×

Hyers-Ulam stability of functional equations in several variables. (English) Zbl 0836.39007

This is a survey about Hyers-Ulam stability of functional equations and systems in several variables. Except for the historical background the paper contains results on the stability of Cauchy, Jensen, quadratic and polynomial equations. The author presents superstability results concerning the multiplicative Cauchy equation and some cosine and sine functional equations. Further, approximately multiplicative linear maps in Banach algebras are considered. Moreover, the stability and the superstability of other equations and systems are discussed. Some final remarks and open problems end the paper. The bibliography contains 120 items.

MSC:

39B62 Functional inequalities, including subadditivity, convexity, etc.
39B32 Functional equations for complex functions
39B22 Functional equations for real functions
39-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to difference and functional equations
39-03 History of difference and functional equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aczél, J.,7. Problem. InReport of the twenty-first International Symposium on Functional Equations. Aequationes Math.26 (1984), 258.
[2] Albert, M. andBaker, J. A.,Functions with bounded n-th differences. Ann. Polon. Math.43 (1983), 93–103.
[3] Alsina, C.,On the stability of a functional equation related to associativity. Ann. Polon. Math.53 (1991), 1–5. · Zbl 0724.39009
[4] Badora, R.,On some generalized invariant means and their application to the stability of the Hyers–Ulam type, I. Manuscript. · Zbl 0787.43003
[5] Baker, J. A.,The stability of the cosine equation. Proc. Amer. Math. Soc.80 (1980), 411–416. · Zbl 0448.39003 · doi:10.1090/S0002-9939-1980-0580995-3
[6] Baker, J., Lawrence, J. andZorzitto, F.,The stability of the equation f(x + y) = f(x)f(y). Proc. Amer. Math. Soc.74 (1979), 242–246. · Zbl 0397.39010
[7] Baster, J., Moszner, Z. andTabor, J.,On the stability of some class of functional equations. [Rocznik Naukovo-Dydaktycziny, WSP w Krakowie, Zeszyt 97, Prace Matematyczne XI]. Cracow, 1985.
[8] Bean, M. andBaker, J. A.,The stability of a functional analogue of the wave equation. Canad. Math. Bull.33 (1990), 376–385. · Zbl 0663.39008 · doi:10.4153/CMB-1990-062-3
[9] Borelli, C.,On Hyers–Ulam stability of Hosszú’s functional equation. Resultate Math.26 (1994), 221–224. · Zbl 0828.39019
[10] Borelli, C. andForti, G. L.,On a general Hyers–Ulam stability result. Internat. J. Math. Math. Sci.18 (1995), 229–236. · Zbl 0826.39009 · doi:10.1155/S0161171295000287
[11] Bourgin, D. G.,Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J.16 (1949), 385–397. · Zbl 0033.37702 · doi:10.1215/S0012-7094-49-01639-7
[12] Bourgin, D. G.,Classes of transformations and bordering transformations. Bull. Amer. Math. Soc.57 (1951), 223–237. · Zbl 0043.32902 · doi:10.1090/S0002-9904-1951-09511-7
[13] Burkill, J. C.,Polynomial approximations to functions with bounded differences. J. London Math. Soc.33 (1958), 157–161. · Zbl 0081.28302 · doi:10.1112/jlms/s1-33.2.157
[14] Cenzer, D.,The stability problem for transformations of the circle. Proc. Roy. Soc. Edinburgh A84 (1979), 279–281. · Zbl 0439.39004
[15] Chmielinski, J. andTabor, J.,On approximate solutions of the Pexider equation. Aequationes Math.46 (1993), 143–163. · Zbl 0801.39007 · doi:10.1007/BF01834004
[16] Cholewa, P. W.,The stability of the sine equation. Proc. Amer. Math. Soc.88 (1983), 631–634. · Zbl 0547.39003 · doi:10.1090/S0002-9939-1983-0702289-8
[17] Cholewa, P. W.,Remarks on the stability of functional equations. Aequationes Math.27 (1984), 76–86. · Zbl 0549.39006 · doi:10.1007/BF02192660
[18] Cholewa, P. W.,The stability problem for a generalized Cauchy type functional equation. Rev. Roumaine Math. Pures Appl.29 (1984), 457–460. · Zbl 0549.39005
[19] Cholewa, P. W.,Almost approximately polynomial functions. InNonlinear analysis. World Sci. Publishing, Singapore, 1987, pp. 127–136.
