On the Lipschitz perturbation of monotonic functions
Let \(I\) be an interval and \(d:I^{2}\rightarrow \mathbb{R}\) be a semimetric. A function \(\ell:I\rightarrow \mathbb{R}\) is called \(d-\)Lipschitz if it satisfies \(\left| \ell(x)-\ell (y)\right| \leq d(x,y)\) for \(x,y\in I\). The authors prove that a function \(p:I\rightarrow \mathbb{R}\) can be decomposed in the form \(p=q+\ell\), where \(q\) is increasing and \(\ell\) is \(d-\)Lipschitz if and only if \[ \sum_{i=1}^{n}\left( p(s_{i})-p(t_{i})-d(t_{i},s_{i})\right) ^{+}\leq \sum_{j=1}^{m}\left( p(v_{j})-p(u_{j})+d(u_{j},v_{j})\right) \] is fulfilled for all real numbers \(t_{1}<s_{1},\dots,t_{n}<s_{n}\) and \(u_{1}<v_{1},\dots,u_{m}<v_{m}\) in \(I\) satisfying the condition \(\sum_{i=1}^{n}\boldsymbol{1}_{[t_{i},s_{i}]}=\sum_{j=1}^{m}\boldsymbol{1}_{[u_{j},v_{j}]},\) where \(\boldsymbol{1}_{[a,b]}\) denotes the characteristic function of the interval \(\left[ a,b\right] .\)
scientific article; zbMATH DE number 5150221