Necessary and Sufficient Conditions for the Lipschitzian Invertibility of the Nonlinear Differential Mapping $d/dt-f$ in the Spaces $L_p({\mathbb R},{\mathbb R})$, $1\le p\le\infty$

We obtain necessary and sufficient conditions for the Lipschitzian invertibility of the differential mapping $d/dt-f$, where $f\colon{\mathbb R}\to{\mathbb R}$ is a continuous mapping, in the spaces $L_p({\mathbb R},{\mathbb R})$, $1\le p\le\infty$.