Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is bounded if there exists some $M>0$ such that for all $x \in X$ we have $$\Vert Tx \Vert_{X} \leq M\Vert x \Vert_{X}$$
If we define a bounded operator of an operator (possibly nonlinear) as an operator that takes bounded sets in $X$ to bounded sets in $Y$. Then if we have an operator $T$ such that there exists an $M > 0$ such that $\Vert Tx \Vert_{X} \leq M\Vert x \Vert_{X}$, then it clearly follows that it coincides with this definition (takes bounded sets to bounded sets).
Question: Can anyone give an example of a nonlinear operator which takes bounded sets to bounded sets but does not satisfy the definition of a bounded linear operator given above, so there does not exists a $M > 0$ such that $\Vert Tx \Vert_{X} \leq M\Vert x \Vert_{X}$.
Thanks for any assistance.