# zbMATH — the first resource for mathematics

Approximate homomorphisms. (English) Zbl 0806.47056
Summary: We present a survey of ideas and results stemming from the following stability problem of S. M. Ulam. Given a group $$G_ 1$$, a metric group $$G_ 2$$ and $$\varepsilon> 0$$, find $$\delta>0$$ such that, if $$f: G_ 1\to G_ 2$$ satisfies $$d(f(xy),f(x)f(y))\leq \delta$$ for all $$x,y\in G_ 1$$, then there exists a homomorphism $$g: G_ 1\to G_ 2$$ such that $$d(f(x),g(x))\leq \varepsilon$$ for all $$x\in G_ 1$$. For Banach spaces the problem was solved by D. Hyers (1941) with $$\delta=\varepsilon$$ and $$g(x)= \lim_{n\to\infty} f(2^ n x)/2^ n$$.
Section 2 deals with the case where $$G_ 1$$ is replaced by an Abelian semigroup $$S$$ and $$G_ 2$$ by a sequentially complete locally convex topological vector space $$E$$. The necessity for the commutativity of $$S$$ and the sequential completeness of $$E$$ 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 the second author 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 semigroup $$S$$ to a class of subsets of a Banach space $$X$$ are dealt with in Section 7. Furthermore, a result by A. Smajdor (1990) on the stability of a functional equation of Pexider type for 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.

##### MSC:
 47J05 Equations involving nonlinear operators (general) 47H04 Set-valued operators 39A10 Additive difference equations
Full Text:
##### References:
  Aczél, J. andDhombres, J.,Functional Equations in Several Variables. Encyclopedia of Mathematics and its Applications, Vol. 31. Cambridge University Press, Cambridge, 1989.  Baker, J. A.,The stability of the cosine equation. Proc. Amer. Math. Soc.80 (1980), 411–416. · Zbl 0448.39003  Baker, J. A.,On some mathematical characters. (Manuscript to appear in Glasnik Matematički). · Zbl 0782.39007  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. · Zbl 0397.39010  Baron, K.,A remark on the stability of the Cauchy equation. Wy\.z. SzkoŁa Ped. Krakow. Rocznik Nauk.-Dydakt. Prace Mat.11 (1985), 7–12.  Baron, K. andKannapan, PL.,On the Pexider difference. Fund. Math.134 (1990), 247–254. · Zbl 0715.39012  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.  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). · Zbl 0870.39011  Cenzer, D.,The stability problem for transformations of the circle. Proc. Roy. Soc. Edinburgh Sect. A84 (1979), 279–281. · Zbl 0439.39004  Cenzer, D.,The stability problem: new results and counterexamples. Lett. in Math. Phys.10 (1985), 155–160. · Zbl 0595.39010  Cholewa, P. W.,The stability of the sine equation. Proc. Amer. Math. Soc.88 (1983), 631–634. · Zbl 0547.39003  Christensen, J. P. R.,On sets of Haar measure zero in Abelian Polish groups. Israel J. Math.13 (1972), 255–260. · Zbl 0249.43002  Van der Corput, J. G.,Goniometrische functies gekarakteriseerd door een functionaal betrekking. Euclides17 (1940), 55–75.  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. · Zbl 0752.47014  Fenyö, I. andForti, G. L.,On the inhomogeneous Cauchy functional equation. Stochastica5 (1981), 71–77.  Forti, G. L.,On an alternative functional equation related to the Cauchy equation. Aequationes Math.24 (1982), 195–206. · Zbl 0517.39007  Forti, G. L.,Remark 11. In:Report of the 22nd Internat. Symp. on Functional Equations. Aequationes Math.29 (1985), 90–91. · Zbl 0593.39007  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  Forti, G. L.,Remark 18. In:Report of the 27th Internat. Symp. on Functional Equations. Aequationes Math39 (1990), 309–310.  Forti, G. L. andSchwaiger, J.,Stability of homomorphisms and completeness. C.R. Math. Rep. Acad. Sci. Canada11 (1989), 215–220. · Zbl 0697.39013  Gajda, Z.,On stability of the Cauchy equation on semigroups. Aequationes Math.36 (1988), 76–79. · Zbl 0658.39006  Gajda, Z.,On stability of additive mappings. Internat. J. Math. Math. Sci.14 (1991), 431–434. · Zbl 0739.39013  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.  