Abstracts

$\begin{abtop} {\sc Baron, Karol}: {\em On the existence of solutions of linear iterative equations in a class of distribution functions} \end{abtop}$
The equation

$\begin{displaymath}F(x) = \sum_{n=1}^{N} p_n F( \tau_n(x))\end{displaymath}$

was considered and results on the existence of its solutions in distribution function classes were presented.

$\begin{abtop} {\sc Bessenyei, Mih\'aly}: {\em Hadamard-type inequalities} \\ (Joint work with Zsolt P\'ales) \end{abtop}$
The classical Hadamard-inequality provides the following lower and upper estimation for a convex function $f: [a,b]\to R$ :

$\begin{displaymath}f\left({a+b\over 2}\right)\le {1\over b-a}{\int\limits_a^b}f(x)dx\le {f(a)+f(b)\over 2}.\end{displaymath}$

Our goal is to generalize this inequality when $f: [a,b]\to R$ is supposed to be n-monotone, that is, for $a\le x_0< \ldots <x_n \le b$ :

$\begin{displaymath}(-1)^n\left\vert\matrix{f(x_o) & \ldots & f(x_n)\cr 1 & \ld... ...ots \cr x_0^{n-1} & \ldots & x_n^{n-1}\cr} \right\vert\ge 0.\end{displaymath}$

For smooth enough function Hadamard-type inequalities are proved by using orthogonal polynomial systems and mean-value theorems. For the general case, a smoothing technique is developed and applied. For instance, for a 3-monotone function $f: [a,b]\to R$ , one can deduce that

$\begin{displaymath}{f(a)+3f\left({a+2b\over 3}\right) \over 4}\le {1\over b-a}{... ...s_a^b}f(x)dx\le {f(b)+3f\left({2a+b\over 3}\right) \over 4}. \end{displaymath}$

$\begin{abtop} {\sc Boros, Zolt\'an}: {\em Decomposition of strongly $\mbox{\bbb Q}$-differentiable functions} \end{abtop}$
A real function is called strongly $\mbox{\bbb Q}$ -differentiable if, for every real number $h \,$ , the limit of the ratio $\left( f(x+rh) - f(x) \right) / r$ exists whenever tends to any fixed real number and tends to zero through the positive rationals. After examining the dependence of strong $\mbox{\bbb Q}$ -derivatives on their parameters, we prove that every strongly $\mbox{\bbb Q}$ -differentiable function can be represented as the sum of an additive mapping and a continuously differentiable function.

$\begin{abtop} {\sc Dar\'oczy, Zolt\'an}: {\em A functional equation on complementary means} \\ (Joint work with Che Tat Ng) \end{abtop}$
We say that a function $M: [a,b]^2 \rightarrow [a,b] \ (a <b; a,b \in \mbox{\bbb R})$ is a mean on if it satisfies the following conditions

$\begin{eqnarray*} &(M1)& \ \ \min\{x,y\} \le M(x,y) \le \max\{x,y\} \mbox{ for ... ... x \not= y; \\ &(M2)& \ \ M \mbox{ is continuous on } [a,b]^2. \end{eqnarray*}$

is a mean on

, then

for all $x \in [a,b]$ , and the function defined by $\hat M (x,y):=x+y-M(x,y) \ (x,y \in [a,b])$ is also a mean on [a,b]. The pair

and $\hat M$ satisfy $A(M,\hat M)=A$ , where

is the arithmetic mean. In this sense, $\hat M$ is complementary to

with respect to the arithmetic mean. Let

be a mean on

. A function $f:[a,b] \rightarrow \mbox{\bbb R}$ is called

-associate if it possesses the following property

$\begin{eqnarray*} (MA) && \mbox{ If } x,y \in [a,b] \mbox{ satisfy } M(x,y) = (... ...2 \mbox{ and } f(x)=f((x+y)/2) \\ && \mbox{ then } f(y)=f(x). \end{eqnarray*}$

We consider the functional equation

$\begin{displaymath} f(M(x,y)) =f (\hat M (x,y)) \hspace{1cm} (x,y \in [a,b]) \end{displaymath}$

with being -associate and continuous.

$\begin{abtop} {\sc Ger, Roman}: {\em Fischer-Musz\'ely additivity of mappings between normed spaces} \end{abtop}$
Generalizing numerous earlier results (see e.g. P. Fischer & Gy Muszély [3], J. Dhombres [2], R. Ger [4], G. Berruti & F. Skof [1], P. Schöpf [6], and R. Ger & B. Koclega [5]) we have obtained, among others, the following two theorems.