[20] Czerwik, St.,On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg62 (1992), 59–64. · Zbl 0779.39003 · doi:10.1007/BF02941618
[21] de la Harpe, P. andKaroubi, M.,Représentations approchées d’un groupe dans une algèbre de Banach. Manuscripta Math.22 (1977), 293–310. · Zbl 0371.22007 · doi:10.1007/BF01172669
[22] Dicks, D.,2. Remark. InReport of the twenty-seventh International Symposium on Functional Equations. Aequationes Math.39 (1990), 310.
[23] Djoković, D. Ž.,A representation theorem for (X 1 1)(X 2 1)X n 1)and its applications. Ann. Pol. Math.22 (1969), 189–198. · Zbl 0187.39903
[24] Drljević, H.,On the stability of the functional quadratic on A-orthogonal vectors. Publ. Inst. Math. (Beograd) (N.S.)36 (50) (1984), 111–118. · Zbl 0598.65029
[25] Drljević, H.,On the stability of a functional which is approximately additive or approximately quadratic on A-orthogonal vectors. InDifferential geometry, calculus of variations, and their applications. [Lecture Notes in Pure and Appl. Math., Vol. 100]. Dekker, New York, 1985, pp. 511–513.
[26] Drljević, H.,The stability of the weakly additive functional. InNonlinear analysis. World Sci. Publishing, Singapore, 1987, pp. 287–306. · Zbl 0676.46025
[27] Drljević, H.,On the representation of functionals and the stability of mappings in Hilbert and Banach spaces. InTopics in mathematical analysis. Ser. Pure Math. II, World Sci. Publishing, Singapore, 1989, pp. 231–245. · Zbl 0752.47014
[28] Drljević, H. andMavar, Z.,About the stability of a functional approximately additive on A-orthogonal vectors. Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka (1982), No. 20, 155–172.
[29] Fenyö, I.,Osservazioni su alcuni teoremi di D. H. Hyers. Istit. Lombardo Accad. Sci. Lett. Rend. A114 (1980), 235–242 (1982).
[30] Fenyö, I.,On an inequality of P. W. Cholewa. InGeneral inequalities, 5. [Internat. Schriftenreihe Numer. Math., Vol. 80]. Birkhäuser, Basel–Boston, MA, 1987, pp. 277–280.
[31] Förg-Rob, W. andSchwaiger, J.,Stability and superstability of the addition and subtraction formulae for trigonometric and hyperbolic functions. InContributions to the Theory of Functional Equations, Proceedings of the Seminar Debrecen-Graz, 1991, Grazer Math. Ber.315 (1991), 35–44.
[32] Förg-Rob, W. andSchwaiger, J.,On the stability of a system of functional equations characterizing generalized hyperbolic and trigonometric functions. Aequationes Math.45 (1993), 285–296. · Zbl 0774.39007 · doi:10.1007/BF01855886
[33] Forti, G. L.,An existence and stability theorem for a class of functional equations. Stochastica4 (1980), 23–30. · Zbl 0442.39005
[34] Forti, G. L.,The stability of homomorphisms and amenability, with applications to functional equations. Abh. Math. Sem. Univ. Hamburg57 (1987), 215–226. · Zbl 0619.39012 · doi:10.1007/BF02941612
[35] Forti, G. L.,Sulla stabilità degli omomorfismi e sue applicazioni alle equazioni funzionali. Rend. Sem. Mat. Fis. Milano58 (1988), 9–25 (1990). · Zbl 0712.39024 · doi:10.1007/BF02925228
[36] Forti, G. L.,18. Remark. InReport of the twenty-seventh International Symposium on Functional Equations. Aequationes Math.39 (1990), 309–310.
