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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
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