Skip to main content
Log in

Approximate homomorphisms

  • Survey Paper
  • Published:
aequationes mathematicae Aims and scope Submit manuscript

Summary

We present a survey of ideas and results stemming from the following stability problem of S. M. Ulam. Given a groupG 1, a metric groupG 2 and ε > 0, find δ > 0 such that, iff: G 1G 2 satisfiesd(f(xy),f(x)f(y)) ⩽ δ for allx, yG 1, then there exists a homomorphismg: G 1G 2 such thatd(f(x),g(x))⩽ε for allx ∈ G l . For Banach spaces the problem was solved by D. Hyers (1941) with δ = ε and

$$g(x) = \mathop {\lim }\limits_{n \to \infty } f(2^n x)/2^n .$$

Section 2 deals with the case whereG 1 is replaced by an Abelian semigroupS andG 2 by a sequentially complete locally convex topological vector spaceE. The necessity for the commutativity ofS and the sequential completeness ofE are also considered.

the method of invariant means is demonstrated in Section 3 for mappings from a right (left) amenable semigroup into the complex numbers.

In Section 4 we present results by Th. Rassias and others, where the Cauchy difference

$$Cf(x,y) = f(x + y) - f(x) - f(y)$$

may be unbounded but satisfies a weaker inequality.

Approximately multiplicative maps are discussed in Section 5, including a stability theorem for homomorphisms of rotations of the circle into itself and approximately multiplicative maps between Banach algebras.

Section 6 is devoted to the work of Z. Moszner (1985) on different definitions of stability.

Results by Z. Gajda and R. Ger (1987) on subadditive set valued mappings from an Abelian semigroupS to a class of subsets of a Banach spaceX are dealt with in Section 7. Furthermore a result by A. Smajdor (1990) on the stability of a functional equation of Pexider type form set valued maps is presented.

Recent works of K. Baron and others on functional congruences, stemming from theorems of J. G. van der Corput (1940), are outlined in Section 8. Section 9 contains remarks and unsolved problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aczél, J. andDhombres, J.,Functional Equations in Several Variables. Encyclopedia of Mathematics and its Applications, Vol. 31. Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  • Baker, J. A.,The stability of the cosine equation. Proc. Amer. Math. Soc.80 (1980), 411–416.

    Google Scholar 

  • Baker, J. A.,On some mathematical characters. (Manuscript to appear in Glasnik Matematički).

  • Baker, J. A., Lawrence, J. andZorzitto, F.,The stability of the equation f(x + y) = f(x)f(y). Proc. Amer. Math. Soc.74 (1979), 242–246.

    Google Scholar 

  • Baron, K.,A remark on the stability of the Cauchy equation. Wyż. SzkoŁa Ped. Krakow. Rocznik Nauk.-Dydakt. Prace Mat.11 (1985), 7–12.

    Google Scholar 

  • Baron, K. andKannapan, PL.,On the Pexider difference. Fund. Math.134 (1990), 247–254.

    Google Scholar 

  • Baron, K. andKannapan, PL.,On the Cauchy difference. (Manuscript submitted for publication).

  • Baron, K. andVolkmann, P.,On the Cauchy equation modulo Z. Fund. Math.131 (1988), 143–148.

    Google Scholar 

  • Baron, K. andVolkmann, P.,On a theorem of van der Corput. (Manuscript submitted for publication).

  • Brzdek, J.,On the Cauchy difference. (Manuscript submitted for publication).

  • Cenzer, D.,The stability problem for transformations of the circle. Proc. Roy. Soc. Edinburgh Sect. A84 (1979), 279–281.

    Google Scholar 

  • Cenzer, D.,The stability problem: new results and counterexamples. Lett. in Math. Phys.10 (1985), 155–160.

    Article  Google Scholar 

  • Cholewa, P. W.,The stability of the sine equation. Proc. Amer. Math. Soc.88 (1983), 631–634.

    Google Scholar 

  • Christensen, J. P. R.,On sets of Haar measure zero in Abelian Polish groups. Israel J. Math.13 (1972), 255–260.

    Google Scholar 

  • Van der Corput, J. G.,Goniometrische functies gekarakteriseerd door een functionaal betrekking. Euclides17 (1940), 55–75.