[37] Forti, G. L. andSchwaiger, J.,Stability of homomorphisms and completeness. C. R. Math. Rep. Acad. Sci. Canada11 (1989), 215–220. · Zbl 0697.39013
[38] Fréchet, M.,Les polynomes abstraits. Journal de Mathématiques8 (1929), 71–92. · JFM 55.0242.03
[39] Gajda, Z.,On stability of the Cauchy equation on semigroups. Aequationes Math.36 (1988), 76–79. · Zbl 0658.39006 · doi:10.1007/BF01837972
[40] Gajda, Z.,Local stability of the functional equation characterizing polynomial functions. Ann. Polon. Math.52 (1990), 119–137. · Zbl 0717.39007
[41] Gajda, Z.,On stability of additive mappings. Internat. J. Math. Math. Sci.14 (1991), 431–434. · Zbl 0739.39013 · doi:10.1155/S016117129100056X
[42] Gajda, Z.,Note on invariant means for essentially bounded functions. Manuscript.
[43] Gajda, Z.,Generalized invariant means and their applications to the stability of homomorphisms. Manuscript.
[44] Gajda, Z.,Invariant means and representations of semigroups in the theory of functional equations. [Prace Naukowe Uniwersytetu Slaskiego w Katowicach, No. 1273]. Uniw. Slask., Katowice, 1992. · Zbl 0925.39005
[45] Gajda, Z. andGer, R.,Subadditive multifunctions and Hyers – Ulam stability. InGeneral inequalities, 5. [Internat. Schriftenreihe Numer. Math., Vol. 80]. Birkhäuser, Basel–Boston, MA, 1987.
[46] Ger, R.,Almost approximately additive mappings. InGeneral inequalities, 3. [Internat. Schriftenreihe Numer. Math., Vol. 64]. Birkhäuser, Basel–Boston, MA, 1983, pp. 263–276.
[47] Ger, R.,Stability of addition formulae for trigonometric mappings. Zeszyty Naukowe Politechniki Slasiej, Ser. Matematyka-Fizyka z.64 (1990), no. 1070, 75–84.
[48] Ger, R.,On functional inequalities stemming from stability questions. InGeneral inequalities, 6. [Internat. Schriftenreihe Numer. Math., Vol. 103]. Birkhäuser, Basel–Boston, Mass., 1992. · Zbl 0770.39007
[49] Ger, R.,Superstability is not natural. InReport of the twenty-sixth International Symposium on Functional Equations. Aequationes Math.37 (1989), 68. · Zbl 0702.46004 · doi:10.1007/BF01836447
[50] Ger, R.,The singular case in the stability behaviour of linear mappings. Grazer Math. Ber.316(1991), 59–70. · Zbl 0796.39012
[51] Greenleaf, F. P.,Invariant means on topological groups. [Van Nostrand Mathematical Studies, Vol. 16]. Van Nostrand, New York–Toronto–London–Melbourne, 1969. · Zbl 0174.19001
[52] Grothendieck, A.,Produits tensoriels topologiques et espace nucléaires. [Memoirs Amer. Math. Soc., No. 16]. A.M.S., Providence, R.I., 1955.
[53] Hyers, D. H.,On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A.27 (1941), 222–224. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[54] Hyers, D. H.,Transformations with bounded n-th differences. Pacific J. Math.11 (1961), 591–602. · Zbl 0099.10501
[55] Hyers, D. H.,The stability of homomorphisms and related topics. InGlobal analysis – analysis on manifolds. [Teubner-Texte Math. 57]. Teubner, Leipzig, 1983, pp. 140–153.
