Krzysztof Ciepliński

On Some Recent Developments in Ulam's Type Stability

We present a survey of some selected recent developments (results and methods) in the theory of Ulams type stability. In particular we provide some information on hyperstability and the fixed point methods.

Hyperstability and Superstability

This is a survey paper concerning the notions of hyperstability and superstability, which are connected to the issue of Ulam’s type stability. We present the recent results on those subjects.

Fixed Point Theory and the Ulam Stability

The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banachs fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

Generalized stability of multi-additive mappings

Abstract In this paper we unify the system of Cauchy functional equations defining multi-additive mapping to obtain a single equation and prove the generalized Hyers–Ulam stability both of this system and this equation using the so-called direct method.

On Ulam’s Type Stability of the Linear Equation and Related Issues

This is a survey paper concerning stability results for the linear functional equation in single variable. We discuss issues that have not been considered or have been treated only briefly in other surveys concerning stability of the equation. In this way, we complement those surveys.

Remarks on the Hyers–Ulam stability of some systems of functional equations

Abstract In this paper we present a method that allows to study the Hyers–Ulam stability of some systems of functional equations connected with the Cauchy, Jensen and quadratic equations. In particular we generalize and extend some already known results.

On Uniqueness of Conjugacy of Continuous and Piecewise Monotone Functions

We investigate the existence and uniqueness of solutions of the functional equation , , where are closed intervals, and , are some continuous piecewise monotone functions. A fixed point principle plays a crucial role in the proof of our main result.

On a System of Schröder Equations on the Circle

Let M be an arbitrary nonempty set and for t ∈ M be continuous mappings of the unit circle . The aim of this paper is to investigate the existence of solutions (Φ, c), where is a continuous function and , of the following system of Schroder equations The particular case when card M = 1 is also considered.

Stability of Multi-Jensen Mappings in Non-Archimedean Normed Spaces

A function f : V n →W, where V and W are normed spaces over a field of characteristic different from 2 and n ≥ 1 is an integer, is called multi-Jensen if it satisfies Jensen’s functional equation in each variable. In this note, we provide a proof of a generalized Hyers–Ulam stability of multi-Jensen mappings in non-Archimedean normed spaces, using the so-called direct method.

Ulam's Type Stability

Quite often (e.g., in applications) we have to do with functions that satisfy some equations only approximately. There arises a natural question what errors we commit when we replace such functions by the exact solutions to those equations. Some tools to evaluate them are provided within the theory of the Ulam (also Hyers-Ulam) type stability. The issue of Ulams type stability of ( rst, functional, but next also di erence, di erential and integral) equations has been a very popular subject of investigations for the last nearly fty years (see, e.g., [3, 8, 9, 10]). The main motivation for it was given by S.M. Ulam in 1940. The following de nition somehow describes the main ideas of such stability notion for equations in n variables (R+ stands for the set of nonnegative reals). De nition 1. Let A be a nonempty set, (X, d) be a metric space, C ⊂ R+ n be nonempty, T map C into R+, and F1,F2 map a nonempty D ⊂ X into X n . We say that the equation F1φ(x1, . . . , xn) = F2φ(x1, . . . , xn) (1) is T stable provided for every ε ∈ C and φ0 ∈ D with