    Google Scholar 

  • Dicks, D.,Thesis. University of Waterloo, Waterloo, Ont., 1990. (Also:Remark 2. In:Report of the 27th Internat. Symp. on Functional Equations. Aequationes Math.39 (1990), 301.)

  • Drljević, H.,On the respresentation of functionals and the stability of mappings in Hilbert and Banach spaces. In: Topics in Math. Analysis (Th. M. Rassias, Ed.). World Sci. Publ., Singapore, 1989, pp. 231–245.

    Google Scholar 

  • Fenyö, I. andForti, G. L.,On the inhomogeneous Cauchy functional equation. Stochastica5 (1981), 71–77.

    Google Scholar 

  • Forti, G. L.,On an alternative functional equation related to the Cauchy equation. Aequationes Math.24 (1982), 195–206.

    Google Scholar 

  • Forti, G. L.,Remark 11. In:Report of the 22nd Internat. Symp. on Functional Equations. Aequationes Math.29 (1985), 90–91.

    Google Scholar 

  • Forti, G. L.,The stability of homomorphisms and amenability with applications to functional equations. Abh. Math. Sem. Univ. Hamburg57 (1987), 215–226.

    Google Scholar 

  • Forti, G. L.,Remark 18. In:Report of the 27th Internat. Symp. on Functional Equations. Aequationes Math39 (1990), 309–310.

    Google Scholar 

  • Forti, G. L. andSchwaiger, J.,Stability of homomorphisms and completeness. C.R. Math. Rep. Acad. Sci. Canada11 (1989), 215–220.

    Google Scholar 

  • Gajda, Z.,On stability of the Cauchy equation on semigroups. Aequationes Math.36 (1988), 76–79.

    Article  Google Scholar 

  • Gajda, Z.,On stability of additive mappings. Internat. J. Math. Math. Sci.14 (1991), 431–434.

    Article  Google Scholar 

  • Gajda, Z.,Generalized invariant means and their application to the stability of homomorphisms. (Manuscript, submitted for publication).

  • Gajda, Z. andGer, R.,Subadditive multifunctions and Hyers-Ulam stability. In: General Inequalities 5 (W. Walter, Ed.). [Internat. Ser. Numer. Math., Vol. 80]. Birkhäuser, Basel, 1987, pp. 281–291.

    Google Scholar 

  • Ger, R.,Superstability is not natural. In:Report on the 26th Internat. Symp. on Functional Equations. Aequationes Math.37 (1989), 68.

    Article  Google Scholar 

  • Ger, R.,On functional inequalities stemming from stability questions. In: General Inequalities 6. (W. Walter, Ed.). [Internat. Ser. Numer. Math., Vol. 103]. Birkhäuser, Basel, 1992, pp. 227–240.

    Google Scholar 

  • Godini, G.,Set-valued Cauchy functional equation. Rev. Roumaine Math. Pures Appl.20 (1975), 1113–1121.

    Google Scholar 

  • Greenleaf, F. P.,Invariant means on topological groups. [Van Nostrand Math. Studies, Vol. 16]. New York, 1969.

  • de laHarpe, P. andKaroubi, M.,Representations approchées d'un groupe dans une algebre de Banach. Manuscripta Math.22 (1977), 293–310.

    Article  Google Scholar 

  • Hewitt, E. andRoss, K. A.,Abstract harmonic analysis. Academic Press, New York, 1963.

    Google Scholar 

  • Hyers, D. H.,On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A.27 (1941), 222–224.

    Google Scholar 

  • Hyers, D. H.,The stability of homomorphisms and related topics. In: Global Analysis — Analysis on Manifolds (Th. M. Rassias, Ed.). Teubner, Leipzig, 1983, pp. 140–153.

    Google Scholar 

  • Isac, G. andRassias, Th. M.,On the Hyers-Ulam stability of Ψ-additive mappings. (Manuscript, to appear in J. Approx. Theory).

  • Johnson, B. E.,Cohomology in Banach algebras [Memoirs Amer. Math. Soc., No. 127]. Amer. Math. Soc., Providence, RI, 1972.

    Google Scholar 

  • Johnson, B. E.,Approximately multiplicative functionals. J. London Math. Soc. (2)34 (1986), 489–510.

    Google Scholar 

  • Johnson, B. E.,Approximately multiplicative maps between Banach algebras. J. London Math. Soc. (2)37 (1988), 294–316.

