All solutions of the equation f(x)+∑i=1naif(x+ρiy)+∑j=1lbjf(σjy)=0\documentclass[12pt]{minimal}
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\begin{document}$${f(x)+\sum_{i=1}^na_if(x+\rho_iy)+\sum_{j=1}^lb_jf(\sigma_jy)=0}$$\end{document} are generalized polynomials of degree at most n. The general solution heavily depends on the parameters ai,ρi,bj,σj\documentclass[12pt]{minimal}
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\begin{document}$${a_i, \rho_i, b_j, \sigma_j}$$\end{document}. Here the stability of this equation is investigated, i. e., for given suitable φ\documentclass[12pt]{minimal}
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\begin{document}$${\varphi}$$\end{document} the inequality ‖f(x)+∑i=1naif(x+ρiy)+∑j=1lbjf(σjy)‖≤φ(x,y)\documentclass[12pt]{minimal}
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\begin{document}$${\Vert f(x)+\sum_{i=1}^na_if(x+\rho_iy)+\sum_{j=1}^lb_jf(\sigma_jy)\Vert\le\varphi(x,y)}$$\end{document} is considered. The method does not seem to standard: At first it is shown that f is “close” to some generalized polynomial P of degree at most n; and then it is shown that P is a solution of the equation above. In this context it is not necessary to know all solutions of the equation. Even more, there is no need to decide whether the equation has non-trivial solutions or not.