[56] Hyers, D. H. andRassias, Th. M.,Approximate homomorphisms. Aequationes Math.44 (1992), 125–153. · Zbl 0806.47056 · doi:10.1007/BF01830975
[57] Isac, G. andRassias, Th. M.,On the Hyers–Ulam stability of {\(\psi\)}-additive mappings. J. Approx. Theory72 (1993), 131–137. · Zbl 0770.41018 · doi:10.1006/jath.1993.1010
[58] Jarosz, K.,Perturbations of Banach algebras. [Lecture Notes in Mathematics, Vol. 1120]. Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1985. · Zbl 0557.46029
[59] Johnson, B. E.,Cohomology in Banach algebras. [Memoirs Amer. Math. Soc., No. 127]. A.M.S., Providence, R.I., 1972.
[60] Johnson, B. E.,Approximately multiplicative functionals. J. London Math. Soc. (2)34 (1986), 489–510. · Zbl 0625.46059 · doi:10.1112/jlms/s2-34.3.489
[61] Johnson, B. E.,Approximately multiplicative maps between Banach algebras. J. London Math. Soc. (2)37 (1988), 294–316. · Zbl 0652.46031 · doi:10.1112/jlms/s2-37.2.294
[62] Kazhdan, D.,On {\(\mu\)}-representations. Israel J. Math.43 (1982), 315–323. · Zbl 0518.22008 · doi:10.1007/BF02761236
[63] Kominek, Z.,On a local stability of the Jensen functional equation. Demonstratio Math.22 (1989), 499–507. · Zbl 0702.39007
[64] Lawrence, J.,The stability of multiplicative semigroup homomorphisms to real normed algebras. I. Aequationes Math.28 (1985), 94–101. · Zbl 0594.46047 · doi:10.1007/BF02189397
[65] Mazur, S. andOrlicz, W.,Grundelegende Eigenschaften der polynomischen Operationen. Erste Mitteilung. Studia Math.5 (1934), 50–68. · JFM 60.1074.03
[66] Mazur, S. andOrlicz, W.,Grundelegende Eigenschaften der polynomischen Operationen. Zweite Mitteilung. Studia Math.5 (1934), 179–189. · JFM 60.1074.04
[67] Moszner, Z.,Sur la stabilité de l’équation d’homomorphisme. Aequationes Math.29 (1985), 290–306. · Zbl 0583.39012 · doi:10.1007/BF02189833
[68] Moszner, Z.,Sur la définition de Hyers de la stabilité de l’équation fonctionnelle. Opuscula Math. (1987), 47–57 (1988). · Zbl 0654.39006
[69] Nashed, M. Z. andVotruba, G. F.,A unified operator theory of generalized inverses. InGeneralized inverses and applications. Academic Press, New York–San Francisco–London, 1976, pp. 1–109.
[70] Nikodem, K.,The stability of the Pexider equation. Ann. Math. Sil.5 (1991), 91–93. · Zbl 0754.39007
[71] Paganoni, L.,Soluzione di una equazione funzionale su dominio ristretto. Boll. Un. Mat. Ital. (5) 17-B (1980), 979–993. · Zbl 0451.39002
[72] Pólya, G. andSzegö, G.,Problems and theorems in analysis, Vol. I, Part One, Ch. 3, Problem 99. [Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 193]. Springer-Verlag, Berlin–Heidelberg–New York, 1972.
[73] Rassias, J. M.,On approximation of approximately linear mappings by linear mappings. J. Funct. Anal.46 (1982), 126–130. · Zbl 0482.47033 · doi:10.1016/0022-1236(82)90048-9
[74] Rassias, J. M.,On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. (2)108 (1984), 445–446. · Zbl 0599.47106
[75] Rassias, J. M.,On a new approximation of approximately linear mappings by linear mappings. Discuss. Math.7 (1985), 193–196. · Zbl 0592.46004
[76] Rassias, J. M.,Solution of a problem of Ulam. J. Approx. Theory57 (1989), 268–273. · Zbl 0672.41027 · doi:10.1016/0021-9045(89)90041-5
[77] Rassias, J. M.,On the stability of the Euler–Lagrange functional equation. C. R. Acad. Bulgare Sci.45 (1992), 17–20. · Zbl 0789.46036
[78] Rassias, Th. M.,On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc.72 (1978), 297–300. · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1
[79] Rassias, Th. M.,The stability of linear mappings and some problems on isometrices. InMathematical analysis and its applications (Kuwait, 1985). [KFAS Proc. Ser., Vol. 3]. Pergamon, Oxford-Elmsford, NY, 1988, pp. 175–184.
