## 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:

 [1] Aczél, J. andDhombres, J.,Functional Equations in Several Variables. Encyclopedia of Mathematics and its Applications, Vol. 31. Cambridge University Press, Cambridge, 1989. [2] Baker, J. A.,The stability of the cosine equation. Proc. Amer. Math. Soc.80 (1980), 411–416. · Zbl 0448.39003 [3] Baker, J. A.,On some mathematical characters. (Manuscript to appear in Glasnik Matematički). · Zbl 0782.39007 [4] 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 [5] 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. [6] Baron, K. andKannapan, PL.,On the Pexider difference. Fund. Math.134 (1990), 247–254. · Zbl 0715.39012 [7] Baron, K. andKannapan, PL.,On the Cauchy difference. (Manuscript submitted for publication). [8] Baron, K. andVolkmann, P.,On the Cauchy equation modulo Z. Fund. Math.131 (1988), 143–148. [9] Baron, K. andVolkmann, P.,On a theorem of van der Corput. (Manuscript submitted for publication). [10] Brzdek, J.,On the Cauchy difference. (Manuscript submitted for publication). · Zbl 0870.39011 [11] Cenzer, D.,The stability problem for transformations of the circle. Proc. Roy. Soc. Edinburgh Sect. A84 (1979), 279–281. · Zbl 0439.39004 [12] Cenzer, D.,The stability problem: new results and counterexamples. Lett. in Math. Phys.10 (1985), 155–160. · Zbl 0595.39010 [13] Cholewa, P. W.,The stability of the sine equation. Proc. Amer. Math. Soc.88 (1983), 631–634. · Zbl 0547.39003 [14] Christensen, J. P. R.,On sets of Haar measure zero in Abelian Polish groups. Israel J. Math.13 (1972), 255–260. · Zbl 0249.43002 [15] Van der Corput, J. G.,Goniometrische functies gekarakteriseerd door een functionaal betrekking. Euclides17 (1940), 55–75. [16] 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.) [17] 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 [18] Fenyö, I. andForti, G. L.,On the inhomogeneous Cauchy functional equation. Stochastica5 (1981), 71–77. [19] Forti, G. L.,On an alternative functional equation related to the Cauchy equation. Aequationes Math.24 (1982), 195–206. · Zbl 0517.39007 [20] Forti, G. L.,Remark 11. In:Report of the 22nd Internat. Symp. on Functional Equations. Aequationes Math.29 (1985), 90–91. · Zbl 0593.39007 [21] 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 [22] Forti, G. L.,Remark 18. In:Report of the 27th Internat. Symp. on Functional Equations. Aequationes Math39 (1990), 309–310. [23] Forti, G. L. andSchwaiger, J.,Stability of homomorphisms and completeness. C.R. Math. Rep. Acad. Sci. Canada11 (1989), 215–220. · Zbl 0697.39013 [24] Gajda, Z.,On stability of the Cauchy equation on semigroups. Aequationes Math.36 (1988), 76–79. · Zbl 0658.39006 [25] Gajda, Z.,On stability of additive mappings. Internat. J. Math. Math. Sci.14 (1991), 431–434. · Zbl 0739.39013 [26] Gajda, Z.,Generalized invariant means and their application to the stability of homomorphisms. (Manuscript, submitted for publication). [27] 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. [28] Ger, R.,Superstability is not natural. In:Report on the 26th Internat. Symp. on Functional Equations. Aequationes Math.37 (1989), 68. · Zbl 0702.46004 [29] 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 [30] Godini, G.,Set-valued Cauchy functional equation. Rev. Roumaine Math. Pures Appl.20 (1975), 1113–1121. · Zbl 0322.39013 [31] Greenleaf, F. P.,Invariant means on topological groups. [Van Nostrand Math. Studies, Vol. 16]. New York, 1969. · Zbl 0174.19001 [32] de laHarpe, P. andKaroubi, M.,Representations approchées d’un groupe dans une algebre de Banach. Manuscripta Math.22 (1977), 293–310. · Zbl 0371.22007 [33] Hewitt, E. andRoss, K. A.,Abstract harmonic analysis. Academic Press, New York, 1963. [34] 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 [35] 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 [36] Isac, G. andRassias, Th. M.,On the Hyers-Ulam stability of {$$\Psi$$}-additive mappings. (Manuscript, to appear in J. Approx. Theory). [37] Johnson, B. E.,Cohomology in Banach algebras [Memoirs Amer. Math. Soc., No. 127]. Amer. Math. Soc., Providence, RI, 1972. [38] Johnson, B. E.,Approximately multiplicative functionals. J. London Math. Soc. (2)34 (1986), 489–510. · Zbl 0625.46059 [39] Johnson, B. E.,Approximately multiplicative maps between Banach algebras. J. London Math. Soc. (2)37 (1988), 294–316. · Zbl 0652.46031 [40] Lawrence, J.,The stability of multiplicative semi-group homomorphisms to real normed algebras. Aequationes Math.28 (1985), 94–101. · Zbl 0594.46047 [41] Moszner, Z.,Sur la stabilité de l’équation d’homomorphisme. Aequationes Math.29 (1985), 290–306. · Zbl 0583.39012 [42] Moszner, Z.,Sur la definition de Hyers de la stabilité de l’équation fonctionelle. Opuscula Math.3 (1987), 47–57 (1988). · Zbl 0654.39006 [43] Rassias, Th. M.,On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc72 (1978), 297–300. · Zbl 0398.47040 [44] 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.) [45] Rassias, Th. M.,On a modified Hyers – Ulam sequence. J. Math. Anal. Appl.158 (1991), 106–113. · Zbl 0746.46038 [46] 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 [47] 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 [48] Rassias, Th. M. andTabor, J.,On approximately additive mappings in Banach spaces. (Manuscript). [49] 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 [50] Sablik, M.,A functional congruence revisited. In:Report on the 28th Internat. Symp. on Functional Equations. Aequationes Math.41 (1991), 273. · Zbl 0986.39500 [51] Schwaiger, J.,Remark 12. In:Report on the 25th Internat. Symp. on Functional Equations. Aequationes Math.35 (1988), 120–121. [52] 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 [53] 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 [54] Steinhaus, H.,Sur les distances des points dans les ensembles de mesure positive. Fund. Math.1 (1920), 93–104. · JFM 47.0179.02 [55] Székelyhidi, L., (a)Note on a stability theorem. Canad. Math. Bull.25 (1982), 500–501. · Zbl 0505.39002 [56] Székelyhidi, L., (b)On a theorem of Baker, Lawrence and Zoritto. Proc. Amer. Math. Soc.84 (1982), 95–96. [57] Székelyhidi, L.,Remark 17. In:Report on the 22nd Intern. Symp. on Functional Equations. Aequationes Math.29 (1985), 95–96. [58] Székelyhidi, L.,Note on Hyer’s theorem. C.R. Math. Rep. Acad. Sci. Canada8 (1986), 127–129. · Zbl 0604.39007 [59] Székelyhidi,Remarks on Hyer’s theorem. Publ. Math. Debrecen34 (1987), 131–135. · Zbl 0627.39006 [60] Turdza, E.,Stability of Cauchy equations. Wy\.z. Szkoła Ped. Krakow. Rocznik Nauk-Dydakt. Prace Mat.10 (1982), 141–145. [61] 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 [62] 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.