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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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