THEOREM 1. Let be an Abelian group and let $(Y, \Vert \cdot \Vert_{Y})$ be a real normed linear space. Let further $f: X \longrightarrow Y$ be a solution to the functional equation

$\begin{displaymath}\Vert f(x+y) \Vert_{Y} = \Vert f(x) + f(y) \Vert_{Y} \,, \quad x,y \in X\,. \eqno{(1)}\end{displaymath}$

Then there exists a nonempty set $T \subset \mbox{\bbb R}^{X}$ , an additive operator $A: X \longrightarrow B(T,\mbox{\bbb R})$ (the Banach space of all bounded functions on , equiped with the uniform convergence norm) and an odd isometry $I: A(X) \longrightarrow Y$ such that

$\begin{displaymath}f(x) = I(A(x))\,, \quad x \in X\,.\end{displaymath}$

Conversely, for an arbitrary real normed linear space $(Z, \Vert \cdot \Vert_{Z})$ , any additive operator $A: X \longrightarrow Z$ and any odd isometry $I: A(X) \longrightarrow Y$ the superposition $f:= I \circ A$ yields a solution of equation .

THEOREM 2. Let $(X, \Vert \cdot \Vert_{X})$ and $(Y, \Vert \cdot \Vert_{Y})$ be two real normed linear spaces. Let further $f: X \longrightarrow Y$ be a solution to the functional equation such that the function $\varphi: X \longrightarrow \mbox{\bbb R}$ defined by the formula

$\begin{displaymath}\varphi (x):= \Vert f(x) \Vert_{Y} \,, \quad x \in X\,,\end{displaymath}$

satisfies any regularity condition that forces a Jensen convex functional to be continuous. Then there exists a nonempty set $T \subset \mbox{\bbb R}^{X}$ , a continuous linear operator $L: X \longrightarrow B(T,\mbox{\bbb R})$ and an odd isometry $I: L(X) \longrightarrow Y$ such that

$\begin{displaymath}f(x) = I(L(x))\,, \quad x \in X\,.\end{displaymath}$

Conversely, for an arbitrary real normed linear space $(Z, \Vert \cdot \Vert_{Z})$ , any continuous linear operator $L: X \longrightarrow Z$ and any odd isometry $I: L(X) \longrightarrow Y$ the superposition $f:= I \circ L$ yields a solution of equation and the corresponding function $\varphi$ is continuous.

REFERENCES

[1]: G. Berruti and F. Skof, Risultati di equivalenza per un'equazione di Cauchy alternativa negli spazi normati, Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 125, Fasc. 5-6 (1991), 154-167.
[2]: J. Dhombres, Some aspects of functional equations, Chulalongkorn Univ., Bangkok, 1979.
[3]: P. Fischer and G. Muszély, On some new generalizations of the functional equation of Cauchy, Canad. Math. Bull. 10 (1967), 197-205.
[4]: R. Ger, On a characterization of strictly convex spaces, Atti Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 127 (1993), 131-138.
[5]: R. Ger and B. Koclega, Isometries and a generalized Cauchy equation, Aequationes Math. 60 (2000), 72-79.
[6]: P. Schöpf, Solutions of $\parallel\! f(\xi +\eta)\!\parallel \,\, =\,\,\parallel\! f(\xi ) + f(\eta )\!\parallel$ , Math. Pannon. 8/1 (1997), 117-127.

$\begin{abtop} {\sc Gil\'anyi, Attila}: {\em On the Dinghas derivative and convex functions of higher order} \end{abtop}$
In this talk Jensen convex functions of higher order are characterized with the help of the lower Dinghas interval derivative.

$\begin{abtop} {\sc H\'azy, Attila}: {\em On approximately convex functions} \\ (Joint work with Zsolt P\'ales) \end{abtop}$
A function $f:D\rightarrow \mbox{\bbb R}$ is called $(\varepsilon,\delta)$ -midconvex if

$\begin{displaymath}f\left(\frac{x+y}{2}\right)\leq \frac12 \left(f(x)+f(y)\right)+\delta +\varepsilon\vert x-y\vert \end{displaymath}$

for all $x,y \in D.$ Our main result shows that if

is locally bounded from above and $(\varepsilon,\delta)$ -midconvex, then

satisfies the following convexity inequality

$\begin{displaymath}f\left(\lambda x+(1-\lambda) y\right) \leq \lambda f(x)+(1-\l... ... )f(y)+2\delta +\varepsilon \varphi(\lambda ) \vert x-y\vert \end{displaymath}$

for every $x,y \in D$ and $\lambda \in [0,1]$ , where $\varphi$ is defined by

$\begin{displaymath}\varphi(\lambda)=\left\{ \begin{array}{ll} \displaystyle{-2... ...\displaystyle{\frac12\leq\lambda\leq 1}. \end{array} \right. \end{displaymath}$

In the case $\varepsilon=0$ the result reduces to that of Nikodem and Ng from 1993.

