Stability of the Euler-Lagrange-Rassias functional equation. (English) Zbl 1113.39034

Pietrzyk, Andrzej

Demonstr. Math. 39, No. 3, 523-530 (2006).

Suppose that \(F\) is a field, \(a_1, a_2\in F\), \(X\) is a linear space over \(F\), \(S\subseteq X\) is non-empty, and \(s \neq 0, 1\) is a real or complex number and \(Y\) is a real or complex Banach space. Under some assumptions on \(S\), the author establishes the stability of the equation \(Q(a_1x+a_2y)+Q(a_2x-a_1y)=s(Q(x)+Q(y))\), where \(Q\) is a mapping from \(S\) into \(Y\). This functional equation is called Euler–Lagrange type in the literature. The author’s result is indeed an extension of that of J.M. Rassias [Demonstr. Math. 29, No. 4, 755–766 (1996; Zbl 0884.47040 )] for the case where \(S=X\) and \(s=a^2+b^2\).

Reviewer: Mohammad Sal Moslehian (Mashhad)

MSC:

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

Keywords:

stability; Euler-Lagrange functional equation; Banach space

Citations:

Zbl 0884.47040