An implicit function theorem in non-smooth case

R. A. Khachatryan

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 4, pp. 86-96

Résumé

In this paper, we consider an equation of the form $F(x,y)=0$, $x\in X$, $y\in M$, where $M$ is a set. By the method of tents (tangent cones), when the set $M$ is given by a nonsmooth restriction of equality type, the existence of a differentiable function $y(\cdot)$ such that $F(x, y(x))=0$, $y(x)\in M$, $y(x_0)=y_0$ is proved. In particular, the existence of smooth local selections for multivalued mappings of the form $a(x) = \{y \in \mathbb{R}^m:\, f_i(x, y) = 0,\, i \in I,\, g(y) = 0\}$, $x \in \mathbb{R}^n$, $y \in \mathbb{R}^m$, is studied by the method of tents. It is assumed that the functions $f_i(x, y)$, $i \in I$, are strictly differentiable, and the function $g (y)$ is locally Lipschitzian. Under certain additional conditions it is proved that through any point of the graph of a set-valued mapping there passes a differentiable selection of this mapping. These assertion can be interpreted as an implicit function theorem in the nonsmooth analysis. Strongly differentiable tents for the sets defined by nonsmooth constraints of the equality type are also constructed in the article. A sufficient condition is provided for the intersection of strictly differentiable tents to be a strictly differentiable tent. It is also shown that the Clark tangent cones are Boltiansky tents for sets defined by locally Lipschitz functions.