ISSN:
1420-8903
Keywords:
47H15
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 1 →G 2 satisfiesd(f(xy),f(x)f(y)) ⩽ δ for allx, y ∈G 1, then there exists a homomorphismg: G 1 →G 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01830975
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