Aequationes Math. 64 (2002) 170–2000001-9054/02/010170-31

c© Birkhauser Verlag, Basel, 2002

Aequationes Mathematicae

Report of Meeting

The Thirty-ninth International Symposium on Functional Equa-tions, August 12 – 18, 2001, Sandbjerg, Denmark

Mathematics Subject Classification (2000). 39–06.

The Thirty-ninth International Symposium on Functional Equations was held inSandbjerg, Denmark, from August 12 through August 18, 2001 and organized bythe Institute of Mathematics of Aarhus University. Professors Henrik Stetkær andPeter de Place Friis were the organizers. The Scientific Committee consisted ofProfessors Zoltan Daroczy (Debrecen), Roman Ger (Katowice), Jurg Ratz (Bern),Ludwig Reich (Graz), and Abe Sklar (Chicago), with Professor Janos Aczel (Wa-terloo, Ontario) as Honorary Chairman. Unfortunately, Professors Daroczy andSklar were unable to attend the symposium. Professor Peter de Place Friis actedas secretary.

The 48 participants came from Austria, Canada, the Czech Republic, Denmark,Israel, Finland, France, Germany, Hungary, Poland, Romania, Switzerland, andthe United States of America.

Professor Jurg Ratz opened the symposium. In his address he proposed todedicate the 39-th ISFE to Professor Walter Benz on the occasion of his 70-thbirthday. Professor Benz was a member of the scientific committee for 25 years.Thededication was enthusiastically accepted by all the participants, who were alsowelcomed by Professor Stetkær.

The scientific talks presented at the symposium focused on the following top-ics: equations in one and several variables, conditional equations, equations onabstract structures, iteration theory, stability, mean values, characterizations offunctions, equations for functions on the complex domain, functional-differentialequations, and functional inequalities. Interesting connections with algebra, analy-sis, geometry, harmonic analysis, number theory, and probability theory, as well asapplications of functional equations to utility theory, psychophysics, and financialmathematics were presented and generated much discussion.

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 171

A number of sessions were as usual devoted to problems and remarks. Duringthis symposium these sessions were particularly lively.

There was a well-received evening concert by Harald Fripertinger (flute) andHans-Heinrich Kairies (piano). The social program also included a half-day excur-sion to Denmark’s oldest town Ribe and its Vikingecenter nearby. After the closingsession, there was a banquet during which Professor Walter Benz expressed thegratitude of the participants to the organizers of the symposium and their staff.

At the closing session conducted by Professor Ludwig Reich the ISFE medal foroutstanding contributions to the meeting was awarded to Professor Attila Gilanyi.

Professors’ Janusz Matkowski and Witold Jarczyk invitation to hold the For-tieth International Symposium on Functional Equations from August 25 throughSeptember 1, 2002 in Gronow, Poland, was gratefully accepted. Professors ZsoltPales and Jaroslav Smıtal announced plans to organize the 41-st and 42-nd Inter-national Symposia in Hungary in 2003 and in the Czech Republic in 2004.

Abstracts of the talks follow in alphabetical order, then problems and remarksin approximately chronological order, and finally a list of participants.

1. Abstracts of talks

Aczel, Janos: A functional equation arising in a theory of noncommutative jointreceipt in utility theory and psychophysics.Joint work with R. Duncan Luce and Che Tat Ng.Utility of joint receipt leads to the functional equation (cf. Remark 7) θ(sw, v) =

H[θ(s, v)w,G(v)K(w)], where θ(s, v) = G(vF [G−1(s)/v]) (s ∈ [0, k[, v ∈]0,∞[,w ∈ [0, 1]; F : [0,∞[→ [1,∞[, G : [0,∞[→ [0, k[, K : [0, 1] → [0, 1] bijections, F,Gstrictly increasing, K strictly decreasing, H strictly increasing in both variables).Luce asked for its general solution(s). This is done through a cascade of functionalequations, yielding

K(w) = (1 − wρ)1/ρ, H(p, q) = (pρ + qρ)1/ρ (ρ > 0).Moreover, G[vF (z)] = A(v)G(vz) + G(v) (v, z ∈ [0,∞[) is the equation for F

and G=Gρ. It has been solved by Ng under the further conditions F ∈C1, G∈C2.The solutions (with ρ = 1) are then

F (z) = (zβ + 1)1/β , G(v) = λ(eκvβ − 1) and

F (z) = (αzβ + 1)1/β , G(v) = γvβ , (α > 0, β > 0, γ > 0, λκ > 0).

Baron, Karol: On the existence of solutions of linear iterative equations in aclass of distribution functionsGiven a sequence (pn)n∈N of nonnegative reals summing up to 1 and a sequence

(τn)n∈N of bijections of R, we consider the equation

F (x) =∞∑

n=1

pnF (τn(x))

172 Report of Meeting AEM

and the problem of the existence of its solutions F in classes of distribution func-tions.

Benz, Walter: Stanilov’s fuctional equation is exorbitantJoint work with Hans-Joachim Samaga and Grosio Stanilov.Let X �= ∅ be a set and D �= ∅ be a subset of X2 := X × X. Determine all

f : D → X such that for all (x, y) ∈ D and z := f(x, y)

(i) (y, z), (x, z), (f(y, z), f(x, z)) are in D and(ii) z = f(f(y, z), f(x, z))

hold true. This is Stanilov’s Functional Equation (SE). One of the main resultspresented is

Theorem. Let Γ �= ∅ be an open subset of R and define X := R and D := Γ× Γ.Then there exist exactly 2ℵ distinct solutions f : D → X of (SE). Here ℵ designatesthe cardinal of R.

Boros, Zoltan: Strong differentiability with respect to a subfieldLet E ⊂ Rn be convex and open. Let K denote a subfield of R. For each

h ∈ Rn, we define Ph = { (x, r) ∈ E× (K \{0}) : x+ rh ∈ E }. We call f : E → Rstrongly K-differentiable if the finite limit

DKh f(x0) = lim

Ph�(x,r)→(x0 , 0)

f(x + rh) − f(x)r

exists for every h ∈ Rn and x0 ∈ E. We present the following characterization ofthis property.

Theorem. f : E → R is strongly K-differentiable, if and only if there existφ : Rn → R and g : E → R such that φ is linear over K, g is continuouslydifferentiable, and f(x) = φ(x) + g(x) for every x ∈ E.

Brillouet-Belluot, Nicole: The ACP-method for solving some compositefunctional equationsWe show the importance of the ACP-method for solving some composite func-

tional equations. This method is based on the well-known theorem of J. Aczel(given also by R. Craigen and Zs. Pales). It gives the representation of a continu-ous cancellative associative operation on a real interval. In particular, we find bythis method all continuous solutions of the functional equation

f(af(x)f(y) + b(f(x)y + f(y)x) + cxy) = f(x)f(y) (x, y ∈ R)

which generalizes both, the Ebanks functional equation and the Baxter functionalequation.

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 173

Chmielinski, Jacek: On some functional equations on spheresFor real inner product spaces we consider the classes of mappings preserving

the inner product or its absolute value on spheres. We compare them with solu-tions of other functional equations on spheres as well as with their unconditionalcounterparts.

Corovei, Ilie: The functional equation f(xy)+ g(xy−1) = h(x)k(y) on nilpotentgroupsConsider the functional equation

f : G → K, f(xy) + g(xy−1) = h(x)k(y) (1)

where G is a group and K a field with charK = 0. In 1919 Wilson has solved theequation (1) when G = (R,+) is the additive group of real numbers and K = R.In 1962 Aczel and Vincze have solved an equation of the type (1) where G is anabelian group and K is a quadratically closed field of characteristic zero.

In our presentation we obtain the general solution of equation (1) when G is anilpotent group, provided all its elements have odd orders and K is a quadraticallyclosed field of characteristic zero.

Choczewski, Bogdan: On S. Rolewicz’s problem connected with Φ-subdiffer-entials in metric spacesJoint work with R. Girgensohn and Z. Kominek.We present R. Girgensohn’s short and simple proof of the result saying that

the only nonnegative and differentiable solution f : R → R of the inequality

f(t) − f(s) − f ′(s)(t − s) ≥ f(t − s), t, s ∈ R, (P)are given by the formula

f(t) = Ct2, t ∈ R, C ≥ 0. (S)This answers the problem met by S. Rolewicz in his study of Φ-subdifferentials

of real valued functions defined on a metric space. For even f an independent,but more involved, proof of the result has been first given by Z. Kominek and thespeaker (who was also able to derive (S) from (P) in a simple way, however, undersome additional assumptions on f , cf. Z. Daroczy and Zs. Pales (eds.) FunctionalEquations – Results and Advances, Kluwer, 2001, 21-24). Solutions of a functionalinequality and a functional equation related to (P), both of Pexider type, arealso discussed. The paper presented will appear online in SIAM Problems andSolutions (http:/www.siam.org/journals/problems).

Davison, Thomas M. K.: Arithmetical Cauchy functionsA function f with domain N0 = {0, 1, 2, . . . } (natural numbers) is an arith-

metical Cauchy function if there are s ≥ 2 non-zero natural numbers a1, a2, . . . , as

with greatest common divisor equal to one such that

f

⎛⎝ s∑

j=1

ajxj

⎞⎠ + (s − 1)f(0) =

s∑j=1

f(ajxj)

174 Report of Meeting AEM

for all xj ∈ N0, j = 1, . . . , s. A function g with domain N0 is singular if g(0) = 0and supp(g) = {n ∈ N0 : g(n) �= 0} is finite. A function h with domain N0 isquasi-periodic (with period p ∈ N ) if h(x + p) + h(0) = h(x) + h(p) for all x ∈ N0.