$\begin{abtop} {\sc Jarczyk, Witold}: {\em Continuous iteration semigroups and cocycles} \\ (Joint work with Grzegorz Guzik and Janusz Matkowski) \end{abtop}$
We reformulated four theorems of M. C. Zdun and gave a unified and simplified description of all continuous iteration semigroups defined on the product of $(0,\infty)$ and an arbitrary closed interval contained in $[-\infty, \infty]$ . Using this result we found all solutions $G: (0, \infty) \times X \to Y$ of the cocycle equation

$\begin{displaymath}G(s+t,x) = G(s,x) G(t,F(s,x)),\end{displaymath}$

where $(Y,\cdot)$ is a given commutative group and $F:(0, \infty) \times X \to X$ is a continuous iteration semigroup. In the proof we also used a result on the form of solutions $\Delta : (0, \infty) \times (p,q) \to Y$ of a conditional triangular equation

$\begin{displaymath}\Delta (s+t,u) = \Delta (s,u) \Delta(t,s+u).\end{displaymath}$

$\begin{abtop} {\sc Kaiser, Zolt\'an}: {\em The stability of the Cauchy equation in $p$-adic fields} \\ (Joint work with Zolt\'an Boros) \end{abtop}$
It is proved that if

is a function from a vector space

over $\mbox{\bbb Q}$ to the

-adic field $\mbox{\bbb Q}_p$ satisfying

$\begin{displaymath} \Vert f(x+y)-f(x)-f(y) \Vert_p \le K \end{displaymath}$

for some fixed

and all $x,y \in X$ (where $\Vert \ \Vert_p$ is the p-adic norm in $\mbox{\bbb Q}_p$ ), then there exists an additive function $g: X \rightarrow \mbox{\bbb Q}_p$ for which

$\begin{displaymath} \Vert f(x) -g(x) \Vert_p \le K \end{displaymath}$

for all $x \in X$ . A similar result is also established for the Jensen equation and for endomorphisms.

$\begin{abtop} {\sc Kapica, Rafa{\l }}: {\em Convergence of sequences of iterates of random-valued vector functions } \end{abtop}$
Given a probability space $(\Omega,{\cal A},P)$ and a closed subset of a Banach lattice we consider functions $f:X\times\Omega\to X$ , their iterates $f^n:X\times\Omega^{\mbox{\bbb N}}\to X$ defined by $f^1(x,\omega)=f(x,\omega_1),\; f^{n+1}(x,\omega)=f(f^n(x,\omega),\omega_{n+1})$ and obtain theorems on the convergence (a.s., in ) of the sequence $(f^n(x,\cdot))$ .

$\begin{abtop} {\sc Kocl{\c e}ga-Kulpa, Barbara}: {\em On a functional inequality in normed spaces} \end{abtop}$
The functional inequality

$\begin{displaymath}\parallel\! f(x + y) \!\parallel \,\, \geq \,\, \parallel\! f(x) + f(y) \!\parallel,\ \ x,y\in G\eqno{(1)}\end{displaymath}$

has been studied by Gy. Maksa and P. Volkmann (Characterization of group homomorphisms having values in an inner product space, Publ. Math. Debrecen 56/1-2 (2000), 197-200) for

mapping a group

into a real or complex inner product space $(X,\Vert\cdot\Vert)$ . It was shown that the inequality (1) implies the Cauchy equation

$\begin{displaymath}f(x+y)=f(x)+f(y),\ \ x,y\in G,\end{displaymath}$

and it was asked if this statement was true also for a strictly convex normed space

. At present, we deal with (1) assuming that

is an arbitrary normed space and the function $f:\mbox{\bbb R}\rightarrow X$ satisfies some regularity conditions. We have the following:

THEOREM. Let $(X,\parallel\!\cdot\!\parallel)$ be a real normed linear space and let $f:\mbox{\bbb R}\longrightarrow X$ be a solution of the functional inequality . If the function $\varphi : \mbox{\bbb R}\longrightarrow \mbox{\bbb R}$ is defined by the formula $\varphi (x):=\, \parallel\! f(x)\!\parallel, \, x \in \mbox{\bbb R}$ and

$\begin{displaymath}({\rm J})\ \ \left\{\begin{array}{ll} \varphi\ \ satisfies\... ...\,\, function\,\, to\,\, be\,\, convex, \end{array} \right.\end{displaymath}$

then

$\begin{displaymath}f(x) = \gamma I(x), \ \ x\in \mbox{\bbb R},\eqno{(2)}\end{displaymath}$

where $I:\mbox{\bbb R}\longrightarrow X$ yields an odd isometry and $\gamma$ is a real constant.

