Nonlinear Analysis: Stability, Approximation, and Inequalities PDF
Preview Nonlinear Analysis: Stability, Approximation, and Inequalities
Springer Optimization and Its Applications Springer Optimization and Its Applications VOLUME68 ManagingEditor PanosM.Pardalos(UniversityofFlorida,Gainesville,USA) Editor-CombinatorialOptimization Ding-ZhuDu(UniversityofTexasatDallas,Richardson,USA) AdvisoryBoard J.Birge(UniversityofChicago,Chicago,IL,USA) C.A.Floudas(PrincetonUniversity,Princeton,NJ,USA) F.Giannessi(UniversityofPisa,Pisa,Italy) H.D.Sherali(VirginiaTech,Blacksburg,VA,USA) T.Terlaky(LehighUniversity,Bethlehem,PA,USA) Y.Ye(StanfordUniversity,Stanford,CA,USA) AimsandScope Optimizationhasbeenexpandinginalldirectionsatanastonishingratedur- ing the last few decades. New algorithmic and theoretical techniques have beendeveloped,thediffusionintootherdisciplineshasproceededatarapid pace,andourknowledgeofallaspectsofthefieldhasgrownevenmorepro- found. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field.Optimizationhasbeenabasictoolinallareasofappliedmathematics, engineering,medicine,economics,andothersciences. The series Springer Optimization and Its Applications publishes under- graduate and graduate textbooks, monographs and state-of-the-art exposi- tory work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics cov- ered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objectiveprogramming,descriptionofsoftwarepackages,approxima- tiontechniquesandheuristicapproaches. Forfurthervolumes: www.springer.com/series/7393 Panos M. Pardalos (cid:2) Pando G. Georgiev (cid:2) Hari M. Srivastava Editors Nonlinear Analysis Stability, Approximation, and Inequalities In honor of Themistocles M. Rassias on the occasion of his 60th birthday Editors PanosM.Pardalos PandoG.Georgiev CenterforAppliedOptimization CenterforAppliedOptimization ISEDepartment UniversityofFlorida UniversityofFlorida Gainesville,FL Gainesville,FL USA USA HariM.Srivastava and DepartmentofMathematics&Statistics LaboratoryofAlgorithmsandTechnologies UniversityofVictoria forNetworksAnalysis(LATNA) Victoria,BritishColumbia NationalResearchUniversity Canada HigherSchoolofEconomics Moscow Russia ISSN1931-6828 SpringerOptimizationandItsApplications ISBN978-1-4614-3497-9 ISBN978-1-4614-3498-6(eBook) DOI10.1007/978-1-4614-3498-6 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012940728 ©SpringerScience+BusinessMedia,LLC2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and 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) With ourdeepestappreciation, wededicatethisvolume totheeminentmathematician ThemistoclesM.Rassias on theoccasion ofhis60thbirthday ThemistoclesM.Rassias Preface This volume is dedicated to Themistocles M. Rassias, on the occasion of his 60th birthday.Thearticlespublishedherepresentsomerecentdevelopmentsandsurveys inNonlinearAnalysisrelatedtothemathematicaltheoriesofstability,approxima- tion,andinequalities. ThemistoclesM.Rassias was borninPellananearSpartainGreecein theyear 1951.HeiscurrentlyaProfessorintheDepartmentofMathematicsattheNational Technical University of Athens. He received his Ph.D. in Mathematics in the year 1976fromtheUniversityofCaliforniaatBerkeleywithStephenSmaleashisthesis advisor. Rassias’workextendsoverseveralfieldsofMathematicalAnalysis.Itincludes Global Analysis, Calculus of Variations, Nonlinear Functional Analysis, Approxi- mationTheory,FunctionalEquations,andMathematicalInequalitiesandtheirAp- plications. Rassias’ work has been embraced by several mathematicians interna- tionallyandsomeofhisresearchhasbeenestablishedwiththescientificterminol- ogy “Hyers–Ulam–Rassias stability”, “Cauchy–Rassias stability”, “Aleksandrov– Rassiasproblemforisometricmappings”. Thestabilitytheoryoffunctionalequationshasitsrootsprimarilyintheinvesti- gationsbyS.M.Ulam,whoposedthefundamentalproblemforapproximatehomo- morphismsintheyear1940,thestabilitytheoremfortheadditivemappingdueto D.H.Hyers(1941),andthestabilitytheoremforthelinearmappingofTh.M.Ras- sias (1978). Much of the modern stability of functional equations has been influ- encedbytheseminalpaperofTh.M.Rassias,entitled“Onthestabilityofthelinear mappinginBanachspaces”[ProceedingsoftheAmericanMathematicalSociety72, 297–300(1978)],whichhasprovidedatheoreticalbreakthrough.Foranextensive discussionofvariousadvancesinstabilitytheoryoffunctionalequations,thereader isreferredtotherecentlypublishedbookofS.