Hyperstability of the generalized polynomial functional equation of degree 5 (Q305834)

Let \(X\) be a normed space and \(Y\) be a Banach space. For a function \(f:X\to Y\) and \(x\in X\) denote \[ \Delta_h f(x):=f(x+h)-f(x)\quad (h \in X) \] and \[ \Delta_h^m f(x):=\Delta_h^{m-1}(\Delta_hf(x))\quad (x,h \in X). \] A function \(f:X\to Y\) satisfying the condition \[ \Delta_h^{m+1}f(x)=0 \] for every \(x,h\in X\) is called a polynomial function of degree \(m\) \((m\in\mathbb N)\). A function \(f:X\to Y\) is called an \(\varepsilon_r\)-approximately polynomial function of degree \(m\) if there exist real numbers \(\varepsilon>0\) and \(r\) such that \[ \|\Delta_h^{m+1} f(x)\|\leq\varepsilon(\| x\|^r +\| y\|^r) \] for all \(x,y\in X\) if \(r\geq 0\) and all \(x,y\in X\setminus\{0\}\) if \(r<0\). The author proves the following results. If \(f: X \to Y\) is an \(\varepsilon_r\)-approximately polynomial function of degree 5, \(f(0)=0\) and \(r<0\), then \(f\) is a polynomial mapping of degree 5. If \(f:X\to Y\) is an \(\varepsilon_r\)-approximately polynomial function of degree 5, \(r\) is nonnegative and different from \(1,2,3,4\) and \(5\), then there exists a unique polynomial mapping \(P:X\to Y\) of degree 5 with \(P(0)=f(0)\) such that \[ \| f(x)-P(x)\|\leq\delta_r(x) \] for all \(x\in X\), where \(\delta_r(x)\) is defined by the formula \[ \delta_r(x)=\left[(628+331\cdot 2^r+7\cdot 4^r)\left(\frac{1}{75|2-2^r|}+\frac{1}{60|2^3-2^r|}+\frac{1}{300|2^5-2^r|}\right)\right.+ \] \[ \left.(4 + 2^r+3{r+1})\left(\frac{1}{|2^2-2^r|}+\frac{1}{|2^4-2^r|}\right)\right]\frac{\varepsilon \| x\|^r}{12}. \] Similar results for \(\varepsilon_r\)-approximately polynomial functions of degree 2 and 3 were proved by \textit{Y.-H. Lee} [``On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2'', J. Changcheong Math. Soc. 22, No. 2, 201--209 (2009); Tamsui Oxf. J. Math. Sci. 24, No. 4, 429--444 (2008; Zbl 1175.39021)].