Rassias, Matina John; Rassias, John Michael On the Ulam stability for Euler-Lagrange type quadratic functional equations. (English) Zbl 1094.39027 Aust. J. Math. Anal. Appl. 2, No. 1, Article 11, 10 p. (2005). 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. Reviewer: Mohammad Sal Moslehian (Leeds) Cited in 14 Documents MSC: 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges Keywords:Banach space; Ulam stability; direct method; Euler–Lagrange type quadratic functional equation Citations:Zbl 1017.39011 PDFBibTeX XMLCite \textit{M. J. Rassias} and \textit{J. M. Rassias}, Aust. J. Math. Anal. Appl. 2, No. 1, Article 11, 10 p. (2005; Zbl 1094.39027)