Solution of the Ulam stability problem for cubic mappings,” Glasnik Matematički

Abstract. 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 stated that this kind of stability problems are of particular interest in probability theory and in the case of functional equations of dierent 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) 2 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. 1. Cubic functional equation Definition 1.1. Let X be a linear space and let Y be a real complete linear space. Then a mapping C: X! Y, is called cubic, if the cubic functional equation (1.1) C(x1 + 2x2) + 3C(x1) = 3C(x1 + x2) + C(x1 x2) + 6C(x2) holds for all 2-dimensional vectors (x1; x2) 2 X2. ( [14-23]). Note that mapping C is called cubic because the following algebraic iden-tit