The stability problem of functional equations originated from a question of Ulam 1 concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’s theorem was generalized by Aoki 3 for additive mappings. In 1978, Rassias 4 generalized Hyers theorem by obtaining a unique linear mapping near an approximate additiv...

In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam-Rassias stability of the following Cauchy-Jensen additive functional equation: begin{equation}label{main} fleft(frac{x+y+z}{2}right)+fleft(frac{x-y+z}{2}right)=f(x)+f(z)end{equation} in various normed spaces. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias’ stability theorem t...

The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R can always be simultaneously bisected by an (n− 1)-dimensional hyperplane. In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and...

Recently, in [5], Najati and Moradlou proved Hyers-Ulam-Rassias stability of the following quadratic mapping of Apollonius type Q(z − x) + Q(z − y) = 1 2 Q(x− y) + 2Q ( z − x + y 2 ) in non-Archimedean space. In this paper we establish Hyers-Ulam-Rassias stability of this functional equation in random normed spaces by direct method and fixed point method. The concept of Hyers-Ulam-Rassias stabi...

This contribution presents a review of results obtained from computations of approximate equations of motion for the Fermi-Pasta-Ulam lattice. These approximate equations are obtained as a finite-dimensional Birkhoff normal form. It turns out that in many cases, the Birkhoff normal form is suitable for application of the KAM theorem. In particular this proves Nishida’s 1971 conjecture stating t...

The Monster group, the biggest of the sporadic groups, is equipped with the highest known number of dimensions and symmetries. Taking into account variants of the Borsuk–Ulam theorem and a novel topological approach cast in a physical fashion that has the potential to be operationalized, the universe can be conceived as a lower-dimensional manifold encompassed in the Monster group. Our universe...

We apply the Luxemburg–Jung fixed point theorem in generalized metric spaces to study the Hyers–Ulam stability for two functional equations in a single variable.