#### Glasnik Matematicki, Vol. 36, No.1 (2001), 63-72.

### SOLUTION OF THE ULAM STABILITY PROBLEM FOR
CUBIC MAPPINGS

### John Michael Rassias

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

*e-mail:* `jrassias@cc.uoa.gr`

**Abstract.** In 1968 S.M. Ulam proposed the general
problem: When is it true taht by changing a little the hypotheses
of a theorem one can still assert that the thesis of the theorem
remains true or spproximately true. In 1978 P.M. Gruber stated that
this kind of stability problems are of particular interest in
probability theory and in the case of functional equations
of different types. In 1982-1998 we solved above Ulam problem
for linear mappings and also established analogous stability problems
for quadratic mappings. In this paper we introduce the new cubic mappings
*C* : *X*
→
*Y*, satisfying the cubic functional equation

*C*(*x*_{1} + 2*x*_{2}) +
3*C*(*x*_{1}) =
3*C*(*x*_{1} + *x*_{2}) +
*C*(*x*_{1} - *x*_{2}) +
6*C*(*x*_{2})

for all 2-dimensional vectors (*x*_{1},*x*_{2})
∈ *X*^{2},
with *X* a linear space (*Y* a real complete linear space),
and then solve the Ulam stability problem for the above-said mappings
*C*.
**1991 Mathematics Subject Classification.**
39B52.

**Key words and phrases.** Cubic mapping, Ulam stability,
approximately cubic, approximately odd mapping.

**Full text (PDF)** (free access)

*Glasnik Matematicki* Home Page