Solution of the Ulam Stability Problem for Euler]Lagrange Quadratic 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. U. Ulam proposed the more 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]1996 we solved the above Ulam problem, or equivalently the Ulam type problem for linear mappings and established analogous stability problems. In this paper we first introduce new quadratic weighted means and fundamental functional equations and then solve the Ulam stability problem for non-linear Euler]Lagrange quadratic mappings Q: XªY, satisfying a mean equation and functional equation m1m2Q a1x1qa2x2.qQ m2a2x1ym1a1x2. s m1a12qm2a22.wm2Q x1.qm1Q x2.x for all 2-dimensional vectors x1, x2.gX2, with X a normed linear space Y[ a real complete normed linear space., and any fixed pair a1, a2. of reals ai and any fixed pair m1, m2.of positive reals mi is1, 2., 0-ms mm11mq2qm21 m1a12qm2a22..