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. 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..
