Approximately Additive and Approximately Linear Mappings

  • Donald H. Hyers
  • George Isac
  • Themistocles M. Rassias
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 34)


We have mentioned the concept of stability and given an example for the Cauchy functional equation (1.1). A definition of stability in the case of homomorphisms between metric groups was suggested by a problem posed by S.M. Ulam in 1940 (see Ulam (1960), p. 64). Given a group (G1, •), a metric group (G2,*) with metric d and a positive number η, suppose that there exists a positive numberε= ε(η) such that, if d(f (xy), f (x) * f (y)) <ε for some f : G1 G2 and all x and y in G1, then a homomorphism h: G1 → G2 exists with d(f (x), h(x)) <η for all x in G1. In this case, the equation of homomorphism h(xy) = h(x) * h(y) is called stable. Theorem 1.1 with G1 = E1, G2 = E2 and with addition as the group operation in each case shows that Cauchy’s equation is stable by this definition with η= ε.


Banach Space Functional Equation Additive Mapping Real Banach Space Topological Index 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Donald H. Hyers
  • George Isac
    • 1
  • Themistocles M. Rassias
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsNational Technical University of AthensAthensGreece

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