On the Ulam stability for Euler-Lagrange type quadratic functional equations. (English) Zbl 1094.39027

Using the direct method the authors prove the stability of the Euler–Lagrange type quadratic functional equation \[ Q(m_1a_1x_1 + m_2a_2x_2) + m_1m_2Q(a_2x_1-a_1x_2) = (m_1a_1^2+m_2a_2^2)[m_1Q(x_1)+m_2Q(x_2)] \] where \(Q\) is a mapping from a real normed space \(X\) into a real Banach space \(Y\), \((a_1, a_2)\) is any fixed pair of reals \(a_i \neq 0~(i =1, 2)\), and \((m_1, m_2)\) is any fixed pair of positive reals \(m_i ~(i =1, 2)\) with \(0 < \frac{m_1+m_2}{m_1m_2+1} (m_1a_1^2+m_2a_2^2) \neq 1\). In the statement of the main theorem, the function \(Q\), which authors want to find is assumed.
The paper is an interesting one in a sequel of papers of the second author on stability of Euler–Lagrange type quadratic equations; see J. M. Rassias [Southeast Asian Bull. Math. 26, No.1, 101-112 (2002; Zbl 1017.39011)] and references therein.


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


Zbl 1017.39011