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Hyers-Ulam stability of an additive-quadratic functional equation. (English) Zbl 1455.39007
Summary: In this paper, we introduce the following \((a, b, c)\)-mixed type functional equation of the form \[ \begin{split} g(ax_1+bx_2+cx_3)-g(-ax_1+bx_2+cx_3) +g(ax_1-bx_2+cx_3)-g(ax_1+bx_2-cx_3) +\\ 2a^2[g(x_1) +g(-x_1)] + 2b^2[g(x_2) +g(-x_2)] + 2c^2[g(x_3) +g(-x_3)] +a[g(x_1)-g(-x_1)] +\\ b[g(x_2)-g(-x_2)] +c[g(x_3)-g(-x_3)] = 4g(ax_1+cx_3) + 2g(-bx_2) + 2g(bx_2) \end{split} \] where \(a\), \(b\), \(c\) are positive integers with \(a >1\), and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods.
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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