On the stability of the functional equation \(f(x+y+xy)=f(x)+f(y)+xf(y)+yf(x)\) (Q1856878)

The authors solve the functional equation (f.e.) \[ L(f;x,y): =f(x+y+xy) -f(x)-f(y)- xf(y)-yf (x)=0 \] and investigate the Hyers-Ulam stability on the interval \((-1,0]\) and the superstability on the interval \([0,\infty)\) of this equation. For the solution the authors prove the following theorem. Let \(X\) be a real (or complex) linear space. A function \(f:(-1,\infty) \to X\) satisfies the cited f.e. for all \(x\in(-1, \infty)\) if and only if there exists a solution \(D:(0, \infty)\to X\) of the f.e. which defines multiplicative derivations in algebras, \[ f(xy)=xf(y) +yf(x), \] such that \(f(x)=D(x+1)\) for all \(x\in (-1,\infty)\). For the Hyers-Ulam stability they prove the theorem: If \(f:(-1,0]\to X\) satisfies the inequality \(\|L(f;x,y) \|\leq\delta\), then there exists a solution \(H:(-1,0]\to X\) of the f.e. \(L(f;x,y)=0\) such that \(\|f(x)-H(x) \|\leq (He)\delta\), \(\forall x\in(-1,0]\). In what concerns the superstability: If \(f:[0, \infty)\to X\) satisfies the inequality \(\|L(f;x,y) \|<\delta\), then \(f\) satisfies the f.e. \(L(f;x,y)=0\) for all \(x,y\in [0,\infty)\).