An open mapping theorem using unbounded generalized Jacobians
The authors present a new open mapping theorem for nonsmooth functions by means of (not necessary bounded) approximate Jacobians. Let \(f: \mathbb{R}^n\to \mathbb{R}^m\) be a continuous function and \(x\in\mathbb{R}^n\). A closed set of \((m\times n)\)-matrices \(Jf(x)\subset L(\mathbb{R}^n, \mathbb{R}^m)\) is called to be an approximate Jacobian of \(f\) at \(x\), iff for every \(u\in \mathbb{R}^n\) and \(v\in \mathbb{R}^m\), one has \[ \limsup_{t\downarrow 0} {(vf)(x+ tu)- (vf)(x)\over t}\leq \sup_{M\in Jf(x)}\langle v, Mu\rangle. \] This notion was introduced by the authors some years ago. If \(f\) is locally Lipschitzian then the Clarke generalized Jacobian is an example of a locally bounded approximate Jacobian. Moreover, a continuous mapping is locally Lipschitzian if and only if it admits a locally bounded approximate Jacobian. A basic tool for the discussion of not Lipschitz functions is the limiting singular approximate Jacobian \[ J^\infty f(x):= \limsup\{t_k Jf(x_k)\mid t_k\downarrow 0\text{ and }x_k\to x\} \] which is nonempty in this case and (if \(Jf\) is assumed to be upper semicontinuous) can be described by the recession cone of the set \(Jf(x)\). After the discussion of properties of recession cones and using a special meanvalue theorem, the authors establish a chain rule for functions with unbounded, but upper semicontinuous approximate Jacobians. On the base of these results they present their main theorem. It is shown that -- assuming a continuous function \(f: \mathbb{R}^n\to \mathbb{R}^m\) with an upper semicontinuous approximate Jacobian mapping \(Jf\) and invertible matrices \(A\in \overline{\text{conv\,}} Jf(x_0)\cup\text{conv}(J^\infty f(x_0)\setminus\{0\})\) -- there exist numbers \(\delta> 0\) and \(\varepsilon> 0\) such that \[ \| f(x_0+ h)- f(x_0)\|\geq \varepsilon\| h\|\quad\text{for all }h\neq 0,\;\| h\|< \delta \] and \[ f(x_0)+ {\varepsilon\delta\over 2}\text{\,int\,} B(0,1)\subseteq f(x_0+ \delta\text{\,int\,}B(0, 1)). \] As corollaries of this result the authors derive associated inverse and implicit function theorems.
scientific article; zbMATH DE number 1788033