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On stability of classes of Lipschitz mappings generated by compact sets of linear mappings. (English. Russian original) Zbl 0959.47008
Sib. Math. J. 41, No. 4, 656-670 (2000); translation from Sib. Mat. Zh. 41, No. 4, 792-810 (2000).
The article is devoted to a further development of the stability theory of classes of Lipschitz mappings in the framework of the concept of $$\omega$$-stability. The main object under study is the class of mappings described below. Suppose that $$G$$ is a nonempty compact subset of the space L$$(\mathbb R^n,\mathbb R^m)$$ of linear mappings from $$\mathbb R^n$$ into $$\mathbb R^m$$. The class of all locally Lipschitz mappings $$g\:\Delta\to\mathbb R^m$$ of domains $$\Delta\subset\mathbb R^n$$, for each of which there is a connected component $$K$$ of $$G$$ such that the differentials $$g'(x)$$ at almost all points $$x\in\text{dom }g$$ belong to $$K$$, is said to be $$(*)$$-generated by $$G$$ and denoted by $$\mathfrak Z(G)$$.
The author starts with proving the theorem claiming that if $$G$$ is a compact convex set then the class $$\mathfrak Z(G)$$ is $$\omega$$-stable. One of the main tools is the theorem about preservation of stability under set-theoretic operations on $$\omega$$-stable classes. As a result, the author completely solves the stability problem for the classes $$\mathfrak Z(G)$$ when either of the dimensions $$n$$ and $$m$$ equals 1. Using these results, the author obtains theorems on stability of classes of Lipschitz solutions to systems of linear partial differential equations and a theorem on $$\omega$$-stability of classes of conformal mappings which may simultaneously contain sense-preserving and sense-reversing mappings.
The author finds a fruitful application of the notion of weak connectedness of sets in vector spaces which was introduced into consideration for studying the structure of the image of the derivative of a differentiable vector-valued mapping.
##### MSC:
 47A55 Perturbation theory of linear operators 47B07 Linear operators defined by compactness properties 47B65 Positive linear operators and order-bounded operators 47F05 General theory of partial differential operators 46G05 Derivatives of functions in infinite-dimensional spaces 46T20 Continuous and differentiable maps in nonlinear functional analysis
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