Theorem. Let f be an arithmetical Cauchy function. Then there are uniquefunctions g, h where g is singular, and h is quasi-periodic such that f = g + h.

Derfel, Gregory: Small solutions of linear nonautonomous delay equationsA “small solution” of a linear delay equation is a solution, which decays more

rapidly than any exponential. The following principal question is still open.

Problem. (Verduyn Lunel, 1992.) Let b : R → R be a continuous function suchthat there exist positive constants m and M such that 0 < m < |b(t)| < M . Canthen the equation

x′(t) = b(t)x(t − 1) (1)

have a nontrivial small solution?

We give a a partial answer to this question. Assume that b(t) > 0 for all t > 0and denote c(t) = 1/b(t). Assume also that∣∣∣c(n)(t)

∣∣∣ ≤ KcDnc nnδ (2)

for large t and some δ > 0. Then any solution x(t) of (1), satisfying the estimate

|x(t)| ≤ C exp(−γt ln t) (3)

for some C > 0, γ > 2 + δ and large t, vanishes identically, i.e. x(t) ≡ 0.

Ebanks, Bruce R.: Solution of some generalized Cauchy difference equationsThis presentation concerns primarily the study of functional equations of the

formf(x) + f(y) − f(x + y) = g(H(x, y)),

where H is a given function of two real variables, and where f and g are unknownfunctions. Let I be a nonvoid real interval with I + I ⊂ I. Suppose there existsa particular solution (f0,g0) on I, either with f0 analytic and not affine or withf0 continuously differentiable and strictly convex. If H and g0 are continuouslydifferentiable, or if g0 is one-to-one, then all solutions of the equation are of theform

g(u) = A1 ◦ g0(u) + c,

f(x) = A1 ◦ f0(x) + A2(x) + c,

where A1, A2 are arbitrary additive functions and c is an arbitrary real constant.Similar results hold for functional equations in which the sum x+ y is replaced

by a smooth quasi-sum θ−1(θ(x) + θ(y)).

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 175

Frank, M. J.: Multivariate Archimedean CopulasThe n-th serial iterate Cn of an Archemedean 2-copula C (= C2), given by

Cn(x1, . . . , xn) = f(f−1(x1) + · · · + f−1(xn)),

is an n-dimensional copula (n-copula) when Cn is an n-increasing function. Theproblem is to find, for a given C, the largest such n. Now Cn is an n-copulafor all n ≥ 2 if and only if f is completely monotonic, in which case simplewell-known criteria can often be applied to solve the problem. When f is notcompletely monotonic, Cn is an n-copula if f is n-monotonic, i.e., if (−1)kf (k) ≥ 0for 0 ≤ k ≤ n. But here the problem is frequently intractable directly.

We present a new solution method which is effective in the latter case; itinvolves careful analysis of the roots of a certain class of recursively defined se-quences of polynomials, many of which are of independent interest. We illustratethis method via some key one-parameter families.

Friis, Peter de Place: A necessary condition in connection with trigonometricfunctional equationsLet K be a compact transformation group acting on the topological group G

by homomorphisms and antihomomorphisms, and let C(G) denote the algebra ofcontinuous complex valued functions on G. If f, g1, . . . gn, h1, . . . hn ∈ C(G) is asolution to ∫

K

f(xk · y)dk =n∑

i=1

gi(x)hi(y), x, y ∈ G, (1)

thenn∑

i=1

gi(x)∫

K

[hi(yk · z) + hi((k · z)y)]dk (2)

=n∑

i=1

[∫K

gi(xk · y)dkhi(z) +∫

K

gi(xk · z)dkhi(y)]

, x, y, z ∈ G.

This necessary condition is applied to the two equations∫

Kf(xk · y)dk =

f(x)g(y)+g(x)f(y) and∫

Kf(xk ·y)dk = f(x)f(y)+g(x)g(y), and this reduces the

problem of solving these two equations to the problem of solving simpler equations.

Fripertinger, Harald: On covariant embeddings of a linear functional equationwith respect to an analytic iteration groupJoint work with Ludwig Reich.Let a(x), b(x), p(x) be formal power series in the indeterminate x over C, such

that ord a(x) = 0, ord p(x) = 1 and p(x) is embeddable into an analytic iterationgroup π = (π(s, x))s∈C in C[[x]]. By a covariant embedding of the linear functionalequation

ϕ(p(x)) = a(x)ϕ(x) + b(x), (L)

176 Report of Meeting AEM

with respect to π we understand families (α(s, x))s∈C and (β(s, x))s∈C with entirecoefficient functions in s, which satisfy a system of two cocycle equations, twoboundary conditions and ϕ(π(s, x)) = α(s, x)ϕ(x) + β(s, x) for all solutions ϕ(x)of (L) and s, t ∈ C. We show how to solve this system of equations and describeunder which conditions there exist covariant embeddings of (L) with respect to π.More information under http://www-ang.kfunigraz.ac.at/˜fripert/.

Ger, Roman: Fischer–Muszely additivity of mappings between normed spacesThe additive Cauchy equation in norm

‖f(x + y)‖ = ‖f(x) + f(y)‖ (∗)had first been studied by P. Fischer and G. Muszely in their paper On some newgeneralizations of the functional equation of Cauchy, (Canad. Math. Bull. 10(1967), 197–205) and then has extensively been examined through the last threedecades by many authors (J. Dhombres, F. Skof, G. Berruti & F. Skof, R. Ger, P.Schopf, R. Ger & B. Koclega and Gy. Maksa & P. Volkmann, among others).

We have obtained the general solution of equation (∗) for mappings with thedomain being an abelian group and having the ranges in quite arbitrary normedlinear spaces. The solutions are expressed in terms of isometries and homomor-phisms.

Gilanyi, Attila: Dinghas-type derivatives and convexity of higher orderJoint work with Zsolt Pales.The nth-order Dinghas derivative of a real valued function f at a point ξ ∈ R

is defined by

Dnf(ξ) = limα≤ξ≤β

β−α↘0

( −n

β − α

)n n∑k=0

(−1)k

(nk

)f

((1 − k

n

)α +

k

nβ

),

if the limit exists. In this talk a mean value inequality is proved for a class of gen-eralized derivatives of the above type. As a consequence of this result, higher-orderconvexity properties (higher order Jensen-convexity, t-Wright-convexity, etc.) arecharacterized in terms of generalized derivatives.

Gronau, Detlef: On the function log Γ(x)/ log x

Joint work with Janusz Matkowski.The function log Γ(x)

log x is characterized as the only convex solution of the func-tional equation

f(x + 1) =log x

log(x + 1)(f(x) + 1), x ∈ (0,∞).

Some relations to the function log Γ(x + 1)/xa, 0 < a ≤ 1 are shown.

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 177

Jarai, Antal: Continuity implies differentiability for solutions of functionalequations – even with few variables

It is proved that under certain conditions the continuous solutions f of thefunctional equation

f(x) = h(x, y, f(g1(x, y)), . . . , f(gn(x, y))), (x, y) ∈ D ⊂ Rs × Rl

are C∞, even if 1 ≤ l < s. As a tool we introduce new function classes which,roughly speaking, interpolate between differentiable and continuous functions.Connections between these classes are also investigated.

Jarczyk, Witold: The arithmetic mean as a linear combination of two quasi-arithmetic means

Joint work with Dorota G�lazowska and Janusz Matkowski.

We determine all the pairs of quasi-arithmetic means having twice continuouslydifferentiable generators and such that the arithmetic mean is a linear combinationof them. Moreover, it turns out that either at least one of these means is arithmetic,or one of the three means is the arithmetic mean of the remaining two.

Kairies, Hans-Heinrich: The action of a Banach space operator on Fourierseries

The operator F , given by F [ϕ](x) :=∑∞

k=0 2−kϕ(2kx), is a continuous auto-morphism on the Banach space C of real functions, which are continuous, 1-periodicand even.

F is known to generate Weierstrass or Takagi type continuous nowhere differ-entiable functions from very simple ϕ ∈ C.

We state general properties of F and investigate in particular the action of Fon Fourier series of elements of C.

For this purpose, the de Rham type functional equation for ψ = F [ϕ]

ψ(x) − 12ψ(2x) = ϕ(x)

is very useful.

Kannappan, Pl.:Normal distributions and the functional equation f(x+y)g(x−y)= f(x)f(y)g(x)g(−y)

Suppose that u and v are independent random variables. If u + v and u − vare independent, then u and v are normally distributed. A functional equationis derived from this, and then we determine the most general solution of thisequation, the regular solutions of which are normal distributions.

178 Report of Meeting AEM

Lajko, Karoly: A functional equation arising in the theory of conditionallyspecified distributionsThe functional equation

G1

(x − a1 − a2y

1 + cy

)+ F1(y) = G2

(y − b1 − b2x

1 + dx

)+ F2(y) (x, y ∈ R (or R+))

is related with the characterization of bivariate distributions. It is investigatedfor functions G1 , F1 , G2 , F2 : R (orR+) → R . Here a1 , a2 , b1 , b2 ∈ R andc, d ∈ R+ are arbitrary constants.

Lesniak, Z.: On an equivalence relation for flows in the planeWe consider an equivalence relation for a given free mapping f of the plane. We

describe the structure of equivalence classes of the relation. Under the assumptionthat f is embeddable in a flow {f t : t ∈ R} we get the theorem which says that frestricted to each equivalence class is a Sperner homeomorphism.

