Table Of ContentSpringer Optimization and Its Applications 124
Janusz Brzdęk
Krzysztof Ciepliński
Themistocles M. Rassias Editors
Developments
in Functional
Equations and
Related Topics
Springer Optimization and Its Applications
VOLUME 124
ManagingEditor
PanosM.Pardalos(UniversityofFlorida)
Editor-CombinatorialOptimization
Ding-ZhuDu(UniversityofTexasatDallas)
AdvisoryBoard
J.Birge(UniversityofChicago)
S.Butenko(TexasA&MUniversity)
F.Giannessi(UniversityofPisa)
H.D.Sherali(VirginiaPolytechnicandStateUniversity)
T.Terlaky(LehighUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimizationhasbeenexpandinginalldirectionsatanastonishingrateduringthe
lastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped,
thediffusionintootherdisciplineshasproceededatarapidpace,andourknowledge
ofallaspectsofthefieldhasgrownevenmoreprofound.Atthesametime,oneof
themoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisonthe
interdisciplinarynatureofthefield.Optimizationhasbeenabasictoolinallareas
ofappliedmathematics,engineering,medicine,economicsandothersciences.
The series Springer Optimization and Its Applications publishes undergraduate
and graduate textbooks, monographs and state-of-the-art expository works that
focusonalgorithmsforsolvingoptimizationproblemsandalsostudyapplications
involvingsuchproblems.Someofthetopicscoveredincludenonlinearoptimization
(convex and nonconvex), network flow problems, stochastic optimization, optimal
control, discrete optimization, multi-objective programming, description of soft-
warepackages,approximationtechniquesandheuristicapproaches.
Moreinformationaboutthisseriesathttp://www.springer.com/series/7393
Janusz Brzde˛k • Krzysztof Cieplin´ski
Themistocles M. Rassias
Editors
Developments in Functional
Equations and Related Topics
123
Editors
JanuszBrzde˛k KrzysztofCieplin´ski
DepartmentofMathematics FacultyofAppliedMathematics
PedagogicalUniversityofCracow AGHUniversityofScienceandTechnology
Kraków,Poland Kraków,Poland
ThemistoclesM.Rassias
DepartmentofMathematics
NationalTechnicalUniversityofAthens
Athens,Greece
ISSN1931-6828 ISSN1931-6836 (electronic)
SpringerOptimizationandItsApplications
ISBN978-3-319-61731-2 ISBN978-3-319-61732-9 (eBook)
DOI10.1007/978-3-319-61732-9
LibraryofCongressControlNumber:2017948656
Mathematics Subject Classification (2010): 39-XX, 43-XX, 41-XX, 46-XX, 51-XX, 47-XX, 60-XX,
65-XX,52-XX
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
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broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
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Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
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Dedicatedtothememoryof
StanisławMarcinUlam(1909–1984)
whomorethan75yearsago
posedaproblemconcerning
approximatehomomorphismsofgroups
whichstimulated
along-lasting interest
instability offunctional equations
Preface
The present book has been composed on the occasion of the 16th International
Conference on Functional Equations and Inequalities (ICFEI) that took place in
the Mathematical Research and Conference Center in Be˛dlewo, Poland, on May
17–23, 2015. We dedicate it to the memory of Stanisław Marcin Ulam (April 13,
1909–May 13,1984),who77yearsago posedaproblemconcerning approximate
homomorphisms of groups that stimulated long-lasting research into stability of
functionalequationsandinequalities(FEI).Severalpapersfeaturedinthisvolume
havebeendevotedfullyorpartlytoUlam’sstabilityproblem.
Thebookconsistsofarticleswrittenbyeminentscientistsfromtheinternational
mathematical community, who present important research works in the field of
FEI as well as related subjects. These works provide an insight into the progress
achievedonthestudyofvariousproblemsofnonlinearflavorandpresentup-to-date
developmentsofselectedtopicsofFEIaswellasofrelatedfieldsofmathematics.
Both old and new results are presented in expository and research papers written
by 17 authors from 8 countries who have been intensively involved in those areas
ofinvestigations.Specialemphasishasbeenplacedonavarietyoftopicsapplying
methodsandtechniquesinvolvingororiginatingfromFEI.
Several of these results have been influenced and inspired by the work of
S.M. Ulam, the well-known mathematician and physicist. Emphasis is placed on
thosequestions,concerningapproximatehomomorphisms,thatheposedin1940.
Weaimforthispublicationtoserveasakindofguidebookformathematicians
andotherresearchers,whoseworksaresomewhatconnectedorrelatedtothefields
ofFEIandinparticulartoUlam’stypestability.
Subjectswhichhavebeentreatedinthisbookinclude(inorderofappearancein
thevolume):
– Somequasi-meansandthebehavioroftheirdifference
– Theisometricapproximationprobleminboundedsetsandsomeapplicationsof
the results related to it to the extension problems for bilipschitz and quasisym-
metricmaps
– Amathematizationmethodofsocialchoice
vii
viii Preface
– One-parametersubgroups(iterationgroups)ofthegroupofallinvertiblepower
seriesinoneindeterminatexoverCandadescriptionoftheirconstruction
– The Fischer-Muszély equation, its pexiderization, and Hyers-Ulam stability, as
wellastwoinequalitiesrelatedtoit
– The“alienationphenomenon”forfunctionalequations(andinequalities)
– HaarmeagersetsandHaarnullsetsandsomeanalogiesbetweenthem
– Different types of stability of a system of two equations related to one-
dimensionaldynamicalsystems
– The role of functional equations in the asymptotic analysis needed to elicit the
characterizationofvariouslawsinprobabilitytheory
– Thetranslationequationanditsstability
– Stochastic convex ordering and some applications of the results related to it to
theHermite-Hadamardtypeinequalities
– Twoconstructionsofthefieldofrealscloselyrelatedtofunctionalequationsand
theirstability
– ThegeneralizedDhombresfunctionalequationandaclassificationofitspossible
solutionsaswellasadescriptionofthestructureofperiodicpointscontainedin
therangeofthesolutions
– Functionalequationsaswellastheirstabilityandsuperstabilityonhypergroups
– Thenonstandardanalysisapproachtosomesystemsoffunctionalequationsand
theirstabilityinthecompact-opentopology
It is a pleasure to express our deepest thanks to all of the mathematicians
who, through their works, participated in this volume. We would also wish to
acknowledge the support of the reviewers and the superb assistance that the staff
ofSpringerhasprovidedforthispublication.
Kraków,Poland JanuszBrzde˛k
Kraków,Poland KrzysztofCieplin´ski
Athens,Greece ThemistoclesM.Rassias
March2017
Contents
1 TheBehavioroftheDifferenceBetweenTwoMeans................... 1
ShoshanaAbramovich
2 IsometricApproximationinBoundedSetsandItsApplications...... 17
PekkaAlestalo
3 OntheIndicatorPluralityFunction...................................... 25
AnnaBahyrycz
4 TheTranslationEquationintheRingofFormalPowerSeries
OverCandFormalFunctionalEquations............................... 41
HaraldFripertingerandLudwigReich
5 Fischer–MuszélyAdditivity:AHalfCenturyStory.................... 71
RomanGer
6 AlienFunctionalEquations:ASelectiveSurveyofResults............ 107
RomanGerandMaciejSablik
7 RemarksonAnalogiesBetweenHaarMeagerSetsandHaar
NullSets ..................................................................... 149
ElizaJabłon´ska
8 OnSomeInequalitiesInspiredbytheStability
ofDynamicalSystem....................................................... 161
ZenonMoszner
9 HomomorphismsfromFunctionalEquationsinProbability.......... 171
AdamJ.Ostaszewski
10 RecentDevelopmentsintheTranslationEquation
andItsStability............................................................. 215
BarbaraPrzebieracz
ix
x Contents
11 OnSomeRecentApplicationsofStochasticConvexOrdering
Theorems to Some Functional Inequalities for Convex
Functions:ASurvey........................................................ 231
TeresaRajba
12 On the Construction of the Field of Reals by Means of
FunctionalEquationsandTheirStabilityandRelatedTopics........ 275
JensSchwaiger
13 GeneralizedDhombresFunctionalEquation............................ 297
JaroslavSmítalandMartaŠtefánková
14 FunctionalEquationsandStabilityProblemsonHypergroups....... 305
LászlóSzékelyhidi
15 Stability of Systems of General Functional Equations
intheCompact-OpenTopology........................................... 333
PavolZlatoš
List of Contributors
Shoshana Abramovich Department of Mathematics, University of Haifa, Haifa,
Israel
PekkaAlestalo DepartmentofMathematicsandSystemsAnalysis,AaltoUniver-
sity,Aalto,Finland
AnnaBahyrycz FacultyofAppliedMathematics,AGHUniversityofScienceand
Technology,Mickiewicza,Kraków,Poland
Harald Fripertinger Institut für Mathematik und wissenschaftliches Rechnen,
NAWI-Graz,Karl-Franzens-UniversitätGraz,Graz,Austria
Roman Ger Instituteof Mathematics, University of Silesia, Bankowa, Katowice,
Poland
Eliza Jabłon´ska Department of Discrete Mathematics, Rzeszów University of
Technology,Rzeszów,Poland
ZenonMoszner InstituteofMathematics,PedagogicalUniversity,Podchora¸z˙ych,
Kraków,Poland
AdamJ.Ostaszewski DepartmentofMathematics,LondonSchoolofEconomics,
London,UK
Barbara Przebieracz Institute of Mathematics, University of Silesia, Bankowa,
Katowice,Poland
TeresaRajba DepartmentofMathematics,UniversityofBielsko-Biala,Willowa,
Bielsko-Biala,Poland
Ludwig Reich Institut für Mathematik und wissenschaftliches Rechnen, NAWI-
Graz,Karl-Franzens-UniversitätGraz,Graz,Austria
MaciejSablik InstituteofMathematics,UniversityofSilesia,Bankowa,Katowice,
Poland
xi
