Remarks on the stability of additive functional equation (Q2770332)

The following theorem is proved.NEWLINENEWLINENEWLINELet \(E_1\) be a real normed space, \(E_2\) a Banach space. Let \(p\geq 0\), \(p\neq 1\) and let \(H:[0,\infty)\times[0,\infty)\to[0,\infty)\) be a mapping satisfying \(H(tx,ty)\leq t^pH(x,y)\) for all \(t,x,y\geq 0\). Suppose that a function \(f:E_1\to E_2\) satisfies \(\bigl\|f(x+y)-f(x)-f(y)\bigr\|\leq H\bigl(\|x\|,\|y\|\bigr)\) for all \(x,y\in E_1\). Then there exists a unique additive mapping \(T:E_1\to E_2\) such that NEWLINE\[NEWLINE \bigl\|f(x)-T(x)\bigr\|\leq {H\bigl(\|x\|,\|x\|\bigr)\over|2-2^p|}\leq {H(1,1)\over|2-2^p|}\|x^p\|NEWLINE\]NEWLINE for all \(x\in E_1\). Moreover, if for every fixed \(x\in E_1\) there exists a real number \(\delta_x>0\) such that the function \(t\mapsto\bigl\|f(tx)\bigr\|\) is bounded on \([0,\delta_x]\), then \(T\) is linear. NEWLINENEWLINENEWLINEBy an appropriate specification of the function \(H\) earlier results of Th. M. Rassias, Z. Gajda and G. Isac/Th. M. Rassias are obtained as corollaries to the Theorem above.