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Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings. (English) Zbl 1125.39025
Let $$X,Y$$ be Banach spaces, $$f:X\to Y$$, $$a_1,a_2$$ fixed nonzero reals, $$m_1,m_2$$ fixed positive reals and $$m_0=\frac{m_1m_2+1}{m_1+m_2}$$. The authors consider the generalized Euler-Lagrange equation:
$\begin{split} m_1m_2f(a_1x_1+a_2x_2)+f(m_2a_2x_1-m_1a_1x_2)\\ =(m_1a_1^2+m_2a_2^2)(m_2f(x_1) +m_1f(x_2)),\quad x_1,x_2\in X\end{split}\tag{1}$ as well as the fundamental Euler-Lagrange functional equation:
$m_1^2m_2f(a_1x)+m_1f(m_2a_2x)=m_0^2m_2f\left(\frac{m_1}{m_0}a_1x\right) +m_0^2m_1f\left(\frac{m_2}{m_0}a_2x\right),\quad x\in X.\tag{2}$ A nonlinear mapping $$f:X\to Y$$ satisfying both the above equations is called generalized Euler-Lagrange quadratic. The aim of the paper is to prove the stability of such mappings, i.e., to show that if $$f$$ satisfies the equations (1) and (2) approximately only (in some precise sense), then it can be approximated by a mapping $$Q$$ which satisfies them exactly. Several results are formulated, also in the realm of quasi-Banach spaces.

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