Maksa, Gyula: Hyperstability of a class of linear functional equationsJoint work with Zsolt Pales.First we investigate the stability properties of the functional equation

ψ(xy) = M(x)ψ(y) + M(y)ψ(x) (x, y ∈]0, 1])

where M is a given multiplicative function which has a value greater than 1, andprove that the stability inequality

|ψ(xy) − M(x)ψ(y) − M(y)ψ(x)| ≤ ε (x, y ∈]0, 1])

(with fixed ε ≥ 0) implies (1). We say shortly that (1) is hyperstable.Next we present generalizations for a class of linear functional equations with

normed space-valued functions defined on a semigroup.

Matkowski, Janusz: Equality of Lagrangean and quasi-arithmetic meansJoint work with Justyna B�lasinska.Let I be a real interval and f, g : I → R be continuous and strictly monotonic

functions. The functions Lf : I2 → I and Qg : I2 → I defined by

Lf (x, y) :=

{f−1

(1

x−y

∫ y

xf(t)dt

), x �= y

x, x = y, Qg(x, y) := g−1

(g(x) + g(y)

2

)

are two variable means, i.e. Lf is a Lagrangean and Qg is a quasi-arithmetic mean,respectively. Assuming some regularity conditions, we determine all functions off and g satisfying the equation

Lf = Qg.

Molnar, Lajos: Transformations on the set of all n-dimensional subspaces of aHilbert space preserving principal anglesWigner’s classical theorem on symmetry transformations plays a fundamental

role in quantum mechanics. It can be formulated, for example, in the following

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 179

way: Every bijective transformation on the set L of all 1-dimensional subspacesof a Hilbert space H which preserves the angle between the elements of L isinduced by either a unitary or an antiunitary operator on H. In this talk wepresent an extension of Wigner’s result from the 1-dimensional case to the case ofn-dimensional subspaces of H with n ∈ N fixed.REFERENCE

[1] L. Molnar, Transformations on the set of all n-dimensional subspaces of a Hilbert spacepreserving principal angles, Commun. Math. Phys. 217 (2001), 409–421.

Morawiec, Janusz: Irregular scaling functions with orthogonal translationsJoint work with Lech Bart�lomiejczyk.Following J. Cnops [Proc. Amer. Math. Soc. 128 (2000), 1975–1979], we

discuss the existence of irregular compactly supported solutions ϕ : R → R of thefunctional equation

ϕ(x) =m−1∑i=0

ϕ(mx − ik),

satisfying ∑i∈Z

ϕ(x + i) = 1 a.e.

andϕ(x)ϕ(x + i) = 0 a.e.

for every i ∈ Z \ {0}.

Moszner, Zenon: Nouveaux resultats sur la fonction d’indiceOn annonce quelques nouveaux resultats au sujet de la fonction d’indice, c. a

d. de la solution de l’equation fonctionnelle conditionnelle

f(x) · f(y) �= 0 ⇒ f(x + y) = f(x) · f(y),

ou f : R(p) = {(x1, . . . , xp) ∈ Rp : xi ≥ 0 pour i = 1, . . . , p} \ {0} → R(p),0 = (0, . . . , 0) ∈ Rp et x + y = (x1 + y1, . . . , xp + yp), x · y = (x1 · y1, . . . , xp · yp)pour x = (x1, . . . , xp) ∈ R(p), y = (y1, . . . , yp) ∈ R(p).

Nikodem, Kazimierz: Remarks on injective additive multifunctionsA multifunction F :< 0,∞) −→ n(Y ) (where n(Y ) is the family of all nonempty

subsets of a vector space Y ) is said to be additive if

F (s + t) = F (s) + F (t) (1)

for all s, t ∈< 0,∞); it is injective if

F (s) ∩ F (t) = ∅ (2)

180 Report of Meeting AEM

for all s, t ∈< 0,∞), s �= t. Some examples and properties of injective additivemultifunctions are presented. In particular the following result is given.

Theorem. Let Y be a real topological vector space and F :< 0,∞) −→ n(Y ) be aninjective additive multifunction. If Y is separable or F is convex-valued or F (x0)is bounded for some x0 ∈ (0,∞), then intF (x) = ∅ for every x ∈ < 0,∞).

Pales, Zsolt: On the solution of the Matkowski–Suto problemJoint work with Zoltan Daroczy.The Matkowski–Suto problem is to solve the functional equation

ϕ−1

(ϕ(x) + ϕ(y)

2

)+ ψ−1

(ψ(x) + ψ(y)

2

)= x + y (x, y ∈ I), (M-S)

where I is an open real interval and ϕ,ψ : I → R are continuous and strictlymonotonic unknown functions. Under analiticity assumptions, (M-S) was firstsolved by Suto in 1914. Matkowski obtained the same set of solutions amongC2-functions in 1999.

In a recent paper (of 2001), we determine the C1-solutions. Our new contribu-tion shows that (M-S) can be solved without any further regularity assumptions.The proofs of these steps make use of the classical theorems of Lebesgue, Baire,Piccard and Jarai and also an extension theorem due to Gy. Maksa and the au-thors.

Ratz, Jurg: On inequalities associated with the Jordan – von Neumann func-tional equationThe result by A. Gilanyi presented at ISFE 38 (Aequationes Math. 61 (2001),

289) is obtained under more general conditions.

Reich, Ludwig: On the general solution of the translation equation in rings offormal power seriesJoint work with W. Jab�lonski.Let (F (t, x))t∈C be a family of formal power series

F (t, x) = x + ϕk(t)xk + . . . (t ∈ C)

with complex coefficients where k ≥ 2, ϕk �= 0, and such that the translationequation

F (t + s, x) = F (t, F (s, x)) (t, s ∈ C) (T)

holds. We prove universal representations of the coefficient functions ϕν(t) (ν ≥ k)as polynomials Φ(k)

ν (a(t), h) in a and h where a : C → C is a nontrivial additivefunction and h = (hn)n≥k is a sequence of complex parameters.

Furthermore, we discuss analogous representations for the congruences

F (t + s, x) ≡ F (t, F (s, x)) (modx) (t, s ∈ C) (T,�)

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 181

for � ≥ 1, and we discuss the problem of extending a solution of (T,�) to a solutionof (T,m) with m > �, or of extending a solution of (T,�) to a solution of T.

Riedel, Thomas: Functional equations characterizing polynomialsThis is a report on joint work with Maciej Sablik and Abe Sklar. Using a

sequence of recursively defined divided differences, we produce and solve two setsof functional equations which characterize polynomials of even and polynomialfunctions of odd degree, respectively.

Sablik, Maciej: On some functional equations of Chini in actuarial mathematicsA. Guerraggio recalled in [1] some functional equations dealt with by M. Chini

at the beginning of 20th century. The equations were connected with actuarialmathematics, and their origin goes back to A. De Morgan. While Chini gave so-lutions under differentiability assumptions, T. Riedel, P. K. Sahoo and the author(cf. [2]) were able to relax regularity requirements in the case of the equation

f(x + y) + f(x + z) = cf(x + h(y, z)),

and to solve it completely for continuous, or even locally bounded functions. In thepresent talk we present some results concerning other Chini functional equations,viz.

f(x + y)f(x + z) = ρf [x + u(y, z)],

andf(x + y)f(x + z) = ρf2[x + u(y, z)].

REFERENCES

[1] A. Guerraggio, Le equazioni funzionali nei fondamenti della matematica finanziaria, Riv.Mat. Sci. Econom. Social. 9 (1986), 33–52.

[2] T. Riedel, M. Sablik and P. K. Sahoo, On a functional equation in actuarial mathematics, J.Math. Anal. Appl. 253 (2001), 16–24.

Sahoo, Prasanna: General solution of a quartic functional equationJoint work with Jukang K. Chung.Motivated by the identity

(x + 2y)4 + (x − 2y)4 + 6x4 = 4[(x + y)4 + (x − y)4 + 6y4]

for all x, y ∈ R, we consider the quartic functional equation

f(x + 2y) + f(x − 2y) + 6f(x) = 4[f(x + y) + f(x − y) + 6f(y)] (1)

for all x, y ∈ R (the set of reals). Using an elementary method, we establish thefollowing result: The function f : R → R satisfies the quartic functional equation(1) for all x, y ∈ R, if and only if f is of the form f(x) = A4(x) where A4(x) =A(x, x, x, x) is the diagonal of a symmetric multiadditive function A : R4 → R infour variables.

182 Report of Meeting AEM

Schleiermacher, Adolf: Some consequences of a theorem of LiouvilleLet S denote the automorphism group of the Euclidean space E of n dimen-

sions comprising Euclidean motions as well as similarities. Let h be a bijectiveC1 mapping of E not contained in S. We are interested in the group G gen-erated by h and S and in its continuous invariants, i.e. continuous real valuedfunctions satisfying f(g(P0), g(P1), . . . , g(Pm)) = f(P0, P1, . . . , Pm) for all g ∈ Gand (P0, P1, . . . , Pm) ∈ D ⊆ Em+1. We shall show that such an invariant mustnecessarily be constant, provided that m ≤ n and D is large enough. This will beachieved by showing that the group G has an orbit which is dense in En+1.

Schwaiger, Jens: On a characterization of trigonometric functionsJoint work with Peter Schopf.Physical observations show that the superposition of two harmonic oscillations

of equal frequency, but of possibly different phase, is also an harmonic oscillationof the same frequency. In mathematical terms this means that for all non-negativereals A,B and all reals α, β there is some non-negative C and some γ such thatfor all real x

A · cos(x + α) + B · cos(x + β) = C · cos(x + γ).

Replacing cos by some unknown function f the corresponding functional equationis solved in the class of real continuous functions with primitive period 2π.

Fixing A,B, α, γ leads to a much more complicated functional equation. It isshown that certain continuous and nowhere differentiable functions of Weierstraßtype are solutions of such equations.

Sikorska, Justyna: On a functional equation related to power meansMarcin E. Kuczma in Aequationes Math. 45 (1993), 300–321, considered ana-

lytic solutions of the functional equation

x + g(y + f(x)) = y + g(x + f(y))

on the real line. In Aequationes Math. 55 (1998), 146–152 the author gave thesolutions in the class of twice differentiable functions. During the 38th ISFE NicoleBrillouet-Belluot presented the proof that (one times) differentiable solutions ofthis equation have the same form as in the previous cases (14. Remark in: Reportof Meeting, Aequationes Math. 61 (2001), 304).

We present solutions of the equation in the class of monotonic convex or concavefunctions. This time we get already some new families of solutions.

Silvennoinen, Heli: On meromorphic solutions of the equation f(p(z)) =R(z, f(z))We consider meromorphic solutions f(z) of the equation

f(p(z)) = R(z, f(z)) =∑m

i=0 ai(z)f(z)i∑nj=0 bj(z)f(z)j

,

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 183

where p(z) is a polynomial, deg p(z) ≥ 2, R(z, f(z)) is an irreducible rationalfunction in f , and the meromorphic functions ai(z) and bj(z) are given.

Smıtal, J.: On a generalized Dhombres functional equationJoint work with P. Kahlig.We consider the functional equation f(xf(x)) = ϕ(f(x)) where ϕ :J→J is a

given increasing homeomorphism of an open interval J⊂(0,∞), and f : (0,∞) → Jis an unknown continuous function. In a previous paper we proved that no con-tinuous solution can cross the line y = p where p is a fixed point of ϕ, with apossible exception for p = 1. The range of any non-constant continuous solutionis an interval whose end-points are fixed by ϕ and which contains in its interiorno fixed point except 1. We also gave a characterization of the class of continuousmonotone solutions and proved a condition sufficient for any continuous functionto be monotone.

In the present paper we give a characterization of the equations (or equivalently,of the functions ϕ) which have all continuous solutions monotone. In particular,all continuous solutions are monotone if either (i) 1 is an end-point of J and Jcontains no fixed point of ϕ, or (ii) 1 ∈ J and J contains no fixed points differentfrom 1.

Stetkær, Henrik: On Jensen’s functional equation on groupsLet G be a group, not necessarily abelian. We prove that the solutions f :G→ C

of Jensen’s functional equation

f(xy) + f(xy−1) = 2f(x), x, y ∈ G,

are functions on the quotient group G/[G, [G,G]]. We give explicit formulas for thesolutions of Jensen’s functional equation in a setting that includes many examples.We show furthermore that the vector space of odd solutions modulo the subspace ofhomomorphisms of G into C is isomorphic to the vector space of homomorphismsof [G,G]/[G, [G,G]] into C.

Our results extend and complement those obtained by I. Corovei, P. Friis andC. T. Ng on Jensen’s functional equation.

Szekelyhidi, Laszlo: Polynomial hypergroups and functional equationsJoint work with Agota Orosz.Hypergroups have been investigated since the pioneer works of C. F. Dunkl,

R. I. Jewett and R. Spector. It turns out that this structure has several propertieswhich make it possible to study functional equations. Such kind of investigationsmay have different applications in the theory of probability on hypergroups whichhas been developed by W. R. Bloom and H. Heyer. In this talk we present someresults concerning classical functional equations on a special kind of hypergroups,namely on polynomial hypergroups.

184 Report of Meeting AEM

Tabor, Jacek: Description of isometries from the real line to a reflexive BanachspaceLet E be a Banach space. By B(0, 1) we denote the unit ball in E. We say

that F is a face of B(0, 1) if there exists a supporting hyperplane H of B(0, 1)such that F = H ∩ B(0, 1).

We show that if E is reflexive, then the following two conditions are equivalent:(i) I : R → E is an isometry,(ii) there exist a face F of B(0, 1), e∈E and a measurable function m :R→F

such that I(t) := e +∫ t

0m(s)ds for t ∈ R.

Tabor, Jozef: Topological relations between the sets of exact and approximatesolutionsJoint work with Jacek Tabor.We study the relation between the sets of exact and approximate solutions of

a given functional equation. Under weak assumptions on the type of the equationthe set of solutions is complete. However, this is not the case for approximatesolutions. It occurs that in the classes of linear functional equations (includingCauchy-type and quadratic) and of superstable equations, this space is complete.The space of solutions of the linear functional equation is a proper subspace of thespace of approximate solutions (in the case of the Cauchy-type equation of infinitecodimension). The situation is different for nontrivial superstable equations wherethe set of solutions has nonempty interior in the space of approximate solutions.

Volkmann, Peter: On stability of functional equations in one variableLet E be a Banach space, Z a set, and ϕ : Z → Z. For f : Z → E and

ε ≥ 0 the following two conditions are equivalent:I) f = h + r, where h(ϕ(x)) = 2h(x), ‖ r(x) ‖≤ ε (x ∈ Z).

II) There is a real η such that

‖ f(ϕn(x)) − 2nf(x) ‖ ≤ 2nε + η (x ∈ Z;n = 1, 2, 3, . . . ).

As a consequence we get a stability result for the functional equationf(ϕ(x)) = 2f(x), and this can be applied to equations having the formf(x ◦ y) = f(x) + f(y).

(Appeared in Sem. LV, http://www.mathematik.uni-karlsruhe.de/˜semlv,No. 11, 6 pp. (2001) (Polish).)

Walorski, Janusz: On some solutions of the Schroder equation in Banach spacesLet K be a cone in a Banach space X. We consider the Schroder equation

φ(f(x)) = ρφ(x),

in which φ : K → X is the unknown function and the function f : K → K isgiven.

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 185

Zdun, Marek Cezary: On the structure of continuous iteration semigroups onthe circleLet S1 = {z ∈ C : |z| = 1}. An iteration semigroup {F t : S1 −→ S1 : t > 0}

is said to be continuous, if every iterate F t is continuous and for every z ∈ S1 themapping t −→ F t(z) is continuous. We give a general description of continuousiteration semigroups on the circle. We also characterize continuous mappingsF : S1 −→ S1 embeddable in a continuous iteration semigroups.

2. Problems and remarks

1. Remark. (A new result on Lorentz transformations.)Let X be a real pre-Hilbert space with dim X ≥ 3, i.e. a real vector space

equipped with an inner product satisfying x2 > 0 for all x �= 0 in X. Moreover,let t ∈ X be a fixed element with t2 = 1. If H := t⊥ = {x ∈ X|xt = 0}, thenX = H ⊕ Rt is a direct sum. Define x ∈ H and x0 ∈ R for x ∈ X by means ofx = x + x0t, and the Lorentz–Minkowski distance of x, y ∈ X by

l(x, y) := (x − y)2 − (x0 − y0)2.

The following theorem which generalizes results of A.D. Alexandrov, F. Cac-ciafesta, E. Schroder holds true.

Theorem. Let f : X → X be a bijection with

∀x, y ∈ X : l(x, y) = 0 =⇒ l(f(x), f(y)) = 0.

Then f is a Lorentz transformation up to a dilatation.

REFERENCE

[1] W. Benz, Lie Sphere Geometry in Hilbert Spaces (Dedicated to S. S. Chern on the occasionof his 90th birthday), to appear.

Walter Benz

2. Remark. (Solution of 13. Problem and 25. Problem from the 37th ISFE.)At the 37th ISFE, Z. Daroczy posed the following as 13. Problem (see Report

of meeting, Aequationes Math. 60 (2000), pp. 191 f.). Let I ⊂ R+ = (0,∞) bean open interval. Find all solutions f : I → R of

f(x) + f(y) = f

(x + y

2

)+ f

(2xy

x + y

).

He conjectured that f = L + c for some logarithmic function L and constant c.Also at the 37th ISFE, J. Matkowski posed the following as 25. Problem. Let

I ⊂ R+ = (0,∞) be an interval. Does there exist a symmetric mean M : I2 → I,

186 Report of Meeting AEM

other than the geometric mean, such that every function f : I → R satisfying thefunctional equation

f(M(x, y)) + f

(xy

M(x, y)

)= f(x) + f(y), x, y ∈ I,

must be of the form f = L + c where L is a logarithmic function and c is a realconstant?

We provide solutions to both problems. Daroczy’s conjecture is true. Moreoverthe answer to Matkowski’s question is “yes” for all means of the form

M(x, y) =(

xp + yp

2

)1/p

with p �= 0. Bruce Ebanks

3. Problem. Let a be a real number greater than 1. Find all functionsh : [0, 1) → R such that

h(x) =1a

[h

(x

a

)+ h

(x + 1

a

)]

for all x ∈ R satisfying 0 ≤ x < min(1, a−1). We note that this functional equationis well known when a = 2. This equation arises from a problem in Markov processesconsidered by Matt Davison of the University of Western Ontario. He considersthe system

xn+1 = 2xn + ε(yn − xn) mod 1yn+1 = 2yn + ε(xn − yn) mod 1,

where ε is a small positive constant. The a above is 2(1 − ε), and the function h

is to be integrable so that∫ 1

0h = 1. Thomas Davison

4. Remark. The following result related to the Hyers–Ulam stability problemfor Wright-convex functions is proved in K. Nikodem, Zs. Pales, On approximatelyJensen-convex and Wright-convex functions, C. R. Math. Rep. Acad. Sci. Canada(to appear).

Theorem. Let X be a vector space, D a convex subset of X with core D �= ∅, f afunction defined on D and ε ≥ 0 a fixed constant. There exist an additive functiona : X → R and a convex function g : D → R such that

|f(x) − a(x) − g(x)| ≤ ε, x ∈ D,

if and only if f satisfies the inequalities

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 187

f

(x1 + · · · + x2n

2n

)≤ f(x1) + · · · f(x2n)

2n+ 2ε, n ∈ N, x1, . . . , x2n ∈ D,

and

f(tx + (1 − t)y) + f((1 − t)x + ty) ≤ f(x) + f(y) + 4ε, x, y ∈ D, t ∈ [0, 1].

Kazimierz Nikodem

5. Problem. We pose a discrete version of a problem of Zs. Pales.We call a sequence of convex sets A1, . . . , An ⊂ R2 a convex chain, if

Ak ∪ Ak+1 ∪ . . . ∪ Al is convex for every k, l ∈ {1, . . . , n}, k ≤ l.Problem. Does there for every R > 0 exist a convex chain A1, . . . , An withdiam(Ak) ≤ 1 for k = 1, . . . , n, and x0 ∈ R2 such that

B(x0, R) ⊂ A1 ∪ . . . An?

One can easily notice that if the answer to the above problem is negative,then the answer to the problem of Zs. Pales is also negative, that is, there is noconvexity preserving continuous function k : [0, 1] → R2 such that the image of khas nonempty interior.

Jacek Tabor

6. Remark. (C’est la reponse a la question posee par D. Gronau apres maconference.) Quelles sont les solutions de l’equation

f(c) · f(d) = 0 =⇒ f(c + d) = f(c) · f(d), (*)

ou f : R(p) := [0,∞)p \ {0} → R(p), 0 = (0, . . . , 0) ∈ Rp et pour c = (c1, . . . , cp)et d = (d1, . . . , dp) on a c + d = (c1 + d1, . . . , cp + dp) et c · d = (c1d1, . . . , cpdp)?

Soit f : R(p) → R(p) et considerons les cas:1) Il existe c, d ∈ R(p) tels que f(c) · f(d) = 0, alors (*) n’est pas satisfaite

puisque f(c + d) �= 0. La fonction f n’est pas une solution de (*) dans ce cas.2) Pour chaque c, d ∈ R(p) on a f(c) · f(d) �= 0, alors (*) a lieu puisque

f(c) ·f(d) = 0 n’a lieu pour aucun c, d. La fonction f est une solution dans ce cas.Nous pouvons donc donner la description suivante des solutions de (*).La fonction f : R(p) → R(p) est une solution de (*) si et seulement si dans

l’ensemble f(R(p)) il n’y a pas d’elements a, b tels que a · b = 0.Z. Moszner

7. Remark. A binary gamble (x,C; y, C) [(x,C; y) for short] is an uncertainalternative with consequence x when event C and y when C = E \ C happens(consequences, and also gambles, have partial order �, events are subsets of a

188 Report of Meeting AEM

set E). Let U [(x,C; y)] = U(x,C; y) (for short) and U(z)[= U(z, C; z)]) denotethe utility of (x,C; y) or of z, respectively. Further, x⊕ z denotes the joint receiptof x and z. It connects to (x,C; y) by (x,C; e) ⊕ y ∼ (x ⊕ y, C; y), (e means “nochange”; e ⊕ y ∼ y). The rank-dependent utility (RDU) is

U(x,C; y) = U(x)W (C) + U(y)(1 − W (C))for x � y (extended to x ≺ y by (x,C; y, C) ∼ (y, C;x,C); W (C) is the “weight”of C). R.D. Luce generalized a commutative ⊕ to the homogeneous V (x ⊕ y) =V (y)F [V (x)/V (y)] (U = G(V )) and RDU to

U(x,C; y) = H(U(x)W (C), U(y)K[W (C)]).All this leads to the functional equation θ(sw, v) = H[θ(s, v)w,G(v)K(w)], whereθ(s, v) = G(vF [G−1(s)/v]) (s ∈ [0, k[, v ∈]0,∞[, w ∈ [0, 1]; F : [0,∞[→ [1,∞[,G : [0,∞[→ [0, k[, K : [0, 1] → [0, 1] are bijections, F,G strictly increasing, Kstrictly decreasing, H strictly increasing in both variables). Luce asked for itsgeneral solution(s). They give equivalents to RDU :

U(x,C; y) = U(x)W (C) + U(y)[1 − W (C)]and either to

U(x ⊕ y) = U(x) + U(y) + δU(x)U(y)with commutative ⊕, or to the new law U(x ⊕ y) = αU(x) + U(y); here ⊕ is notcommutative, if α �= 1, but is bisymmetric: (x⊕x′)⊕ (y⊕y′) = (x⊕y)⊕ (x′⊕y′).

Janos Aczel

8. Remark. (Mappings preserving two hyperbolic distances.)Let X be a real pre-Hilbert space with dim X ≥ 2, i.e. a real vectorspace

equipped with an inner product satisfying x2 > 0 for all x �= 0 in X. According tothe Weierstrass model of hyperbolic geometry, the elements of X are called points,and the hyperbolic distance of x, y ∈ X is defined by h(x, y) ≥ 0 with

cosh h(x, y) =√

1 + x2√

1 + y2 − xy.(X,h) is a metric space. In Journ. Geom. 70, 2001, we prove the

Theorem. Let ρ > 0 be a fixed real number and N > 1 be a fixed integer. Iff : X → X satisfies

∀x, y ∈ X : h(x, y) = ρ =⇒ h(f(x), f(y)) ≤ ρ,

∀x, y ∈ X : h(x, y) = Nρ =⇒ h(f(x), f(y)) ≥ Nρ,

then f must be a hyperbolic isometry of X, i.e.

∀x, y ∈ X : h(x, y) = h(f(x), f(y))

holds true.

Problem. Find a real number α, as small as possible, such that the Theoremholds for every real number N > α. It turns out that α ≥ 1.

Walter Benz

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 189

9. Remark. (In connection with the talk of Zoltan Boros.)Denote by F(R) the family of all Q-differentiable self-mappings of the real line

R. Then, according to Z. Boros’ main result, we have

F(R) = {f : R → R | f = c1 + awith c1 ∈ C1(R) and a : R → R additive}Bearing in mind a well-known theorem of N. G. de Bruijn stating that the classC1(R) does have the difference property, we easily see that

F(R) = {f : R → R | for every h ∈ R one has Δhf ∈ C1(R)}(Δh stands here for the difference operator with increment h).

This observation suggests now the following possible definition of Q-differenti-ability of higher orders. Namely, we will say that a function f : R → R is n-timesQ-differentiable provided that for every h ∈ R one has Δn

hf ∈ Cn(R) (Δnh stands

here for the n-th iterate of the operator Δh).With the aid of a theorem of Z. Gajda we may then conclude that a function

f : R → R is n-times Q-differentiable, if and only if f admits a representation ofthe form

f = cn + pn

where cn ∈ Cn(R) and pn is a polynomial function of n-order, i.e.

Δn+1h f(x) = 0 for allx, h ∈ R.

It is not hard to replace here the domain R by an open interval contained in R.Roman Ger

10. Remark. Let X be a real pre-Hilbert space with dim X ≥ 2, i.e. a realvector space equipped with an inner product satisfying x2 > 0 for all nonzero xin X.

If c designates the speed of light, the space V of velocities in special relativitytheory is given by {x ∈ R3|x2 < c2} and the impulse for p ∈ V and a mass m0 > 0at rest, by

m0√1 − p2

c2

· p.

Replacing R3 by X and putting c = 1 and m0 = 1, we getCase (R). V := {x ∈ X|x2 < 1} and impx := x√

1−x2 for x ∈ V .In classical mechanics the space of velocities is R3 and the impulse for p ∈ V

and a mass m > 0 is m · p. Again, replacing R3 by X and putting m = 1, we getCase (C). V := X and impx := x for x ∈ V .

Theorem. Assume (R) or (C) and let f : V × V → V satisfy(i) f(p, x) = f(p, y) implies x = y for all p, x, y ∈ V ,(ii) If p ∈ V and l is a line of X with l ∩ V �= ∅, there exists a line g with

{f(p, x)|x ∈ l ∩ V } = g ∩ V ,

190 Report of Meeting AEM

(iii) ‖ f(p, 0) ‖=‖ p ‖ for all p ∈ V ,(iv) impf(p, q) − imp q ∈ R>0 · imp p for all p, q ∈ V .

Then

f(p, q) =p + q

1 + pq+

1

1 +√

1 − p2

(pq)p − p2q

1 + pq

in the case (R), and f(p, q) = p + q in the case (C) for all p, q ∈ V .

According to an earlier result of the author in Abh. Math. Sem. Univ. Hamb.70 (2000) 251–258, assumptions (i), (ii) can be replaced in the case (R) by thefunctional equation

S(x, y) = S(f(z, x), f(z, y)), ∀x, y, z ∈ V,

where√

1 − x2√

1 − y2S(x, y) := 1 − xy.Walter Benz

11. Problem and Remark. It is sometimes said that the explicit solutions offunctional equations in a single variable are not interesting or not very interest-ing. Maybe so theoretically. For applications, such individual explicit solutionsare definitely important. Functional equations in a single variable of the formf(kx) = g[f(x)] seem to have applications in image processing. See e.g. SteveMann’s paper in the IEEE Trans. on Image Processing, Vol 9, No. 8, 2000, inparticular the Table 1 of equations (called comparametric equations there) and oftheir solutions. The editor’s words about the author at the end of the paper arealso worth reading (the author is inventor of “wearable computers”). The solu-tions given there are clearly not the most general (arbitrary periodic functions aremissing). Anyway, it is the explicit solutions that are applied.Problem. Under what (regularity, convexity or assumptotic) assumptions arethose the general solutions?

