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Ulam stability problem for a mixed type of cubic and additive functional equation. (English) Zbl 1132.39022

Summary: It is the aim of this paper to obtain the generalized Hyers-Ulam stability result for a mixed type of cubic and additive functional equation
\[ \begin{split} f\biggl(\biggl( \sum_{i=1}^l x_i\biggr)+ x_{x+1}\biggr)+ f\biggl(\biggl( \sum_{i=1}^l x_i\biggr)- x_{l+1}\biggr)+ 2\sum_{i=1}^l f(x_i)\\ =2f \biggl( \sum_{i=1}^l x_i\biggr)+ \sum_{i=1}^l [f(x_i+x_{l+1})+ f(x_i-x_{l+1})] \end{split} \]
for all \((x_1,\dots, x_l,x_{l+1})\in X^{l+1}\), where \(l\geq 2\).

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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