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On the stability problem for a mixed type of quartic and quadratic functional equation. (English) Zbl 1106.39027

The problem “If we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality must be close to the solutions of the given equation?” is the essence of Hyers-Ulam-Rassias stability theory; cf. Th. M. Rassias [Acta Appl. Math. 62, No. 1, 23–130 (2000; Zbl 0981.39014)].

For a mapping f:E 1 E 2 between real vector spaces, let us define x 2 f(x 1 ) to be f(x 1 +x 2 )+f(x 1 -x 2 ) and x 2 ,,x n+1 n f(x 1 )= x n+1 ( x 2 ,,x n n-1 f(x 1 )) (n).

In the paper under review, the author determines the general solution for the mixed type functional equation

x 2 ,,x n n-1 f(x 1 )+2 n-1 (n-2) i=1 n f(x i )=2 n-2 1i<jn x j f(x j ),

and proves its Hyers-Ulam-Rassias stability by using the Hyers type sequences; see Th. M. Rassias [J. Math. Anal. Appl. 158, No. 1, 106–113 (1991; Zbl 0746.46038)].

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges