##
Solution of a quadratic stability Ulam type problem

##
*John Michael Rassias*

**Address.**

National and Capodistrian University of Athens,

Pedagogical Department E.E.

Section of Mathematics and Informatics

4, Agamemnonos Str., Aghia Paraskevi

Attikis 15342, GREECE
**E-mail.** jrassias@primedu.uoa.gr

**Abstract.**

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

problem.

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

problem.

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.