Hahng-Yun Chu

On the stability of the generalized cubic set-valued functional equation

Abstract In this article, we focus on the set-valued dynamics related to the theory of functional equations. We study the n -dimensional cubic set-valued functional equation f ( ∑ j = 1 n − 1 x j + 2 x n ) ⊕ f ( ∑ j = 1 n − 1 x j − 2 x n ) ⊕ ∑ j = 1 n − 1 f ( 2 x j ) = 2 f ( ∑ j = 1 n − 1 x j ) ⊕ 4 ∑ j = 1 n − 1 ( f ( x j + x n ) ⊕ f ( x j − x n ) ) where n ≥ 2 is an integer. We also prove the Hyers–Ulam stability for the set-valued functional equation.

A Mazur-Ulam problem in non-Archimedean n-normed spaces

In this article, we study the notions of n-isometries in non-Archimedean n-normed spaces over linear ordered non-Archimedean fields and prove the Mazur-Ulam theorem in the spaces. Furthermore, we obtain some properties for n-isometries in non-Archimedean n-normed spaces.MSC:46B20, 51M25, 46S10.

On the Envelopes of Homotopies

This paper is indented to explain a dynamics on homotopies on the compact metric space, by the envelopes of homotopies. It generalizes the notion of not only the envelopes of maps in discrete geometry ([3]), but the envelopes of flows in continuous geometry ([5]). Certain distinctions among the homotopy geometry, the ow geometry and the discrete geometry will be illustrated. In particular, it is shown that any -limit set, as well as any attractor, for an envelope of homotopies is an empty set (provided the homotopies that we treat are not trivial), whereas it is nonempty in general in discrete case.

VOLUME PRESERVING DYNAMICS WITHOUT GENERICITY AND RELATED TOPICS

In this article, we focus on certain dynamic phenomena in volume-preserving manifolds. Let be a compact manifold with a volume form and be a diffeomorphism of class that preserves . In this paper, we do not assume is -generic. We prove that satisfies the chain transitivity and we also show that, on , the -stable shadowability is equivalent to the hyperbolicity.

Dynamics of Riemannian

In this paper we study several dynamical properties of the riemannian

1

1

-foliations on

-dimensional foliation

3

\mathcal{L}

-manifolds

on an oriented closed 3-manifold

A note on envelopes of homotopies

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