Solution of the Ulam Stability Problem for an Euler Type Quadratic Functional Equation

In 1968 S.M. Ulam proposed the 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 according to P.M. Gruber this kind of problems is of particular interest in probability theory and in the case of functional equations of different types. In 1997 W. Schuster established a new vector quadratic identity on the basis of the well-known Euler type theorem on quadrilaterals: If ABCD is a quadrilateral and M, N are the mid-points of the diagonals AC, BD as well as A′, B′, C′, D′ are the mid-points of the sides AB, BC, CD, DA, then |AB|² + |BC|² + |CD|² + |DA|² = 2|A′C′|² + 2|B′D′|² + 4|MN|². Employing in this paper the above geometric identity we introduce the new Euler type quadratic functional equation

$begin{array}{l}2{[}Q(x_{0} - x_{1}+Q(x_{1}-x_{2})+Q(x_{2}- x_{3})+Q(x_{3}-x_{0}){]}qquad = Q(x_{0}-x_{1}-x_{2}+x_{3})+Q(x_{0} + x_{1}-x_{2}-x_{3})+2Q(x_{0}-x_{1}+ x_{2}-x_{3})end{array}$

for all vectors (x₀, x₁, x₂, x₃) X⁴, with X and Y linear spaces. For every x ∈ R set Q(x) = x². Then the mapping Q : X → Y is quadratic. Note also that if Q : R → R is quadratic, then we have Q(x) = Q(1)x². Besides note that the geometric interpretation of the special example

$begin{array}{l}2{[}(x_{0} - x_{1})^{2}+ (x_{1}-x_{2})^{2}+ (x_{2}-x_{3})^{2}+(x_{3}-x_{0})^{2}{]}qquad = (x_{0}-x_{1}-x_{2} + x_{3})^{2}+(x_{0} + x_{1}-x_{2}-x_{3})^{2} + 2(x_{0}-x_{1}+ x_{2}-x_{3})^{2}end{array}$

leads to the above-mentioned Euler type theorem on quadrilaterals ABCD with position vectors x₀, x₁, x₂, x₃ of vertices A, B, C, D, respectively. Then we solve the Ulam stability problem for the afore-mentioned equation, with non-linear Euler type quadratic mappings Q : X → Y.