Springer Optimization and Its Applications VOLUME52 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 Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques havebeendeveloped,thediffusionintootherdisciplineshasproceededata rapidpace,andourknowledgeofallaspectsofthefieldhasgrownevenmore profound.Atthe sametime, oneofthe moststriking trendsin optimization is the constantly increasing emphasis on the interdisciplinary nature of the field.Optimizationhasbeenabasictoolinallareasofappliedmathematics, engineering,medicine,economicsandothersciences. The series Springer Optimization and Its Applications publishes under- graduate and graduate textbooks, monographs and state-of-the-art exposi- tory works that focus on algorithms for solving optimization problems and alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objectiveprogramming,descriptionofsoftwarepackages,approxima- tiontechniquesandheuristicapproaches. Forfurthervolumes: http://www.springer.com/series/7393 Themistocles M. Rassias • Janusz Brzde¸k Editors Functional Equations in Mathematical Analysis 123 Editors ThemistoclesM.Rassias JanuszBrzde¸k DepartmentofMathematics DepartmentofMathematics NationalTechnicalUniversityofAthens PedagogicalUniversity ZografouCampus 30084Krakow 15780Athens Poland Greece [email protected] [email protected] ISSN1931-6828 ISBN978-1-4614-0054-7 e-ISBN978-1-4614-0055-4 DOI10.1007/978-1-4614-0055-4 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2011935375 MathematicsSubjectClassification(2010):39-XX,46-XX,33-XX ©SpringerScience+BusinessMedia,LLC2012 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Dedicatedtothememoryof StanisławMarcin Ulam(1909–1984) ontheoccasionofthe100thanniversary ofhisbirth Preface Thevolumeconsistsofarticleswrittenbyeminentscientistsfromtheinternational mathematical community, who present important research works in the field of MathematicalAnalysisand relatedsubjects, in particular,in FunctionalEquations andInequalities.Theseworksprovideaninsightinaprogressinthestudyofvarious problemsofnonlinearcharacter. Several of these results have been influenced by the work of the well-known mathematician and physicist Stanisław Marcin Ulam (April 3, 1909 to May 13, 1984). An emphasis is given to one of his questions concerning approximate homomorphisms. The book is dedicated to the memory of Ulam on the occasion of the 100th anniversaryofhisbirth. Itisdividedintotwoparts.PartIfocusesonseveralaspectsoftheUlamstability theory.PartIIcontainspapersonvarioussubjectsofMathematicalAnalysis. Itis a pleasureto expressourdeepestthankstoall ofthe mathematicianswho, throughtheirworks,participatedinthisvolume. We would also wish to acknowledge the superb assistance that the staff of Springerhasprovidedinthepreparationofthispublication. AthensandCracow ThemistoclesM.Rassias JanuszBrzde¸k vii Contents PartI StabilityinMathematicalAnalysis 1 StabilityPropertiesofSomeFunctionalEquations..................... 3 RomanBadora 2 NoteonSuperstabilityofMikusin´ski’sFunctionalEquation.......... 15 BogdanBatko 3 A General Fixed Point Method for the Stability ofCauchyFunctionalEquation........................................... 19 LiviuCa˘dariuandViorelRadu 4 OrthogonalityPreservingPropertyanditsUlamStability............ 33 JacekChmielin´ski 5 OntheHyers–UlamStabilityofFunctionalEquations withRespecttoBoundedDistributions .................................. 59 Jae-YoungChung 6 StabilityofMulti-JensenMappingsinNon-Archimedean NormedSpaces.............................................................. 79 KrzysztofCieplin´ski 7 OnStabilityoftheEquationofHomogeneousFunctions onTopologicalSpaces...................................................... 87 StefanCzerwik 8 Hyers–UlamStabilityoftheQuadraticFunctionalEquation......... 97 ElhoucienElqorachi,YoussefManar, andThemistoclesM.Rassias 9 IntuitionisticFuzzyApproximatelyAdditiveMappings ............... 107 M.Eshaghi-Gordji,H.Khodaei,H.Baghani, andM.Ramezani ix