#### Glasnik Matematicki, Vol. 34, No.2 (1999), 243-252.

### SOLUTION OF THE ULAM STABILITY PROBLEM FOR
QUARTIC MAPPING

### John Michael Rassias

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

**Abstract.** In 1940, S. M. Ulam proposed at the
University of Wisconsin the problem: "Give conditions in order for a
linear mapping near an approximately linear mapping to exist."
In 1968, S. M. Ulam proposed the general problem:
"When is it true that by changing a little the hypotheses of a
theorem one can still assert that the thesis of the theorem
remains true or approximately true?" In 1978, P. M. Gruber
proposed the Ulam type problem: "Suppose a mathematical object
satisfies a certain property approximately. Is it then possible
to approximate this object by objects, satisfying the property
exactly?" According to P. M. Gruber, this kind of stability problems
is of particular interest in probability theory and in the case of
functional equations of different types. In 1982-1998, we solved
the above Ulam problem, or equivalently the Ulam type problem
for linear mappings and also established analogous stability
problems for quadratic and cubic mappings. In this paper we introduce
the new quartic mappings *F* : *X*
→
*Y*, satisfying the new quartic functional equation

*F*(*x*_{1} + 2*x*_{2}) +
*F*(*x*_{1} - 2*x*_{2}) +
6*F*(*x*_{1}) =
4[*F*(*x*_{1} + *x*_{2}) +
*F*(*x*_{1} - *x*_{2}) +
6*F*(*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 mappings *F*.
**1991 Mathematics Subject Classification.**
39B.

**Key words and phrases.** Ulam problem, Ulam type problem,
quartic mappings, quartic functional equation, quartic functional
inequality, approximately quartic, stability problem.

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