aequationes mathematicae

, Volume 44, Issue 2–3, pp 125–153 | Cite as

Approximate homomorphisms

  • Donald H. Hyers
  • Themistocles M. Rassias
Survey Paper


We present a survey of ideas and results stemming from the following stability problem of S. M. Ulam. Given a groupG1, a metric groupG2 and ε > 0, find δ > 0 such that, iff: G1G2 satisfiesd(f(xy),f(x)f(y)) ⩽ δ for allx, yG1, then there exists a homomorphismg: G1G2 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 whereG1 is replaced by an Abelian semigroupS andG2 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.

AMS (1991) subject classification



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Copyright information

© Birkhäuser Verlag 1992

Authors and Affiliations

  • Donald H. Hyers
    • 1
    • 2
  • Themistocles M. Rassias
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsUniversity of LaVerneKifissiaGreece

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