Conversely, for an arbitrary odd isometry $I:\mbox{\bbb R}\longrightarrow X$ and for every constant $\gamma\in\mbox{\bbb R}$ , the function $f:\mbox{\bbb R}\longrightarrow X$ given by the formula yields a solution to the inequality and the corresponding function $\varphi$ is continuous and convex.

$\begin{abtop} {\sc Lajk\'o, K\'aroly}: {\em Functional Equations in Probability Theory (solved and unsolved problems)} \end{abtop}$
Functional equations have many interesting applications in the characterization problems of probability theory (e.g. in the characterizations of univariate probability distributions by independent statistics and in the characterizations of bivariate distributions from conditional distributions). In these characterizations the functional equations with measurable unknown functions are satisfied for all or for almost all pairs from an open set of $\mbox{\bbb R}^2$ (or $\mbox{\bbb R}^n$ ) respectively. Several solved and unsolved problems were presented in this talk.

$\begin{abtop} {\sc Maksa, Gyula}: {\em Hyperstability of a class of linear functional equations} \\ (Joint work with Zsolt P\'ales) \end{abtop}$
First we investigate the stability properties of the functional equation

$\begin{displaymath} \psi(xy)=M(x)\psi(y)+M(y)\psi(x) \qquad (x,y\in]0,1]) \eqno(1) \end{displaymath}$

where

is a given multiplicative function which has a value greater than

and prove that the stability inequality

$\begin{displaymath} \vert\psi(xy)-M(x)\psi(y)-M(y)\psi(x)\vert\le\varepsilon \qquad (x,y\in]0,1]) \end{displaymath}$

(with any fixed $\varepsilon\ge0$ ) implies (1). We say shortly that (1) is hyperstable.

Next we present the following generalization.

Let $S=(S,\cdot)$ and

denote a semigroup and a real normed space, respectively. In addition, let $\phi_1,\dots,\phi_n:S\to S$ be pairwise distinct automorphisms of

such that the set $\{\phi_1,\dots,\phi_n\}$ is a group with respect to the composition as group operation. THEOREM. Let $\varepsilon:S\times S\to\mbox{\bbb R}$ be a function such that there exists a sequence $(u_k):\mbox{\bbb N}\to S$ satisfying

$\begin{displaymath} \lim_{k\to\infty} \varepsilon(u_ks,t)=0 \qquad (s,t\in S). \end{displaymath}$

Assume that $f:S\to X$ satisfies

$\begin{displaymath} \Bigl\Vert f(s)+f(t)-\frac{1}{n}\sum_{i=1}^n f(s\phi_i(t))\Bigr\Vert \le \varepsilon(s,t) \qquad (s,t\in S). \end{displaymath}$

Then is a solution of

$\begin{displaymath} f(s)+f(t)=\frac{1}{n}\sum_{i=1}^n f(s\phi_i(t)) \qquad (s,t\in S). \end{displaymath}$

$\begin{abtop} {\sc Matkowski, Janusz}: {\em A solution of a problem of H.\ Haruki and Th.\ M.\ Rassias} \end{abtop}$
We prove the following THEOREM. A function $f: (0, \infty) \times (0, \infty) \to \mbox{\bbb R}$ , continuous on the diagonal $\{(x,x): \, x>0\}$ , satisfies the functional equation

$\begin{displaymath}f\left( \frac{x+y}{2}\;, \frac{2xy}{x+y} \right) = f(x,y),\;\;\;x,y >0\end{displaymath}$

if and only if there exists a single variable and continuous function $F : (0,\infty) \to \mbox{\bbb R}$ such that

$\begin{displaymath}f(x,y) = F(xy), \;\;x,y > 0.\end{displaymath}$

This solves an open problem posed by H. Haruki and Th. M. Rassias in [1]. A

-xal generalization of this result is also presented.

REFERENCES

[1]: H. Haruki and Th. M. Rassias, A new analogue of Gauss' functional equation, Internat. J. Math. Sci. 18 (1995), 749-756.

