# zbMATH — the first resource for mathematics

On the Hyers-Ulam-Rassias stability of a general cubic functional equation. (English) Zbl 1032.39015
Let $$X$$ be a real vector space and let $$Y$$ be a real Banach space. For a function $$f:X\to Y$$ the difference operator is given by $$\Delta_yf(x)=f(x+y)-f(x)$$. Let $$\Delta_y^1f(x)=\Delta_yf(x)$$, $$\Delta_y^{k+1}f(x)=\Delta_y\bigl(\Delta_y^kf(x)\bigr)$$. A general solution of the functional equation $$\Delta_y^nf(x)=0$$ was given by D. Z. Djoković [Ann. Polon. Math. 22, 189-198 (1969; Zbl 0187.39903)] and by S. Mazur and W. Orlicz [Studia Math. 5, 50-68, 179-189 (1935; Zbl 0013.21002)]. Observe that $$\Delta_y^4f(x)=f(x+4y)-4f(x+3y)+6f(x+2y)-4f(x+y)+f(x)$$. By replacing $$x+2y$$ by $$x$$ the functional equation $$\Delta_y^4f(x)=0$$ can be written in the form $f(x+2y)+f(x-2y)+6f(x)=4f(x+y)+4f(x-y).\tag{1}$ This functional equation is considered in the present paper. Its general solution is reproved. The main purpose of the paper is the investigation of the Hyers-Ulam-Rassias stability problem for the equation (1) (for stability results concerning the equation $$\Delta_y^nf(x)=0$$ cf. M. A. Albert and J. A. Baker [Ann. Polon. Math. 43, 93-103 (1983; Zbl 0436.39005)] and M. Kuczma [An introduction to the theory of functional equations and inequalities, Cauchy’s equation and Jensen’s inequality (1985; Zbl 0555.39004)]). Namely, the functional inequality $\bigl\|f(x+2y)+f(x-2y)+6f(x)-4f(x+y)-4f(x-y)\bigr\|\leq\phi(x,y)\tag{2}$ is considered for suitably chosen function $$\phi:X^2\to[0,\infty)$$. It is proved that if $$f$$ fulfils (2) and the approximately cubic condition $\bigl\|f(2x)+8f(-x)\bigr\|\leq\delta\tag{3}$ for some $$\delta\geq 0$$, then close to $$f$$ there exists a unique function $$T:X\to Y$$ satisfying (1). The function $$T$$ is given by $$T(x)=\lim_{n\to\infty}\frac{f(3^nx)}{27^n}$$. If (3) is replaced by the approximately quadratic condition $$\bigl\|f(2x)-4f(-x)\bigr\|\leq\delta$$, it is proved that close to $$f$$ there exists a unique quadratic function $$Q:X\to Y$$ given by $$Q(x)=\displaystyle\lim\limits_{n\to\infty}\frac{f(3^nx)}{9^n}$$. If (3) is replaced by the approximately odd condition $$\bigl\|f(2x)+2f(-x)\bigr\|\leq\delta$$, it is proved that close to $$f$$ there exists a unique additive function $$A:X\to Y$$ given by $$A(x)=\lim_{n\to\infty}\frac{f(3^nx)}{3^n}$$. Similar results are proved for mappings between Banach modules over a unital Banach algebra.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 39B72 Systems of functional equations and inequalities
Full Text: