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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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