$\begin{abtop} {\sc P\'ales, Zsolt}: {\em Stability of generalized monomial functional equations} \end{abtop}$
The stability problem and selection theorems concerning the generalized monomial functional equation

$\begin{displaymath} p_0f(x)+p_1f(xy)+\cdots+p_nf(xy^n)=g(y) \qquad (x,y\in S) \end{displaymath}$

are investigated with the help of the so-called invariant mean technique, where
$p_0,p_1,\dots,p_n\in\mbox{\bbb R}$ , $p_n\neq0$ , $p_0+p_1+\cdots+p_n=0$ ,

is a commutative semigroup, and

maps

into a locally convex space. If

$\begin{displaymath} p_0+p_1t+\cdots+p_nt^n=(t-1)^n \qquad (t\in\mbox{\bbb R}), \end{displaymath}$

then the results reduce to that of obtained jointly with R. Badora, R.Ger, and L. Székelyhidi in some recent papers.

REFERENCES

[1]: R. Badora, R.Ger, and Zs. Páles, Additive selections and the stability of the Cauchy functional equation, Bull. Austr. Math. Soc., accepted.
[2]: R. Badora, Zs. Páles, and L. Székelyhidi, Monomial selection of set-valued maps, Aequationes Math. 58(3) (1999), 214-222.

$\begin{abtop} {\sc Sablik, Maciej}: {\em On compatibility of the social development indices} \end{abtop}$
We discuss the question of compatibility of some indices used by the United Nations Development Program to determine the level of human development. Our goal is to restrict the range of arbitrariness in choosing quasi-arithmetic means to measure the development in different countries. Various indices used by UNDP are usually aggregated from some basic subindices with the help of quasiarithmetic means. However, the arbitrariness in choosing the aggregating means leads to a non-compatibility of two ways of determining the national index, first one consisting in aggregating subindices on the national scale, and the other in counting regional indices first, and then accumulating them into a national index. We show that compatibility assumption leads to a variant of generalized bisymmetry equation, which was solved in a pretty general setting by J. Aczél, Gy. Maksa and M. Taylor. However, since we are looking for means as solutions, we are able to get directly some results with assumptions slightly relaxed. In particular, we prove the following. THEOREM. Let $N\geq 2$ be a positive integer, let $I\subset \mbox{\bbb R}$ be a non-degenerate interval, and suppose that $M:I^3\to I, \, S:I^3\to I,\, A:I^N\to I$ and $B:I^N\to I$ are means satisfying

$\begin{eqnarray*} \lefteqn{M(A(x_{1,1},\dots,x_{1,N}),A(x_{2,1},\dots,x_{2,N}),... ...,1},S(t_1,u_1,v_1)), \dots,M(x_{1,N},x_{2,N},S(t_N,u_N,v_N))), \end{eqnarray*}$

for all $x_{i,j}, t_j, u_j, v_j \in I, \, i\in \{1,2,3\}, \, j\in \{1,\dots,N\}.$ If , or , or is a quasi-arithmetic weighted mean with an increasing and continuous generating function $\varphi$ then all the remaining means are also weighted quasi-arithmetic means with the same generating function $\varphi.$

$\begin{abtop} {\sc Sz\'ekelyhidi, L\'aszl\'o}: {\em Functional Equations on Hypergroups} \end{abtop}$
The concept of DJS-hypergroup (according to the initials of C.F.Dunkl, R.I.Jewett and R.Spector) is due to R.Lasser (see e.g. [1]). One begins with a locally compact Haussdorff space , with the space $\cal M(K)$ of all finite complex regular measures on , and with the space $\cal M^1(K)$ of all probability measures in $\cal M(K)$ . The point mass concentrated at is denoted by $\delta_x$ . Suppose that we have the following:

$(H^{*})$ There is a continuous mapping $(x,y)\mapsto \delta_x*\delta_y$ from $K\times K$ into $\cal M^1(K)$ , the latter endowed with the weak topology with respect to the space of compactly supported complex valued continuous functions on . This mapping is called convolution.
$(H^{\vee})$ There is an involutive homeomorphism $x\mapsto x^{\vee}$ from to . This mapping is called involution.
There is a fixed element in . This element is called identity.

