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(x1 + 2x2) + 3C(x1) = 3C(x1 + x2) + C(x1 - x2) + 6C(x2)

for all 2-dimensional vectors (x1,x2) ∈ X2, 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.

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