Solution of a quadratic stability Ulam type problem

John Michael Rassias

National and Capodistrian University of Athens,
Pedagogical Department E.E.
Section of Mathematics and Informatics
4, Agamemnonos Str., Aghia Paraskevi
Attikis 15342, GREECE


In 1940 S.~M.~Ulam (Intersci.~Publ., Inc., New York 1960)
imposed at the University of Wisconsin the problem:
``Give conditions in order for a linear mapping near an
approximately linear mapping to exist".
According to  P.~M.~Gruber (Trans.~Amer.~Math.~Soc.~245 (1978),
263--277) the afore-mentioned problem of S.~M.~Ulam belongs to
the following  general problem or Ulam type problem:
``Suppose a mathematical object satisfies a certain
property approximately. Is it then possible to approximate this
objects\ by objects, satisfying the property exactly?"
In 1941 D.~H.~Hyers (Proc.~Nat.~Acad.~Sci.~27 (1941), 411--416)
established the stability Ulam problem with Cauchy inequality
involving a non-negative constant.
Then in 1989 we (J.~Approx.~Theory, 57 (1989), 268--273) solved
Ulam problem with Cauchy functional inequality, involving a
product of powers of norms. Finally we (Discuss.~Math.~12 (1992),
95--103) established the general version of this stability
In this paper we solve a stability Ulam type problem for a general
quadratic functional inequality. Moreover, we introduce
an approximate eveness on approximately quadratic mappings of this
   These problems, according to P.~M.~Gruber (1978), are of particular
interest in probability theory and in the case of functional
equations of different types.
Today there are applications in
actuarial and financial mathematics, sociology and psychology, as
well as in algebra and geometry.

AMSclassification. 39B.

Keywords. Ulam problem, Ulam type problem, stability, quadratic,
approximate eveness, approximately quadratic, quadratic mapping
near an approximately quadratic mapping.