[80] Rassias, Th. M.,On the stability of mappings. Rend. Sem. Mat. Fis. Milano58 (1988), 91–99 (1990). · Zbl 0711.47002 · doi:10.1007/BF02925233
[81] Rassias, Th. M.,On a modified Hyers–Ulam sequence. J. Math. Anal. Appl.158 (1991), 106–113. · Zbl 0746.46038 · doi:10.1016/0022-247X(91)90270-A
[82] Rassias, Th. M. andŠemrl, P.,On the behaviour of mappings which do not satisfy Hyers–Ulam stability. Proc. Amer. Math. Soc.114 (1992), 989–993. · Zbl 0761.47004 · doi:10.1090/S0002-9939-1992-1059634-1
[83] Rassias, Th. M. andŠemrl, P.,On the Hyers–Ulam stability of linear mappings. J. Math. Anal. Appl.173 (1993), 325–338. · Zbl 0789.46037 · doi:10.1006/jmaa.1993.1070
[84] Rätz, J.,On approximately additive mappings. InGeneral inequalities, 2. [Internat. Schriftenreihe Numer. Math., Vol. 47]. Birkhäuser, Basel–Boston, MA, 1980, pp. 233–251.
[85] Scott, W. R.,Group theory. Dover Publications, New York, 1987.
[86] Šemrl, P.,The stability of approximately additive functions. Manuscript. · Zbl 0844.39005
[87] Šemrl, P.,Isomorphisms of standard operator algebras. To appear in Proc. Amer. Math. Soc. · Zbl 0824.47037
[88] Šemrl, P.,The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral Equations Operator Theory18 (1994), 118–122. · Zbl 0810.47029 · doi:10.1007/BF01225216
[89] Schwaiger, J.,On the stability of a functional equation for homogeneous functions. In Report of the twenty-second International Symposium on Functional Equations. Aequationes Math.29 (1985), 80. · Zbl 0578.39006 · doi:10.1007/BF02189812
[90] Shapiro, H. N.,Note on a problem in number theory. Bull. Amer. Math. Soc.54 (1948), 890–893. · Zbl 0032.26202 · doi:10.1090/S0002-9904-1948-09090-5
[91] Shtern, A. I.,On stability of homomorphisms in the group \(\mathbb{R}\)*. Vestnik MGU Ser. Matem. Mech.37 (1982), 29–32. – English translation in Moscow Univ. Math. Bull.37 (1982), 33–36. · Zbl 0494.39007
[92] Shtern, A. I.,Quasirepresentations and pseudorepresentations. Funktsional. Anal. Prilozhen.25 (1991), 70–73. – English translation in Funct. Anal. Appl. 25 (1991), 140–143. · Zbl 0737.22003
[93] Skof, F.,Sull’approssimazione delle applicazioni localmente {\(\delta\)}-additive. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.117 (1983), 377–389 (1986).
[94] Skof, F.,Proprietà locali e approssimazione di operatori. InGeometry of Banach spaces and related topics (Milan, 1983). Rend. Sem. Mat. Fis. Milano53 (1983), 113–129 (1986). · Zbl 0599.39007 · doi:10.1007/BF02924890
[95] Skof, F.,Approssimazione di funzioni {\(\delta\)}-quadratiche su dominio ristretto. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.118 (1984), 58–70.
[96] Skof, F.,On approximately quadratic functions on a restricted domain. InReport of the third International Symposium on Functional Equations and Inequalities, 1986. Publ. Math. Debrecen38 (1991), 14.
[97] Smajdor, A.,Hyers–Ulam stability for set valued functions. InReport of the twenty-seventh International Symposium on Functional Equations. Aequationes Math.39 (1990), 297. · Zbl 0706.39006 · doi:10.1007/BF01833142
[98] Székelyhidi, L.,The stability of linear functional equations. C. R. Math. Rep. Acad. Sci. Canada3 (1981), 63–67.