Janos Aczel

12. Remark. (To J. Ratz’s talk.)In J. Ratz’s talk the following generalization of the main result of A. Gilanyi,

Eine zur Parallelogrammgleichung aquivalente Ungleichung, Aequationes Math.62 (2001), 303–309, was presented.

Theorem. Let G be a group, E be an inner product space over R or C. If afunction f : G → E satisfies

f(xy) = f(yx) (x, y ∈ G), (C’)

f(xyxy−1) = f(x2) (x, y ∈ G) (C”)

and ‖2f(x) + 2f(y) − f(xy−1)‖ ≤ ‖f(xy)‖ (x, y ∈ G),

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 191

then it fulfills the functional equation

f(xy) + f(xy−1) − 2f(x) − 2f(y) = 0 (x, y ∈ G).

We prove that this statement is also valid, if we omit assumption (C’) above.

Attila Gilanyi

13. Remark. Connected with J. Ratz’s and R. Ger’s talks at this meeting andin connection with K. Nikodem’s problem presented during the 38th InternationalSymposium on Functional Equations, Noszvaj, Hungary, 2000 ([1]; cf. also [2]), weprove the following statement.

Theorem. Let G be an abelian group divisible by 2, H be a Hilbert space and εbe a non-negative real number. If a function f : G → H satisfies

‖f(x − y) − 2f(x) − 2f(y)‖ ≤ ‖f(x + y)‖ + ε (x, y ∈ G),

then there exists a uniquely determined function g : G → H for which

g(x + y) + g(x − y) − 2g(x) − 2g(y) = 0 (x, y ∈ G)

and‖f(x) − g(x)‖ ≤ 5

2ε (x ∈ G).

REFERENCES[1] K. Nikodem, 7. Problem, Report of Meeting, Aequationes Math. 61 (2001), 301.[2] Jacek Tabor and Jozef Tabor, 19. Remark, (Solution of 7. Problem by K. Nikodem), Report

of Meeting, Aequationes Math. 61 (2001), 307–309.

Attila Gilanyi

14. Problems. In connection with R. Ger’s and J. Ratz’s talks we present thefollowing problems.

Let S be a non-empty set and X be a linear normed space. Consider thefunctional equations and inequalities

F1[f(g1(x, y)), . . . , f(gk(x, y))] = F2[f(gk+1(x, y)), . . . , f(gn(x, y))], (1)‖F1[f(g1(x, y)), . . . , f(gk(x, y))]‖=‖F2[f(gk+1(x, y)), . . . , f(gn(x, y))]‖, (2)‖F1[f(g1(x, y)), . . . , f(gk(x, y))]‖≤‖F2[f(gk+1(x, y)), . . . , f(gn(x, y))]‖, (3)‖F1[f(g1(x, y)), . . . , f(gk(x, y))]‖≤‖F2[f(gk+1(x, y)), . . . , f(gn(x, y))]‖ + ε.

(4)

Find functions gi : S × S → S, (i = 1, . . . , n), and Fj : X → X, (j = 1, 2), forwhich, f : S → X being the unknown function

– equation (1) is equivalent to equation (2) or

192 Report of Meeting AEM

– equation (1) is equivalent to inequality (3) or– equation (2) is equivalent to equation (3).

Find functions g1, . . . , gn, F1, F2 for which the following stability property holds:there exists a real constant c such that, for an arbitrary non-negative real numberε and for an arbitrary solution f : S → X of inequality (4), there exists a solutionh : S → X of (1) (or (2) or (3)) for which ‖f(x) − h(x)‖ ≤ cε, (x ∈ S).

We note that the problems can be formulated in more general forms (e.g. con-sidering different functions f1, . . . , fn instead of f in the equations and inequalitiesabove, or, as it was also remarked by R. Ger during the discussion about the prob-lems, writing another given function instead of the norm on both sides of (2), (3)and (4)).

Concerning the solution of the problems, in his talk, R. Ger made a contributionto the equivalence of equations (1) and (2) in the case of the Cauchy equation,J. Ratz presented a result on the equivalence of (1) and (3) for the square-normfunctional equation, furthermore, in Remark 2 a stability theorem was provedconnecting (1) and (4) for the square-norm equation. Further investigations of theproblems presented can be found in the references below.

REFERENCES

[1] J. Aczel, K. Fladt, M. Hosszu, Losungen einer mit dem Doppelverhaltnis zusammenhangen-den Funktionalgleichung, MTA Mat. Kut. Int. Kozl. 7 A (1962), 335–352.

[2] G. Berruti, F. Skof, Risultati di equivalenza per un’equazione di Cauchy alternativa neglispazi normati, (Equivalence results for an alternative Cauchy equation in normed spaces,Italian), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 125 (1991), 154–167.

[3] P. Fischer, Gy. Muszely A Cauchy-fele fuggvenyegyenlet bizonyos tıpusu altalanosıtasai,(On certain generalizations of the functional equation of Cauchy type, Hungarian), Mat.Lapok 16 (1965), 67–75.

[4] P. Fischer, Gy. Muszely On some new generalizations of the functional equation of Cauchy,Canad. Math. Bull. 10 (1967), 197–205.

[5] P. Fischer, Sur l’equivalence des equations fonctionelles, Ann. Fac. Sci. Univ. Toulouse 30(1966/1968), 71–74.

[6] R. Ger, On a characterization of strictly convex spaces, Atti. Accad. Sci. Torino Cl. Sci.Fis. Mat. Natur. 127 (1993), 131–138.

[7] R. Ger, A Pexider-type equation in normed linear spaces, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997/1998), 291–303.

[8] R. Ger, B. Koclega Isometries and a generalized Cauchy equation, Aequationes Math. 60(2000), 72–79.

[9] A. Gilanyi, On inequalities derived from the square-norm equation, Abstract, Report ofMeeting, Aequationes Math. 61 (2001), 289.

[10] A. Gilanyi, Eine zur Parallelogrammgleichung aquivalente Ungleichung, AequationesMath., 62 (2001), 303–309.

[11] M. Hosszu, Egy alternatıv fuggvenyegyenletrol, (On an alternative functional equation,Hungarian), Mat. Lapok 14 (1963), 98–102.

[12] M. Hosszu, Eszrevetelek a relativitaselmeleti idofogalom Reichenbach-fele ertelmezesehez,NME Magyarnyelvu Kozlemenyei, Miskolc, 1964, 223–233.

[13] S. Kurepa, On P. Volkmann’s paper, Glas. Mat. Ser. III, 22(42) (1987), 371–374.

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 193

[14] Gy. Maksa, Remark on the talk of P. Volkmann, Proc. of the Twenty-third InternationalSymposium on Functional Equations (Gargnano, Italy, 1985), Faculty of Mathematics,University of Waterloo, Waterloo, Ontario, Canada, 72–73.

[15] Gy. Maksa, P. Volkmann, Characterization of group homomorphisms having values in aninner product space, Publ. Math. Debrecen 56 (2000), 197–200.

[16] K. Nikodem, 7. Problem, Report of Meeting, Aequationes Math. 61 (2001), 301.[17] P. Schopf, Solutions of ‖f(ξ + η)‖ = ‖f(ξ) + f(η)‖, Math. Pannonica 8 (1997), 117–127.[18] F. Skof, On the functional equation ‖f(x + y) − f(x)‖ = ‖f(y)‖, Atti Accad. Sci. Torino

Cl. Sci. Fis. Mat. Natur. 127 (1993), 229–237.[19] F. Skof, On two conditional forms of the equation ‖f(x+y)‖ = ‖f(x)+f(y)‖, Aequationes

Math. 45 (1993), 167–178.[20] F. Skof, On some alternative quadratic equations, Results Math. 27 (1995), 402–411.[21] F. Skof, On some alternative quadratic equations in inner-product spaces, Atti Sem. Mat.

Fis. Univ. Modena 46 (1998), 951–962.[22] F. Skof, V. Marcella, On the functional equation |f(x + y) + f(x − y)| = |2f(x) + 2f(y)|,

Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 130 (1996), 153–162.

[23] H. Swiatak On the equation ϕ(x + y)2 = [ϕ(x)g(y) + ϕ(y)g(x)]2, Zeszyty Naukowe Uniw-ersytetu Jagiellonskego, Nr. II, Prace Matematyczne 10 (1965), 97–104.

[24] Jacek Tabor, Jozef Tabor, 19. Remark, (Solution of 7. Problem by K. Nikodem), Reportof Meeting, Aequationes Math. 61 (2001), 307–309.

[25] Jacek Tabor, Jozef Tabor, 20. Problem, Report of Meeting, Aequationes Math. 61 (2001),309.

[26] E. Vincze, Alternatıv fuggvenyegyenletek megoldasairol, (Solutions of alternative functionalequations, Hungarian), Mat. Lapok 15 (1964), 179–195.

[27] E. Vincze, Beitrag zur Theorie der Cauchyschen Funktionalgleichungen, Arch. Math. 15(1964), 132–135.

[28] E. Vincze, Uber eine Verallgemeinerung der Cauchyschen Funktionalgleichung, FunkcialajEkvacioj 6 (1964), 55–62.

