Necessary and sufficient conditions of Lipschitz reversibility of nonlinear differential operator \(d/dt-f\) in the space of bounded functions on axis (Q1605595)

Let \(f:\mathbb R\to\mathbb R\) be a continuous function. Let \[ M=\sup_{u,v\in{\mathbb R},u\not=v}\left|\frac{f(u)-f(v)}{u-v}\right|, \qquad m=\inf_{u,v\in{\mathbb R},u\not=v}\left|\frac{f(u)-f(v)}{u-v}\right|. \] The author deal with the Lipschitz differential operator \({\mathcal L}: C^1\to C^0\) determined by the equation \(({\mathcal L}x)(t)=\frac{dx(t)}{dt}-f(x(t)),\;t\in\mathbb R\). The main theorems of the paper are the following: Let \(M<\infty\). The operator \({\mathcal L}\) has an inverse Lipschitz operator \({\mathcal L}^{-1}\) iff \(m>0\). Let \(0<m\leq M<\infty\). Then for all \(h_1,h_2\in C^0\) \[ \|{\mathcal L}^{-1}h_1-{\mathcal L}^{-1}h_2\|_{C^0}\leq \frac{1}{m}\|h_1-h_2\|_{C^0},\;\|{\mathcal L}^{-1}h_1-{\mathcal L}^{-1}h_2\|_{C^1}\leq \frac{M+m+1}{m}\|h_1-h_2\|_{C^0} \] and \[ \|{\mathcal L}^{-1}h\|_{C^0}\leq \frac{1}{m}\|h+f(0)\|_{C^0},\;\|{\mathcal L}^{-1}h\|_{C^1}\leq \frac{M+m+1}{m}\|h+f(0)\|_{C^0}. \] for all \(h\in C^0\).