Ger, R.,Superstability is not natural. In:Report on the 26th Internat. Symp. on Functional Equations. Aequationes Math.37 (1989), 68. · Zbl 0702.46004  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. · Zbl 0770.39007  Godini, G.,Set-valued Cauchy functional equation. Rev. Roumaine Math. Pures Appl.20 (1975), 1113–1121. · Zbl 0322.39013  Greenleaf, F. P.,Invariant means on topological groups. [Van Nostrand Math. Studies, Vol. 16]. New York, 1969. · Zbl 0174.19001  de laHarpe, P. andKaroubi, M.,Representations approchées d’un groupe dans une algebre de Banach. Manuscripta Math.22 (1977), 293–310. · Zbl 0371.22007  Hewitt, E. andRoss, K. A.,Abstract harmonic analysis. Academic Press, New York, 1963.  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  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. · Zbl 0517.22001  Isac, G. andRassias, Th. M.,On the Hyers-Ulam stability of {$$\Psi$$}-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.  Johnson, B. E.,Approximately multiplicative functionals. J. London Math. Soc. (2)34 (1986), 489–510. · Zbl 0625.46059  Johnson, B. E.,Approximately multiplicative maps between Banach algebras. J. London Math. Soc. (2)37 (1988), 294–316. · Zbl 0652.46031  Lawrence, J.,The stability of multiplicative semi-group homomorphisms to real normed algebras. Aequationes Math.28 (1985), 94–101. · Zbl 0594.46047  Moszner, Z.,Sur la stabilité de l’équation d’homomorphisme. Aequationes Math.29 (1985), 290–306. · Zbl 0583.39012  Moszner, Z.,Sur la definition de Hyers de la stabilité de l’équation fonctionelle. Opuscula Math.3 (1987), 47–57 (1988). · Zbl 0654.39006  Rassias, Th. M.,On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc72 (1978), 297–300. · Zbl 0398.47040  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{$$\deg$$}. (SameReport, p. 309.)  Rassias, Th. M.,On a modified Hyers – Ulam sequence. J. Math. Anal. Appl.158 (1991), 106–113. · Zbl 0746.46038  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). · Zbl 0761.47004  Rassias, Th. M. andŠemrl, P.,On the Hyers – Ulam stability of linear mappings. (Manuscript 2, to appear in J. Math. Anal. Appl.). · Zbl 0894.39012  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. · Zbl 0433.39014  Sablik, M.,A functional congruence revisited. In:Report on the 28th Internat. Symp. on Functional Equations. Aequationes Math.41 (1991), 273. · Zbl 0986.39500  Schwaiger, J.,Remark 12. In:Report on the 25th Internat. Symp. on Functional Equations. Aequationes Math.35 (1988), 120–121.  Skof, F.,On the approximation of locally {$$\delta$$}-additive mappings (Italian). Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat.117 (1983), 377–389. · Zbl 0794.39008  Smajdor, A.,Hyers – Ulam stability for set-valued functions. In:Report on the 27th Internat. Symp. on Functonal Equations. Aequationes Math.39 (1990), 297. · Zbl 0706.39006  Steinhaus, H.,Sur les distances des points dans les ensembles de mesure positive. Fund. Math.1 (1920), 93–104. · JFM 47.0179.02  Székelyhidi, L., (a)Note on a stability theorem. Canad. Math. Bull.25 (1982), 500–501. · Zbl 0505.39002  Székelyhidi, L., (b)On a theorem of Baker, Lawrence and Zoritto. Proc. Amer. Math. Soc.84 (1982), 95–96.  Székelyhidi, L.,Remark 17. In:Report on the 22nd Intern. Symp. on Functional Equations. Aequationes Math.29 (1985), 95–96.  Székelyhidi, L.,Note on Hyer’s theorem. C.R. Math. Rep. Acad. Sci. Canada8 (1986), 127–129. · Zbl 0604.39007  Székelyhidi,Remarks on Hyer’s theorem. Publ. Math. Debrecen34 (1987), 131–135. · Zbl 0627.39006  Turdza, E.,Stability of Cauchy equations. Wy\.z. Szkoła Ped. Krakow. Rocznik Nauk-Dydakt. Prace Mat.10 (1982), 141–145.  Ulam, S. M.,A collection of mathematical problems. Interscience Publ., New York, 1960. (Also:Problems in modern mathematics. Wiley, New York, 1964.) · Zbl 0086.24101  Ulam, S. M.,Sets, numbers and universes. Mass. Inst. of Tech. Press, Cambridge, MA, 1974. · Zbl 0558.00017
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.