious.


  [1]: http://en.wikipedia.org/wiki/Lipschitz_continuity
  [2]: http://de.wikipedia.org/wiki/Michael_Crandall
  [3]: http://en.wikipedia.org/wiki/Thomas_M._Liggett
  [4]: http://en.wikipedia.org/wiki/Banach_fixed_point_theorem