Janusz Brzdęk

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.

Functional equations in mathematical analysis

Abstract We prove stability results for a family of functional equations. Keywords Functional equation • Stability Mathematics Subject Classification (2000): Primary 39B82 1.1 Introduction This paper deals with the stability properties of functional equations which are jointgeneralizations of some classical equations (e.g., of Cauchy, Jensen, d’Alembert,Wilson, exponential and quadratic). For information concerning the stability werefer to Forti [10], Ger [11]andHyersetal.[13].Throughout this note N, R and C stand, as usual, for the set of positive integers,reals and complex numbers, respectively. Moreover,K is either the field R or C, X is an abelian group and Λ is a finite subgroup of the automorphism group of X (theaction of λ ∈ Λ on x ∈X is denoted by λ x )and N is the cardinality of Λ.The stability problem for the functional equation1 N ∑ λ ∈ Λ f ( x +λ y )= f ( x ) g ( y ) , (1.1)where f,g : X → K, was posed and solved by Badora in [3](seealso[2]).Equation (1.1) is a joint generalization of the exponential functional equation(Λ =

A hyperstability result for the Cauchy equation

We prove a hyperstability result for the Cauchy functional equation \(f(x+y)=f(x)+f(y)\), which complements some earlier stability outcomes of J.M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function \(f\), mapping a normed space \(E_1\) into a normed space \(E_2\), and for every real numbers \(r,s\) with \(r+s>0\) one of the following two conditions must be valid: \begin{align*} \sup_{x,y\in E_1} \|f(x+y)-f(x)-f(y)\| \|x\|^r \|y\|^s=\infty,\\ \sup_{x,y\in E_1} \|f(x+y)-f(x)-f(y)\|\, \|x\|^r \,\|y\|^s=0. \end{align*} In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem. 10.1017/S0004972713000683

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.

Subgroups of the groupZn and a generalization of the Gołąb-Schinzel functional equation

SummaryLetn be a positive integer and letX be a linear space over a commutative fieldK. In the setĀ = (K\{0}) × X we define a binary operation ·:Ā × Ā → Ā by

On orthogonally exponential and orthogonally additive mappings

(a,x) \cdot (b,y) = (ab,y + b^{n - 1} x),(a,x),(b,y) \in \bar A.

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

Then