On the Hadamard lemma and the Lipschitz condition

We prove that the difference of values of a nonlinear Lipschitz continuous differential operator is always representable as the difference of values of some linear Lipschitz continuous differential operator with the same Lipschitz constant. The proof is based on the Hadamard lemma, provided that, in addition to the above requirements, the nonlinearity is continuously differentiable in spatial variables. In general case the proof is based on various criteria of the weak compactness and on various approximating statements obtained by the Steklov averaging technique.