Handbook of functional equations. Stability theory PDF
Preview Handbook of functional equations. Stability theory
Springer Optimization and Its Applications 96 Themistocles M. Rassias Editor Handbook of Functional Equations Stability Theory Springer Optimization and Its Applications Volume 96 ManagingEditor PanosM.Pardalos(UniversityofFlorida) Editor–CombinatorialOptimization Ding-ZhuDu(UniversityofTexasatDallas) AdvisoryBoard J.Birge(UniversityofChicago) C.A.Floudas(PrincetonUniversity) F.Giannessi(UniversityofPisa) H.D.Sherali(VirginiaPolytechnicandStateUniversity) T.Terlaky(McMasterUniversity) Y.Ye(StanfordUniversity) AimsandScope Optimizationhasbeenexpandinginalldirectionsatanastonishingrateduringthe lastfewdecades.Newalgorithmicandtheoreticaltechniqueshavebeendeveloped, thediffusionintootherdisciplineshasproceededatarapidpace,andourknowledge ofallaspectsofthefieldhasgrownevenmoreprofound.Atthesametime,oneof themoststrikingtrendsinoptimizationistheconstantlyincreasingemphasisonthe interdisciplinarynatureofthefield. Optimizationhasbeenabasictoolinallareas ofappliedmathematics,engineering,medicine,economics,andothersciences. The series Springer Optimization and ItsApplications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that fo- cus on algorithms for solving optimization problems and also study applications involvingsuchproblems.Someofthetopicscoveredincludenonlinearoptimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control,discreteoptimization,multi-objectiveprogramming,descriptionofsoftware packages,approximationtechniquesandheuristicapproaches. Moreinformationaboutthisseriesathttp://www.springer.com/series/7393 Themistocles M. Rassias Editor Handbook of Functional Equations Stability Theory 2123 Editor ThemistoclesM.Rassias DepartmentofMathematics NationalTechnicalUniversityofAthens Athens,Greece ISSN1931-6828 ISSN1931-6836(electronic) ISBN978-1-4939-1285-8 ISBN978-1-4939-1286-5(eBook) DOI10.1007/978-1-4939-1286-5 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2014952594 Mathematics Subject Classification (2010): 39B05, 39B22, 39B52, 39B62, 39B82, 40A05, 41A30, 54C60,54C65. © SpringerScience+BusinessMedia,LLC2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface HandbookofFunctionalEquations:StabilityTheoryconsistsof17chapterswritten byeminentscientistsfromtheinternationalmathematicalcommunity,whopresent importantresearchworksinthefieldofmathematicalanalysisandrelatedsubjects, particularlyintheUlamstabilitytheoryoffunctionalequations.Theseworkspro- vide an insight in a large domain of research with emphasis to the discussion of severaltheories,methodsandproblemsinapproximationtheory,influencedbythe seminalworkofthewell-knownmathematicianandphysicistStanislawUlam(1909– 1984).Emphasisisgiventooneofhisfundamentalproblemsconcerningapproximate homomorphisms. The chapters of this book focus mainly on both old and recent developments ontheequationofhomomorphismforsquaresymmetricgroupoids, thelinearand polynomialfunctionalequationsinasinglevariable,theDrygasfunctionalequation on amenable semigroups, monomial functional equation, the Cauchy–Jensen type mappings, differential equations and differential operators, operational equations andinclusions,generalizedmodulelefthigherderivations,selectionsofset-valued mappings,D’Alembert’sfunctionalequation,characterizationsofinformationmea- sures, functionalequationsinrestricteddomains, aswellasgeneralizedfunctional stabilityandfixedpointtheory.Itisapleasuretoexpressourdeepestthankstoallthe mathematicianswho, throughtheirworks, participatedinthispublication. Iwould like to thank Dr. Michael Batsyn for his invaluable help during the preparation of thisbook. Iwouldalsowishtoacknowledgethesuperbassistancethatthestaffof Springerhasprovidedforthepublicationofthiswork. Athens,Greece ThemistoclesM.Rassias v Contents OnSomeFunctionalEquations ..................................... 1 MarcinAdam,StefanCzerwikandKrzysztofKról RemarksonStabilityoftheEquationofHomomorphismforSquare SymmetricGroupoids ............................................. 37 AnnaBahyryczandJanuszBrzde¸k On Stability of the Linear and Polynomial Functional Equations inSingleVariable.................................................. 59 JanuszBrzde¸kandMagdalenaPiszczek Selections of Set-valued Maps Satisfying Some Inclusions andtheHyers–UlamStability....................................... 83 JanuszBrzde¸kandMagdalenaPiszczek GeneralizedUlam–HyersStabilityResults:AFixedPointApproach .... 101 LiviuCa˘dariu On a Weak Version of Hyers–Ulam Stability Theorem inRestrictedDomains ............................................. 113 JaeyoungChungandJeongwookChang On the Stability of Drygas Functional Equation on Amenable Semigroups....................................................... 135 ElhoucienElqorachi,YoussefManarandThemistoclesM.Rassias Stability of Quadratic and Drygas Functional Equations, withanApplicationforSolvinganAlternativeQuadraticEquation ..... 155 GianLuigiForti AFunctionalEquationHavingMonomialsandItsStability ............ 181 M.E.Gordji,H.KhodaeiandThemistoclesM.Rassias vii viii Contents Some Functional Equations Related to the Characterizations ofInformationMeasuresandTheirStability ......................... 