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Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings. (English) Zbl 0928.39014
Let \(X\) be a normed linear space and \(Y\) be a real Banach space. The author considers the Ulam stability problem for nonlinear Euler-Lagrange quadratic mappings, that is for the mappings \(Q:X\to Y\) satisfying the following system of functional equations \[ m_1m_2 Q(a_1x_1+ a_2 x_2) +Q(m_2a_2x_1- m_1a_1x_2)= (m_1a_1^2+ m_2a_2^2)\bigl[ m_2 Q(x_1)+ m_1Q(x_2) \bigr], \] for all vectors \((x_1,x_2)\in X^2\) and any fixed pair \((a_1, a_2)\) of reals \(a_i\) and any fixed pair \((m_1,m_2)\) of positive reals \(m_i\) \((i=1,2)\) and \[ m^2_1m_2Q(a_1x)+ m_1Q(m_2a_2x) =m^2_0m_2 Q\left({m_1\over m_0} a_1x\right)+ m_0^2m_1Q \left({m_2\over m_0} a_2x\right), \] with \(m_0={m_1 m_2+1 \over m_0}\) for all \(x\in X\) and fixed above reals \(a_i\) and positive reals \(m_i\). The problem is solved under some special assumptions on the mapping \(Q\). The author solves the problem separately for the two cases \({m_1a^2_1 +m_2a^2_2\over m_0}= :m>1\) and \(m<1\).
Reviewer: M.C.Zdun (Kraków)

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
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