    Google Scholar 

  • Lawrence, J.,The stability of multiplicative semi-group homomorphisms to real normed algebras. Aequationes Math.28 (1985), 94–101.

    Google Scholar 

  • Moszner, Z.,Sur la stabilité de l'équation d'homomorphisme. Aequationes Math.29 (1985), 290–306.

    Google Scholar 

  • Moszner, Z.,Sur la definition de Hyers de la stabilité de l'équation fonctionelle. Opuscula Math.3 (1987), 47–57 (1988).

    Google Scholar 

  • Rassias, Th. M.,On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc72 (1978), 297–300.

    Google Scholar 

  • Rassias, Th. M.,The stability of mappings and related topics. In:Report on the 27th Internat. Symp. on Functional Equations. Aequationes Math.39 (1990), 292–293.Problem 16, 2°. (SameReport, p. 309.)

  • Rassias, Th. M.,On a modified Hyers — Ulam sequence. J. Math. Anal. Appl.158 (1991), 106–113.

    Article  Google Scholar 

  • Rassias, Th. M. andŠemrl, P.,On the behavior of mappings which do not satisfy Hyers — Ulam stability. (Manuscript 1, to appear in Proc. Amer. Math. Soc., 1992).

  • Rassias, Th. M. andŠemrl, P.,On the Hyers — Ulam stability of linear mappings. (Manuscript 2, to appear in J. Math. Anal. Appl.).

  • Rassias, Th. M. andTabor, J.,On approximately additive mappings in Banach spaces. (Manuscript).

  • Rätz, J.,On approximately additive mappings. In: General Inequalities 2 (E. F. Beckenbach, Ed.). [Internat. Ser. Numer. Math., Vol. 47]. Birkhäuser, Basel, 1980, pp. 233–251.

    Google Scholar 

  • Sablik, M.,A functional congruence revisited. In:Report on the 28th Internat. Symp. on Functional Equations. Aequationes Math.41 (1991), 273.

    Google Scholar 

  • Schwaiger, J.,Remark 12. In:Report on the 25th Internat. Symp. on Functional Equations. Aequationes Math.35 (1988), 120–121.

    Google Scholar 

  • Skof, F.,On the approximation of locally δ-additive mappings (Italian). Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat.117 (1983), 377–389.

    Google Scholar 

  • Smajdor, A.,Hyers — Ulam stability for set-valued functions. In:Report on the 27th Internat. Symp. on Functonal Equations. Aequationes Math.39 (1990), 297.

    Google Scholar 

  • Steinhaus, H.,Sur les distances des points dans les ensembles de mesure positive. Fund. Math.1 (1920), 93–104.

    Google Scholar 

  • Székelyhidi, L., (a)Note on a stability theorem. Canad. Math. Bull.25 (1982), 500–501.

    Google Scholar 

  • Székelyhidi, L., (b)On a theorem of Baker, Lawrence and Zoritto. Proc. Amer. Math. Soc.84 (1982), 95–96.

    Google Scholar 

  • Székelyhidi, L.,Remark 17. In:Report on the 22nd Intern. Symp. on Functional Equations. Aequationes Math.29 (1985), 95–96.

    Google Scholar 

  • Székelyhidi, L.,Note on Hyer's theorem. C.R. Math. Rep. Acad. Sci. Canada8 (1986), 127–129.

    Google Scholar 

  • Székelyhidi,Remarks on Hyer's theorem. Publ. Math. Debrecen34 (1987), 131–135.

    Google Scholar 

  • Turdza, E.,Stability of Cauchy equations. Wyż. Szkoła Ped. Krakow. Rocznik Nauk-Dydakt. Prace Mat.10 (1982), 141–145.

    Google Scholar 

  • Ulam, S. M.,A collection of mathematical problems. Interscience Publ., New York, 1960. (Also:Problems in modern mathematics. Wiley, New York, 1964.)

    Google Scholar 

  • Ulam, S. M.,Sets, numbers and universes. Mass. Inst. of Tech. Press, Cambridge, MA, 1974.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hyers, D.H., Rassias, T.M. Approximate homomorphisms. Aeq. Math. 44, 125–153 (1992). https://doi.org/10.1007/BF01830975

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01830975

AMS (1991) subject classification

Navigation