In 1968, Efimov proved the following remarkable theorem: Letf: R2→ R2∈ C1be such thatdet f′(x) < 0 for allx∈ R2and let there exist a function a(x) > 0 and constantsC1⩾ 0 , C2⩾ 0 such that the inequalities| 1 / a(x) - 1 / a(y) | ⩽ C1| x- y| + C2and| det f′(x) | ⩾ a(x) | curl f(x) | + a2(x) hold true for allx, y∈ R2. Thenf(R2) is a convex domain andf mapsR2ontof(R2) homeomorphically. Here curl f(x) stands for the curl of f at x∈ R2. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of global inverse functions, as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.
Предметные области OECD FOS+WOS
- 1.01 МАТЕМАТИКА