An Inverse Function Theorem for Continuous Mappings

An Inverse Function Theorem for Metrically Regular Mappings

We prove that if a mapping F : X → → Y , where X and Y are Banach spaces, is metrically regular at x̄ for ȳ and its inverse F−1 is convex and closed valued locally around (x̄, ȳ), then for any function G : X → Y with lipG(x̄) · regF (x̄ | ȳ)) < 1, the mapping (F + G)−1 has a continuous local selection x(·) around (x̄, ȳ + G(x̄)) which is also calm.

The Fashoda Meet Theorem for Continuous Mappings

The articles [21], [25], [2], [20], [26], [5], [27], [6], [3], [1], [24], [10], [18], [16], [9], [4], [13], [11], [19], [23], [17], [7], [8], [22], [12], [15], and [14] provide the terminology and notation for this paper. We use the following convention: n is a natural number, p1, p2 are points of E T , and a, b, c, d are real numbers. Let us consider a, b, c, d. One can verify that ClosedInsid...

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue’s dominated convergence theorem and Ekeland’s variational principle. As a consequence, the assumptions are substantially weakened: th...

An invariant for continuous mappings

The Beckman - Quarles theorem for continuous mappings from R

Let φ : Cn×Cn → C, φ((x1, ..., xn), (y1, ..., yn)) = (x1−y1)+ ...+ (xn − yn). We say that f : Rn → Cn preserves distance d ≥ 0 if for each x, y ∈ Rn φ(x, y) = d implies φ(f(x), f(y)) = d. We prove that if x, y ∈ Rn (n ≥ 3) and |x − y| = ( √ 2 + 2/n)k · (2/n)l (k, l are nonnegative integers) then there exists a finite set {x, y} ⊆ Sxy ⊆ Rn such that each unit-distance preserving mapping from Sxy...