[29] P. Volkmann, Pour une fonction reelle f l’inequation |f(x)+f(y)| ≤ |f(x+y)| et l’equationde Cauchy sont equivalentes, Abstract, Proc. of the Twenty-third International Symposiumon Functional Equations (Gargnano, Italy, 1985), Faculty of Mathematics, University ofWaterloo, Waterloo, Ontario, Canada, 43.

Attila Gilanyi

15. Problem. (On convexity preserving mappings.)Let X be a linear space and define, for x, y ∈ X, the closed and open segments

[x, y] and ]x, y[ by

[x, y] := {tx + (1 − t)y | t ∈ [0, 1]}, ]x, y[:= {tx + (1 − t)y | t ∈]0, 1[}.A mapping T : D → Y (where D is a convex subset of X and Y is also a

linear space) is called convexity preserving if, for all convex subset K of D, theimage T (K) is convex. Analogously, the inverse relation T−1 is called convexitypreserving if, for all convex subset K ⊂ Y , the inverse image T−1(K) is convex.

One can easily obtain the following characterizations.

Let T : D → Y . Then T is convexity preserving, if and only if

T ([x, y]) ⊃ [T (x), T (y)] (x, y ∈ D), (1)

194 Report of Meeting AEM

and T−1 is convexity preserving, if and only if

T ([x, y]) ⊂ [T (x), T (y)] (x, y ∈ D). (2)

In the case X = Y = R it is immediate to see that T is convexity preserving, ifand only if it has the so-called Darboux property, and T−1 is convexity preserving,if and only if it is monotonic.

In the case when the image of T has dimension greater than 2 we make insteadof (2) the stronger assumption

T (]x, y[) ⊂]T (x), T (y)[ (x, y ∈ D). (3)

Then the following characterization is possible.

Theorem. Let T : D → Y be such that dim T (D) ≥ 2. Then T satisfies (3), ifand only if there exist linear maps A : X → Y , a : X → R and constants B ∈ Yand b ∈ R such that

a(x) + b > 0 and T (x) =A(x) + B

a(x) + b(x ∈ D).

Problem. Find analogous characterizations for convexity preserving mappings(i.e. mappings that satisfy (1)) or for mappings satisfying

T (]x, y[) ⊃]T (x), T (y)[ (x, y ∈ D). (4)

It follows from the above theorem that if dimT (D) ≥ 2 and T satisfies (3),then

T ([x, y]) = [T (x), T (y)] (x, y ∈ D), (5)

hence T is also convexity preserving. It is not clear whether (1) or (4) are alsoequivalent to (5).

Zsolt Pales

16. Remark. (Bernstein–Doetsch-type theorems for quasiconvexity.)A function f : I → R is called quasiconvex, if

f(tx + (1 − t)y) ≤ max(f(x), f(y)) (x, y ∈ I, t ∈ [0, 1]). (1)

If, for x �= y, the inequality is strict, then f is said to be strictly quasiconvex.Analogously, if

f(x + y

2

)≤ max(f(x), f(y)) (x, y ∈ I), (2)

then f is called Jensen-quasiconvex (and strictly Jensen-quasiconvex, if this in-equality is strict for x �= y).

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 195

Clearly, (strictly) quasiconvex functions are always (strictly) Jensen-quasicon-vex. The problem dealt with in a recent paper [1] is to find regularity conditions onf such that (strict) Jensen-quasiconvexity together with these conditions implies(strict) quasiconvexity. In [1], the following result was established.

Theorem. Let f : I → R be upper semicontinuous. Then f is (strictly) Jensen-quasiconvex, if and only if it is (strictly) quasiconvex.

This result is analogous to that found by Bernstein and Doetsch in 1915 on theconnection of Jensen-convexity and convexity. While, in the case of convexity, veryweak regularity conditions (such as, measurability, local upper boundedness at apoint, etc.) are enough to obtain the equivalence of convexity and Jensen-convexity,it is not clear whether, for the quasiconvexity setting, the upper semicontinuity canbe weakened. One can also study the stability of (1) and (2), that is the connectionbetween the functional inequalities

f(tx + (1 − t)y) ≤ max(f(x), f(y)) + ε (x, y ∈ I, t ∈ [0, 1]) (3)

andf(x + y

2

)≤ max(f(x), f(y)) + ε (x, y ∈ I), (4)

whose solutions are called ε-quasiconvex and ε-Jensen-quasiconvex, respectively.Results analogous to that known for the convex setting (due to C. T. Ng and K.

Nikodem) are not valid in this case, because, for all c > 0, a ε-Jensen-quasiconvexfunction can be constructed which is not (cε)-quasiconvex.

REFERENCE

[1] A. Gilanyi, K. Nikodem, and Zs. Pales, Bernstein–Doetsch type results for quasiconvex func-tions, submitted for publication.

Zsolt Pales

17. Remark. (On the equality of means.)Recently, Zoltan Daroczy and myself have dealt with the equality problem (of

n variable) means belonging to different classes.Let I ⊂ R be a nonvoid open interval and let n ≥ 2 be a given natural number.

Let CM(I) denote the set of all continuous and strictly monotonic real functionsdefined on the interval I.

A function M : In → I is called a conjugate-arithmetic mean of n variables onI, if there exists ϕ ∈ CM(I) for which

M(x1, . . . , xn) = ϕ−1

⎛⎜⎜⎝

ϕ(x1) + · · · + ϕ(xn) − ϕ

(x1 + · · · + xn

n

)n − 1

⎞⎟⎟⎠

for all x1, . . . , xn ∈ I.

196 Report of Meeting AEM

The following problem seems to be natural: Which conjugate-arithmetic meansof n variables are also quasi-arithmetic means of n variables on the interval I?

In the case n = 2, this problem is equivalent to the Matkowski–Suto problem,which has been solved by the authors. For n ≥ 3, we have obtained the followingresult.Theorem. Let n ≥ 3 be fixed and let M : In → I be a conjugate-arithmetic meanof n variables on I. Then M is a quasi-arithmetic mean of n variables on I, ifand only if M is the arithmetic mean of n variables on I.

Zsolt Pales

18. Problem. Determine the general increasing bijections F : [0,∞[→ [1,∞[,G : [0,∞[→ [0, k[ (k ∈]0,∞]) which satisfy

G[vF (z)] = A(v)G(vz) + G(v) (v ∈ [0,∞[, z ∈ [0,∞[)

(from the equation, A(v) = G[vF (1)]G(v) − 1 for v ∈]0,∞[; A(0) arbitrary).

Note 1: As mentioned in my talk, C. T. Ng proved that, under the furtherconditions F ∈ C1, G ∈ C2, the solutions are given either by

F (z) = (αzβ + 1)1/β , G(v) = γvβ (v, z ∈ [0,∞[)

or byF (z) = (zβ + 1)1/β , G(v) = λ[exp(κvβ) − 1] (v, z ∈ [0,∞[)

(α > 0, β > 0, γ > 0, λκ > 0).Note 2: The above functional equation seems to be important, maybe even

unavoidable for determining rank-(ordering-)dependent utility of joint receipt inthe theory of gambles and its application to psychophysics.

Janos Aczel

19. Remark. (To the 11. Problem and Remark by J. Aczel.)Most of iterative functional equations listed (together with their particular

solutions) by S. Mann [IEEE Trans. Image Proc. 9 (2000), 1389–1406] are nonlin-ear ones. The equation claimed by the author to be “most commonly used . . . inquantigraphic image processing” reads

ϕ(kx) = kacϕ(x)[1 + (ka − 1)ϕ(x)1c ]−c, x ∈ R+ := [0,+∞), (1)

and it has the solution

ϕ(x) = pcxac[1 + pxa]−c, x ∈ R+. (2)

Claim. If 0 < k < 1, a > 0, c > 0, then formula (2) represents, for every p > 0,the only solution ϕ : R+ → R+ of equation (1) which is continuous on R+ andsatisfies the condition

limx→0+

x−acϕ(x) = p. (3)

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 197

Sketch of proof. Rewrite equation (1) in the equivalent form

ϕ(x) = ϕ(kx)[ka + (1 − ka)ϕ(kx)1c , x ∈ R+, (4)

and observe that the function ψ : R+ → R+ defined by

ψ(x) := x−acϕ(x), x ∈ R+ \ {0}; ψ(0) := p, (5)

is a solution, continuous on R+, of the equation

ψ(x) = H(x, ψ(kx)), x ∈ R+, (6)

whereH(x, y); = y [1 + (1 − ka)xay

1c ]−c, (x, y) ∈ R2

+,

if and only if ϕ is continuous on R+, and it satisfies equation (4) and condition(3).

We have H(0, p) = 0 and we may find a neighbourhood [0, α) × [p − β, p + β]of (0, p) ∈ R2

+ in which |∂H∂y (x, y)| ≤ 1.

Take now two solutions ψ1, ψ2 : R+ → R+ of (6), continuous on R+, ψ1(0) =ψ2(0) = p. With x ∈ [0, α), by the Mean Value Theorem and by induction wearrive at

|ψ1(x) − ψ2(x)| ≤ |ψ1(knx) − ψ2(knx)|, n ∈ N.

When n → ∞ (since 0 < k < 1) this yields ψ1(x) = ψ2(x) for x ∈ [0, α). Bythe Extension Theorem (cf. M. Kuczma, B. Choczewski and R. Ger, IterativeFunctional Equations, Cambridge University Press, 1990) we get ψ1 = ψ2. Conse-quently, ϕ1 = ϕ2, these functions being defined by (5) with ψ replaced by ψ1 andψ2, respectively. Thus equation (1) has the unique solution satisfying (3), hencenecessarily given by (2).

