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Generalized stability of isometries in p-homogeneous F-spaces

Żołdak M.

Demonstratio Mathematica

2011

Vol. 44, nr 2

335--348

Let X, Y be p-homogeneous, complete F-spaces, r > 0, δ > 0. A function ƒ: X→Y is an (r, δ)-isometry, if (…) for x, y ∈ X. We consider the following problem: does for each ε > 0, r > 0 there exist a δ > 0 such that for each surjective (r, δ)-isometry ƒ there exists an isometry U : X → Y such that (…).

On the Mazur-Ulam theorem in metric groups

Żołdak M.

Demonstratio Mathematica

2009

Vol. 42, nr 1

123-130

Let X, Y be abelian uniquely 2-divisible groups with metrics dx,dy respectively, invariant with respect to the translations and let there exist a constant c > 1 such that dy(2y, 0) > cdy(y, 0) for y is an element of Y. We prove that each surjective isometry U : X -> Y has a form U (x) = a(x) + U (0) for x is an element of X, where a : X -> Y is a homomorphism.

Iterative equations in p-normes spaces

Żołdak M.

Demonstratio Mathematica

2006

Vol. 39, nr 3

571-578

Let X be a real p-normed, complete space and V be a nonempty closed subset of X. We consider the functional equation.