Solution of the Ulam stability problem for quartic mappings

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. Jvl. 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. 1\1. 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. lvl. 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 ---t Y , satisfying the new quartic functional equation F(XI + 2X2) + F(XI 2X2) + 6F(Xl) "" 4 [F(XI + X2) + F(XI X2) + 6F(X2)] for a1l2-dimensional vectors (Xl,X2) E X2, with X a linear space (Y: "" a real complete linear space), and then solve the Ulam stability problem for the above mappings F. 1. QUARTIC FUNCTIONAL EQUATION DEFINITION 1.1. Let X be a linear space and let Y be a real complete linear space. Then a mapping F : X --t Y, is called quartic, if the new quartic 1991 Mathematics Subject Classification. 39B.