Several other nonlinear equations from S. Mann’s TABLE I can be treatedsimilarly.

Bogdan Choczewski

(Compiled by Peter de Place Friis)

List of Participants

Aczel, Janos, Department of Pure Mathematics, University of Waterloo, Wa-terloo, Ontario, Canada N2L 3G1; e-mail: jdaczel@math.uwaterloo.ca

Baron, Karol, Institute of Mathematics, Silesian University, ul. Bankowa 14,PL-40-007 Katowice, Poland; e-mail: baron@us.edu.pl

Benz, Walter Anton, Mathematisches Seminar, Universitat Hamburg, Bun-desstr. 55, D-20146 Hamburg, Germany; e-mail: benz@math.uni-hamburg.de

198 Report of Meeting AEM

Boros, Zoltan, Institute of Mathematics and Informatics, University of Debre-cen, Pf. 12, H-4010 Debrecen, Hungary; e-mail: boros@math.klte.hu

Brillouet-Belluot, Nicole, Service des Mathematiques, Ecole Centrale deNantes, 1 rue de la Noe, F-44072 Nantes-Cedex 03, France;e-mail: nicole.belluot@ec-nantes.fr

Chmielinski, Jacek, Instytut Matematyk8, Akademia Pedagogiczna, ul. Pod-chorazych 2, PL-30-084 Krakow, Poland; e-mail: jacek@wsp.krakow.pl

Choczewski, Bogdan, Faculty of Applied Mathematics, University of Miningand Metallurgy, ul. Mickiewicza 30, PL-30-059 Krakow, Poland;e-mail: smchocze@cyf-kr.edu.pl

Corovei, Ilie, Department of Mathematics, Technical University, C. DaicoviciuNo. 15, R-3400 Cluj-Napoca, Romania; e-mail: Ilie.Corovei@math.utcluj.ro

Davison, Thomas M. K., Department of Mathematics, McMaster University,1280 Main Street West, Burke Science Building, L8S 4K1 Hamilton, Ontario,Canada; e-mail: davison@mcmaster.ca

Derfel, Gregory, Department of Mathematics, Ben-Gurion University, PO Box653, 84105 Beer-Sheva, Israel; e-mail: derfel@cs.bgu.ac.il

Ebanks, Bruce R., Department of Mathematics & Statistics, Mississippi StateUniversity, Mississippi State, MS 39762, USA; e-mail: ebanks@math.msstate.edu

Frank, Jerry, Department of Mathematics, Illinois Institute of Technology,Chicago IL 60616, USA; e-mail: frankm@iit.edu

Friis, Peter de Place, Department of Mathematics, Eastern Washington Uni-versity, 216 Kingston Hall, Cheney, WA 99004, USA

Fripertinger, Harald, Institut fur Mathematik, Karl Franzens Universitat,Heinrichstr. 36/4, A-8010 Graz, Austria; e-mail: fripert@kfunigraz.ac.at

Ger, Roman, Institute of Mathematics, Silesian University, ul. Bankowa 14,PL-40-007 Katowice, Poland; e-mail: romanger@us.edu.pl

Gilanyi, Attila, Institute of Mathematics and Informatics, University of Debre-cen, Pf. 12, H-4010 Debrecen, Hungary; e-mail: gil@math.klte.hu

Gronau, Detlef, Institut fur Mathematik, Karl Franzens Universitat, Heinrich-str. 36, A-8010 Graz, Austria; e-mail: gronau@kfunigraz.ac.at

Jarai, Antal, Department of Numerical Analysis, Eotvos Lorand University,Pazmany Peter Setany. 1/D., H-1117 Budapest, Hungary;e-mail: ajarai@moon.inf.elte.hu

Jarczyk, Witold, Institute of Mathematics, Technical University, Podgorna 50,PL-65-246 Zielona Gora, Poland; e-mail: wjarc@lord.wsp.zgora.pl

Kairies, Hans-Heinrich, Institut fur Mathematik, Technische Universitat Claus-thal, Erzstr. 1, D-38678 Clausthal-Zellerfeld 1, Germany;e-mail: kairies@math.tu-clausthal.de

Kannappan, Palaniappan, Department of Pure Mathematics, University of Wa-terloo, Waterloo, Ontario, Canada N2L 3G1;e-mail: plkannap@math.uwaterloo.ca

Vol. 64 (2002) The Thirty-ninth International Symposium on Functional Equations 199

Kominek, Zygfryd, Institute of Mathematics, Silesian University, ul. Bankowa14, PL-40-007 Katowice, Poland; e-mail: zkominek@ux2.math.us.edu.pl

Lajko, Karoly, Institute of Mathematics and Informatics, University of Debre-cen, Pf. 12, H-4010 Debrecen, Hungary; e-mail: lajko@math.klte.hu

Lesniak, Zbigniew, Instytut Matematyki, Akademia Pedagogiczna, ul. Pod-chorazych 2, PL-30-084 Krakow, Poland; e-mail: zlesniak@wsp.krakow.pl

Maksa, Gyula, Institute of Mathematics and Informatics, University of Debre-cen, Pf. 12, H-4010 Debrecen, Hungary; e-mail: maksa@math.klte.hu

Matkowski, Janusz, Institute of Mathematics, Technical University, Podgorna50, PL-65-246 Zielona Gora, Poland; e-mail: matkow@omega.im.wsp.zgora.pl

Molnar, Lajos, Institute of Mathematics and Informatics, University of Debre-cen, Pf. 12, H-4010 Debrecen, Hungary; e-mail: molnarl@math.klte.hu

Morawiec, Janusz, Institute of Mathematics, Silesian University, ul. Bankowa14, PL-40-007 Katowice, Poland; e-mail: morawiec@gate.math.us.edu.pl

Moszner, Zenon: Instytut Matematyki, WSP, ul. Podchorazych 2, PL-30-084Krakow, Poland; e-mail: zmoszner@wsp.krakow.pl

Nikodem, Kazimierz, Department of Mathematics, Technical University, ul. Wil-lowa 2, PL-43-309 Bielsko-Bia�la, Poland; e-mail: knik@pb.bielsko.pl

Pales, Zsolt, Institute of Mathematics and Informatics, University of Debrecen,Pf. 12, H-4010 Debrecen, Hungary; e-mail: pales@math.klte.hu

Ratz Jurg: Mathematisches Institut, Universitat Bern, Sidlerstr. 5, CH-3012Bern, Switzerland; e-mail: math@math-stat-unibe.ch

Reich, Ludwig, Institut fur Mathematik, Karl Franzens Universitat, Heinrichstr.36, A-8010 Graz, Austria; e-mail: ludwig.reich@kfunigraz.ac.at

Riedel, Thomas, Department of Mathematics, University of Louisville, LouisvilleKY 40292, USA; e-mail: Thomas.Riedel@louisville.edu

Sablik, Maciej, Institute of Mathematics, Silesian University, ul. Bankowa 14,PL-40-007 Katowice, Poland; e-mail: mssablik@us.edu.pl

Sahoo, Prasanna, Mathematics Department, University of Louisville, LouisvilleKY 40292, USA; e-mail: sahoo@louisville.edu

Schleiermacher, Adolf, Rablstrasse 18., D-81669 Munchen, Germany; e-mail:ADSLE@aol.com

Schwaiger, Jens, Institut fur Mathematik, Karl Franzens Universitatm Hein-richstr. 36, A-8010 Graz, Austria; e-mail: jens.schwaiger@kfunigraz.ac.at

Sikorska, Justyna, Institute of Mathematics, Silesian University, ul. Bankowa14, PL-40-007 Katowice, Poland; e-mail: sikorska@ux2.math.us.edu.pl

Silvennoinen, Heli, Department of Mathematics, University of Joensuu, POBox 111, Fin-80101 Joensuu, Finland; e-mail: hsilven@cc.joensuu.fi

Smital, Jaroslav, Institute of Mathematics, Silesian University, Bezrucovo nam.13, CZ-74601 Opava, Czech Republic; e-mail: Jaroslav.Smital@math.slu.cz

Stetkær, Henrik, Department of Mathematics, University of Aarhus, NyMunkegade, DK-8000 Aarhus C, Denmark; e-mail: stetkaer@imf.au.dk

200 Report of Meeting AEM

Szekelyhidi, Laszlo, Institute of Mathematics and Informatics, University ofDebrecen, Pf. 12, H-4010 Debrecen, Hungary; e-mail: szekely@math.klte.hu

Tabor, Jacek Jacub, Institute of Mathematics, Jagiellonian University, Rey-monta 4 St., PL-30-059 Krakow, Poland; e-mail: tabor@im.uj.edu.pl

Tabor, Jozef, Institute of Mathematics, Wyzsza Szko�la Pedagogiczna wRzeszowie, ul. Rejtana 16 a, PL-35-310, Rzeszow, Poland;e-mail: tabor@univ.rzeszow.pl

Volkmann, Peter, Mathematisches Institut I, Universitat Karlsruhe, D-76128Karlsruhe, Germany; no e-mail

Walorski, Janusz, Institute of Mathematics, Silesian University, ul. Bankowa14, PL-40-007 Katowice, Poland; e-mail: walorski@gate.math.us.edu.pl

Zdun, Marek Cezary, Instytut Matematiki, Akademia Pedagogiczna, ul. Pod-chorazyc 2, PL-30-084 Krakow, Poland; e-mail: mczdun@wsp.krakow.pl

Manuscript received: March 11, 2002.

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