199 EszterGselmannandGyulaMaksa Approximate Cauchy–Jensen Type Mappings in Quasi-β-Normed Spaces ........................................................... 243 Hark-MahnKim,Kil-WoungJunandEunyoungSon AnAQCQ-FunctionalEquationinMatrixParanormedSpaces ......... 255 JungRyeLee,ChoonkilPark,ThemistoclesM.Rassias andDongYunShin On the Generalized Hyers–Ulam Stability of the Pexider Equation onRestrictedDomains ............................................. 279 YoussefManar,ElhoucienElqorachiandThemistoclesM.Rassias Hyers-Ulam Stability of Some Differential Equations andDifferentialOperators ......................................... 301 DorianPopaandIoanRas¸a Results and Problems in Ulam Stability of Operatorial Equations andInclusions .................................................... 323 IoanA.Rus Superstability of Generalized Module Left Higher Derivations onaMulti-BanachModule ......................................... 353 T.L.ShateriandZ.Afshari D’Alembert’s Functional Equation and Superstability Problem inHypergroups ................................................... 367 D.Zeglami,A.RoukbiandThemistoclesM.Rassias Contributors MarcinAdam Institute of Mathematics, Silesian University of Technology, Gli- wice,Poland Z.Afshari DepartmentofMathematicsandComputerSciences,Sabzevar,Iran Anna Bahyrycz Department of Mathematics, Pedagogical University, Kraków, Poland Janusz Brzde¸k Department of Mathematics, Pedagogical University, Kraków, Poland LiviuCa˘dariu DepartmentofMathematics, PolitehnicaUniversityofTimis¸oara, Timis¸oara,Romania Jeongwook Chang Department of Mathematics Education, Dankook University, Yongin,RepublicofKorea JaeyoungChung DepartmentofMathematics,KunsanNationalUniversity,Kun- san,RepublicofKorea Stefan Czerwik Institute of Mathematics, Silesian University of Technology, Gliwice,Poland ElhoucienElqorachi DepartmentofMathematics,FacultyofSciences,University IbnZohr,Agadir,Morocco Gian Luigi Forti Dipartimento di Matematica, Università degli Studi di Milano, Milano,Italy M.E.Gordji DepartmentofMathematics,SemnanUniversity,Semnan,Iran EszterGselmann DepartmentofAnalysis,InstituteofMathematics,Universityof Debrecen,Debrecen,Hungary Kil-Woung Jun Department of Mathematics, Chungnam National University, Daejeon,Korea H.Khodaei DepartmentofMathematics,MalayerUniversity,Malayer,Iran ix x Contributors Hark-Mahn Kim Department of Mathematics, Chungnam National University, Daejeon,Korea Krzysztof Król Institute of Mathematics, Silesian University of Technology, Gliwice,Poland JungRyeLee DepartmentofMathematics,DaejinUniversity,Daejin,Korea Gyula Maksa Department of Analysis, Institute of Mathematics, University of Debrecen,Debrecen,Hungary Youssef Manar Department of Mathematics, Faculty of Sciences, University Ibn Zohr,Agadir,Morocco Choonkil Park Research Institute for Natural Sciences, Hanyang University, Hanyang,Korea Magdalena Piszczek Department of Mathematics, Pedagogical University, Kraków,Poland Dorian Popa Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca,Romania IoanRas¸a TechnicalUniversityofCluj-Napoca,DepartmentofMathematics,Cluj- Napoca,Romania ThemistoclesM.Rassias DepartmentofMathematics,NationalTechnicalUniver- sityofAthens,ZografouCampus,Athens,Greece A.Roukbi DepartmentofMathematics,IbnTofailUniversity,Kenitra,Morocco IoanA.Rus DepartmentofMathematics,Babes¸-BolyaiUniversity,Cluj-Napoca, Romania T.L.Shateri DepartmentofMathematicsandComputerSciences,Sabzevar,Iran DongYunShin DepartmentofMathematics,UniversityofSeoul,Seoul,Korea Eunyoung Son Department of Mathematics, Chungnam National University, Daejeon,Korea D. Zeglami Department of Mathematics, E.N.S.A.M., Moulay Ismail University, AlMansour,Meknes,Morocco On Some Functional Equations MarcinAdam,StefanCzerwikandKrzysztofKról SubjectClassifications:39B22,39B52,39B82,40A05,41A30. Abstract Thischapterconsistsofthreeparts.Inthefirstpartweconsiderso-called Adomian’spolynomialsandpresenttheproofoftheconvergenceofthesequenceof suchpolynomialstothesolutionoftheequation.Thesecondpartisdevotedtopresent several approximation methods for finding solutions of so-called Kordylewski– Kuczma functional equation. Finally, in the last one we present a stability result inthesenseofUlam–Hyers–Rassiasforgeneralizedquadraticfunctionalequation ontopologicalspaces. Keywords Adomian’spolynomials·Decompositionmethod·ConvergenceofAdo- mian’s iterations · Approximate solutions of functional equations · Generalized quadraticfunctionalequation·Stability 1 OntheConvergenceofAdomian’sMethod 1.1 Introduction G.Adomianinseveralpapers(seee.g.[6–9])developedanumericaltechniqueusing specialkindsofpolynomials(calledAdomianpolynomials)forsolvingnon-linear functionalequations. Inthismethodthesolutionisgivenbyaserieshavingterms whichareAdomian’spolynomials.Unfortunately,theproblemsofconvergenceare notsatisfactorilysolved. M.Adam((cid:2))·S.Czerwik·K.Król InstituteofMathematics,SilesianUniversityofTechnology, Kaszubska23,44-100Gliwice,Poland e-mail:[email protected] S.Czerwik e-mail:[email protected] K.Król e-mail:[email protected] ©SpringerScience+BusinessMedia,LLC2014 1 T.M.Rassias(ed.),HandbookofFunctionalEquations, SpringerOptimizationandItsApplications96,DOI10.1007/978-1-4939-1286-5_1
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