Identifying

by $\delta_x$ the mapping in $(H^{*})$ has a unique extension to a continuous bilinear mapping from $\cal M(K)\times \cal M(K)$ to $\cal M(K)$ . The involution on

extends to an involution on $\cal M(K)$ . Then a DJS-hypergroup is a quadruple $(K,*,\vee,e)$ satisfying the axioms: for any

we have

(H1) $\delta_x*(\delta_y*\delta_z)=(\delta_x*\delta_y)*\delta_z$ ;
(H2) $(\delta_x*\delta_y)^{\vee}=\delta_{y^{\vee}}*\delta_{x^{\vee}}$ ;
(H3) $\delta_x*\delta_e=\delta_e*\delta_x=\delta_x$ ;
(H4) is in the support of $\delta_x*\delta_{y^{\vee}}$ if and only if ;
(H5) the support of $\delta_x*\delta_y$ is compact;
(H6) the mapping $(x,y)\mapsto$ supp $(\delta_x*\delta_y)$ from $K\times K$ into the space of nonvoid compact subsets of is continuous, the latter being endowed by the Michael-topology.

If $\delta_x*\delta_y=\delta_y*\delta_x$ for all

, then we call the hypergroup commutative. For instance, if

is a locally compact Haussdorff-group, $\delta_x*\delta_y=\delta_{xy}$ for all

, $x^{\vee}$ is the inverse of

, and

is the identity of

, then we obviously have a hypergroup $(K,*,\vee,e)$ , which is commutative if and only if the group

is commutative. However, not every hypergroup originates in this way. Let $0<\theta\leq 1$ be arbitrary and let $K=\{0,1\}$ . We define

and involution as the identity map. The products $\delta_0*\delta_0=\delta_0$ , $\delta_0*\delta_1=\delta_1*\delta_0 =\delta_1$ are obvious, and we let

$\begin{displaymath}\delta_1*\delta_1 =\theta\delta_0+(1-\theta)\delta_1.\end{displaymath}$

It is easy to see that we get a hypergroup for any $\theta$ in

. For $\theta=1$ we get the two-element group of integers modulo

. We identify

by $\delta_x$ and we define the translation operator

by the element

according to the formula:

$\begin{displaymath}T_yf(x)=\int_Kfd(\delta_x*\delta_y),\end{displaymath}$

for any

integrable with respect to $\delta_x*\delta_y$ . In particular,

is defined for any continuous complex valued function on

. In other words we have

$\begin{displaymath}f(x*y)=\int_Kfd(\delta_x*\delta_y)\end{displaymath}$

for any

. Having translation operators we may consider the classical functional equations on hypergroups. On commutative hypergroups one can study

exponentials:

$\begin{displaymath}m(x*y)=m(x)m(y),\end{displaymath}$

with $m(x)\ne 0$ for all in , which are common eigenfunctions of all translation operators;
additive functions:

$\begin{displaymath}a(x*y)=a(x)+a(y)\end{displaymath}$

for all in ;
polynomial functions:

$\begin{displaymath}(T_y-I)^{n+1}f(x)=0\end{displaymath}$

for all in , where is a nonnegative integer, is arbitrary in and is the identity operator;
d'Alembert-equation:

$\begin{displaymath}f(x*y)+f(x*y^{\vee})=2f(x)f(y)\end{displaymath}$

for all in ;
stability problems for the classical equations;
spectral synthesis problems for translation invariant function spaces; etc.

The main goal of this work is to call attention to hypergroups and to the possibility of studying functional equations on hypergroups. It seems that some of the classical methods can be adopted to the hypergroup-case but in some cases new ideas are needed.

REFERENCE

[1]: J. M. Anderson, G. L. Litvinov, K. A. Ross, A. I. Singh, V. S. Sunder, and N. J. Wildberger (eds.), Harmonic Analysis and Hypergroups, Birkhäuser, Boston, Basel, Berlin (1998).

$\begin{abtop} {\sc Szostok, Tomasz}: {\em On a generalized orthogonal additivity} \end{abtop}$
Logical connections between the modified version of orthogonal Cauchy equation and the following unconditional equation

$\begin{displaymath}f(x+y)=g\left(\frac{\Vert x-y\Vert}{\Vert x+y\Vert}\right)[f(x)+f(y)]\end{displaymath}$

are examined. Namely, it is proved that under some assumptions this equation preserves the solutions of orthogonal Cauchy equation. Further the Cauchy equation with the right-hand side multiplied by some constant is considered. This equation is assumed for all

satisfying the equality $\frac{\Vert x-y\Vert}{\Vert x+y\Vert}=\alpha.$ Finally solutions of this conditional equation in the case of odd functions defined on inner product spaces and $\alpha$ lying in some interval are determined.

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