[99] Székelyhidi, L.,On a stability theorem. C. R. Math. Rep. Acad. Sci. Canada 3 (1981), 253–255.
[100] Székelyhidi, L.,On a theorem of Baker, Lawrence and Zorzitto. Proc. Amer. Math. Soc.84 (1982), 95–96. · Zbl 0485.39003
[101] Székelyhidi, L.,The stability of d’Alembert-type functional equations. Acta Sci. Math. (Szeged)44 (1982), 313–320 (1983). · Zbl 0517.39008
[102] Székelyhidi, L.,Note on a stability theorem. Canad. Math. Bull.25 (1982), 500–501. · Zbl 0505.39002 · doi:10.4153/CMB-1982-074-0
[103] Székelyhidi, L.,Note on Hyers’s theorem. C. R. Math. Rep. Acad. Sci. Canada8 (1986), 127–129. · Zbl 0604.39007
[104] Székelyhidi, L.,Remarks on Hyers’s theorem. Publ. Math. Debrecen34 (1987), 131–135. · Zbl 0627.39006
[105] Székelyhidi, L.,Fréchet’s equation and Hyers theorem on noncommutative semigroups. Ann. Polon. Math.48 (1988), 183–189. · Zbl 0656.39005
[106] Székelyhidi, L.,Stability of some functional equations in economics. Rend. Sem. Mat. Fis. Milano58 (1988), 169–176 (1990). · Zbl 0713.39006 · doi:10.1007/BF02925239
[107] Székelyhidi, L.,An abstract superstability theorem. Abh. Math. Sem. Univ. Hamburg59 (1989), 81–83. · Zbl 0711.39004 · doi:10.1007/BF02942317
[108] Székelyhidi, L.,Stability properties of functional equations describing the scientific laws. J. Math. Anal. Appl.150 (1990), 151–158. · Zbl 0708.39006 · doi:10.1016/0022-247X(90)90202-Q
[109] Székelyhidi, L.,The stability of the sine and cosine functional equations. Proc. Amer. Math. Soc.110 (1990), 109–115. · Zbl 0718.39004
[110] Székelyhidi, L.,Stability properties of functional equations in several variables. Manuscript. · Zbl 0859.39015
[111] Tabor, J.,Ideal stability of the Cauchy and Pexider equations. In Report of the twenty-second International Symposium on Functional Equations, Aequationes Math.29 (1985), 82.
[112] Tabor, J.,Ideal stability of the Cauchy equation. InProceedings of the twenty-third International Symposium on Functional Equations. Centre for Inf. Th., Univ. of Waterloo, Waterloo, Ont., 1985.
[113] Tabor, J.,On functions behaving like additive functions. Aequationes Math.35 (1988), 164–185. · Zbl 0652.39013 · doi:10.1007/BF01830942
[114] Tabor, J.,Quasi-additive functions. Aequationes Math.39 (1990), 179–197. · Zbl 0708.39005 · doi:10.1007/BF01833149
[115] Tabor, J.,Approximate endomorphisms of the complex field. J. Natur. Geom.1 (1992), 71–86. · Zbl 0757.47033
[116] Ulam, S. M.,A collection of mathematical problems. Interscience Publ., New York, 1961.Problems in Modern Mathematics, Wiley, New York, 1964. · Zbl 0106.43102
[117] Ulam, S. M.,Sets, numbers, and universes. M.I.T. Press, Cambridge, 1974. · Zbl 0558.00017
[118] Whitney, H.,On functions with bounded n-th differences. J. Math. Pures Appl. (9)36 (1957), 67–95. · Zbl 0077.06901
[119] Whitney, H.,On bounded functions with bounded n-th differences. Proc. Amer. Math. Soc.10 (1959), 480–481. · Zbl 0089.27601
[120] Zorzitto, F.,31. Problem. InReport of the twenty-sixth International Symposium on Functional Equations. Aequationes Math.37 (1989), 118.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.