-M.Jung,“Hyers–Ulam–RassiasSta- bilityofFunctionalEquationsinNonlinearAnalysis”,©Springer,NewYork,2011. In the formulation as well as in the solution of stability problems of functional equations, one frequently encounters the interplay of Mathematical Analysis, Ge- ometry,Algebra,andTopology.Rassiashascontributedalsotoothersubjectssuch as Minimal Surfaces (Plateau problem), Isometric Mappings (Aleksandrov prob- ix x Preface lem),ComplexAnalysis(PoincaréinequalityandMöbiustransformations),andAp- proximationtheory(Extremalproblems).Hehaspublishedmorethan230scientific researchpapers,6researchbooksandmonographs,and30editedvolumesoncur- rent research topics in Mathematics. He has also published 4 textbooks in Mathe- maticsforGreekuniversitystudents. Some of the honors and positions that he has received include “Membership” at the School of Mathematics of the Institute for Advanced Study at Princeton for the academic years 1977–1978 and 1978–1979 (which he did not accept for fam- ily reasons); “Research Associate” at the Department of Mathematics of Harvard University (1980) invited by Raoul Bott, “Visiting Research Professor” at the De- partment of Mathematics of the Massachusetts Institute of Technology (1980) in- vited by F.P. Peterson; “Accademico Ordinario” of the Accedemia Tiberina Roma (since1987);“Fellow”oftheRoyalAstronomicalSocietyofLondon(since1991); “Teacheroftheyear”(1985–1986and1986–1987)and“OutstandingFacultyMem- ber”(1989–1990,1990–1991,and1991–1992)oftheUniversityofLaVerne,Cal- ifornia (Athens Campus); “Ulam Prize in Mathematics” (2010). In addition to the above,during the last few years, Th.M. Rassias had been bestowed with honorary degrees “Doctor Honoris Causa” from the University of Alba Iulia in Romania (2008)andan“HonoraryDoctorate”fromtheUniversityofNišinSerbia(2010).In 2003,avolumeentitled“StabilityofFunctionalEquationsofUlam–Hyers–Rassias Type” was dedicated to the 25 years since the publication of Th. M. Rassias’ sta- bilitytheorem(editedbyS.Czerwik,Florida,USA).In2009,aspecialissueofthe Journal of Nonlinear Functional Analysis and Applications (Vol. 14, No. 5) was dedicated to the 30th Anniversary of Th.M. Rassias’ stability theorem. In 2007, a special volume of the Banach Journal of Mathematical Analysis (Vol. 1, Issues 1 & 2) was dedicated to the 30th Anniversary of Th.M. Rassias’ stability theorem. He is an “editor” or “advisory editor” of several international mathematical jour- nals published in the USA, Europe, and Asia. He has delivered lectures at several universitiesinNorthAmericaandEurope,includingHarvardUniversity,MIT,Yale University,PrincetonUniversity,StanfordUniversity,UniversityofMichigan,Uni- versityofMontréal,ImperialCollegeLondon,Technion—IsraelInstituteofTech- nology(Haifa),TechnischeUniversitätBerlin,andtheUniversitätGöttingen. Thecontributedpapersinthepresentvolumehighlightsomeofthemostrecent achievementsthathavebeenmadeinMathematicalAnalysis. Rassias’curiosity,enthusiasmaswellashispassionfordoingresearchaswellas teachingareunlimited.HehasservedasamentorinMathematicstoseveralstudents atuniversitieswherehehastaught. His research work has received up-to-date more than 7,000 citations (see, e.g., theGoogleScholar).Thatisanimpressivenumberofcitationsforamathematician. Thus,Rassiashasachievedinternationaldistinctioninthebroadestsense. ThereaderisreferredtothearticleofPerEnfloandM.SalMoslehian,Aninter- view with Themistocles M. Rassias, Banach Journal of Mathematical Analysis 1, 252–260(2007)[seealsowww.math.ntua.gr/~trassias/]. Inwhatfollows,wepresentabriefoutlineofthecontributedpapersinthisvol- ume,whicharecollectedinanalphabeticalorderofthecontributors. Preface xi In Chap.1, S. Abramovich deals with Jensen’s type inequality, its bounds and refinements,andwitheigenvaluesoftheSturm–Liouvillesystem. In Chap.2, M. Adam and S. Czerwik consider some quadraticdifference oper- ators(e.g.,Lobaczewskidifferenceoperators)andquadratic-lineardifferenceoper- ators (e.g., d’Alembert difference operators and quadratic difference operators) in some special function spaces. They prove a stability result in the sense of Ulam– Hyers–Rassias for the quadratic functional equation in a special class of differen- tiablefunctions. InChap.3,C.Affane-AjiandN.K.Govilpresentastudyconcerningthelocation ofthezerosofapolynomialstartingfromtheresultsofGaussandCauchytosome ofthemostrecentinvestigationonthetopic. Chapter4 by D. Andrica and V. Bulgarean is devoted to isometry groups Iso (Rn) for p(cid:2)1, p(cid:2)=2 and p=∞, where the metric d is appropriately de- dp p fined. In Chap.5, I. Biswas, M. Logares, and V. Muñoz prove that the moduli spaces M (r,Λ)are,inmanycases,rational.Herethemodulispacesaredefinedbyusing τ aconceptofτ-stablepairsofrankr andfixeddeterminantΛ. In Chap.6, D. Breaz, Y. Polatog˜lu, and N. Breaz investigate a subclass of gen- eralized p-valent Janowski type convex functions and its application to harmonic mappings. InChap.7,J.Brzde¸k,D.Popa,andB.Xupresentsomeobservationsconcerning stabilityofthefollowinglinearfunctionalequation: (cid:2) (cid:3) (cid:4)m (cid:2) (cid:3) ϕ fm(x) = a (x)ϕ fm−i(x) +F(x) i i=1 intheclassoffunctions ϕ mappinganonemptyset S intoaBanachspace X over afieldK∈{R,C},wheremisafixedpositiveintegerandthefunctionsf :S→S, F :S→Xanda :S→K(i=1,...,m)aregiven. i In Chap.8, M.J. Cantero and A. Iserles examine the limiting behavior of solu- tionstoaninfinitesetofrecursionsinvolvingq-factorialtermsasq→1. In Chap.9, E.A. Chávezand P.K. Sahoodeterminethe general solutionsof the followingfunctionalequations: f (x+y)+f (x+σy)=f (x) and 1 2 3 (cid:2) (cid:3) f (x+y)+f (x+σy)=f (x)+f (y), x,y∈Sn , 1 2 3 4 where f ,f ,f ,f :Sn →G are unknown functions, S is a commutative semi- 1 2 3 4 group, σ :S →S is an endomorphism of order 2, G is a 2-cancellative abelian group,andnisapositiveinteger. InChap.10,W.-S.Cheung,G.Leng,J.Pecˇaric´,andD.Zhaopresentrecentde- velopmentsofBohr-typeinequalities. InChap.11,S.DingandY.Xingestablishsomebasicnorminequalities,includ- ing the Poincaré inequality, weak reverse Hölder inequality, and the Caccioppoli xii Preface inequality,forconjugateharmonicforms.TheyalsoprovetheCaccioppoliinequal- itywiththeOrlicznormforconjugateharmonicforms. InChap.12,S.S.Dragomirpresentsasurveyaboutsomerecentinequalitiesre- latedtothecelebratedJensen’sresultforpositivelinearorsublinearfunctionalsand convexfunctions. In Chap.13, A. Ebadian and N. Ghobadipour prove the generalized Hyers– ∗ Ulam–Rassias stability of bi-quadratic bi-homomorphisms in C -ternary algebras andquasi-Banachalgebras. InChap.14,E.ElhoucienandM.Youssefapplyafixedpointtheoremtoprove theHyers–Ulam–Rassiasstabilityofthefollowingquadraticfunctionalequation: (cid:2) (cid:3) f(kx+y)+f kx+σ(y) =2k2f(x)+2f(y). InChap.15,M.Fujii,M.SalMoslehian,andJ.Mic´ic´ surveyseveralsignificant results on the Bohr inequality and present its generalizations involving some new approaches. In Chap.16, P. Ga˘vru¸ta and L. Ga˘vru¸ta provide an introduction to the Hyers– Ulam–Rassiasstabilityoforthogonallyadditivemappings. InChap.17,M.EshaghiGordji,N.Ghobadipour,A.Ebadian,M.BavandSavad- kouhi, and C. Park investigate ternary Jordan homomorphisms on Banach ternary algebrasassociatedwiththefollowingfunctionalequation: (cid:5) (cid:6) x 1 f 1 +x +x = f(x )+f(x )+f(x ). 2 3 1 2 3 2 2 In Chap.18, F. Habibian, R. Bolghanabadi, and M. Eshaghi Gordji investigate theHyers–Ulam–Rassiasstabilityofcubicn-derivationsfromnon-archimedeanBa- nachalgebrasintonon-archimedeanBanachmodules. In Chap.19, S.-S. Jin and Y.-H. Lee investigate a fuzzy version of stability for thefollowingfunctionalequation: 2f(x+y)+f(x−y)+f(y−x)−f(2x)−f(2y)=0 inthesenseofM.MirmostafaeeandM.S.Moslehian. InChap.20,K.-W.Jun,H.-M.Kim,andE.-Y.SonprovethegeneralizedHyers– UlamstabilityofthefollowingCauchy–Jensenfunctionalequation: (cid:5) (cid:6) x+y f(x)+f(y)+nf(z)=nf +z , n inann-divisibleabeliangroupGforanyfixedpositiveintegern(cid:2)2. In Chap.21, S.-M. Jung applies the fixed point method for proving the Hyers– Ulam–Rassiasstabilityofthegammafunctionalequation. In Chap.22, H.A. Kenary proves the generalized Hyers–Ulam stability in ran- dom normed spaces of the following additive-quadratic-cubic-quartic functional equation:
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