Hyers-Ulam stability of the quadratic equation of Pexider type (Q2731060)

The following theorem is proved: NEWLINENEWLINENEWLINELet \(E_1\) be a real normed space and~\(E_2\) be a~Banach space. If the functions \(f_1,f_2,f_3,f_4:E_1\to E_2\) satisfy the inequality \(\bigl\|f_1(x+y)+f_2(x-y)-f_3(x)-f_4(y)\bigr\|\leq\varepsilon\) for some \(\varepsilon\geq 0\) and for all \(x,y\in E_1\), then there exists a~unique quadratic function \(Q:E_1\to E_2\) and exactly two additive functions \(A_1,A_2:E_1\to E_2\) such that NEWLINE\[NEWLINE \begin{aligned} \bigl\|f_1(x)-Q(x)-A_1(x)-A_2(x)-f_1(0) \bigr\|&\leq{137\over 3}\varepsilon,\\ \bigl\|f_2(x)-Q(x)-A_1(x)+A_2(x)-f_2(0) \bigr\|&\leq{125\over 3}\varepsilon,\\ \bigl\|f_3(x)-2Q(x)-2A_1(x)-f_3(0) \bigr\|&\leq{136\over 3}\varepsilon,\\ \bigl\|f_4(x)-2Q(x)-2A_2(x)-f_4(0) \bigr\|&\leq{124\over 3}\varepsilon \end{aligned} NEWLINE\]NEWLINE for all \(x\in E_1\). Moreover, if \(f_3(tx)\) and \(f_4(tx)\) are continuous in~\(t\in\mathbb{R}\) for each \(x\in E_1\), then \(Q(tx)=t^2Q(x)\) for all \(x\in E_1\) and \(A_1,A_2\) are linear.