Solution of the Ulam stability problem for cubic mappings (Q2748016)

A functional equation NEWLINE\[NEWLINE f(x+2y)+3f(x)=3f(x+y)+f(x-y)+6f(y)\tag{1} NEWLINE\]NEWLINE is introduced. The following theorem is proved. Let \(X\) be a normed space, let~\(Y\) be a~real Banach space and let \(c\geq 0\). If \(f:X\to Y\) fulfils the inequality NEWLINE\[NEWLINE\|f(x+2y)+3f(x)-(3f(x+y)+f(x-y)+6f(y))\|\leq c NEWLINE\]NEWLINE for all \(x,y\in X\), then there exists the unique mapping \(C:X\to Y\) satisfying~(1) for all \(x,y\in X\), such that \(\|f(x)-C(x)\|\leq {11\over 42}c\) for all \(x\in X\).