On the stability of a quadratic Jensen type functional equation
For real linear spaces \(X\) and \(Y\), \(f: X\to Y\) satisfies the functional equation \[ 9f\Biggl({x+ y+ z\over 3}\Biggr)+ f(x)+ f(y)+ f(z)= 4\Biggl[f\Biggl({x+ y\over 2}\Biggr)+ f\Biggl({y+ z\over 2}\Biggr)+ f\Biggl({z+ x\over 2}\Biggr)\Biggr] \] if and only if \(f(x)= Q(x)+ A(x)+ B\) for an additive function \(A: X\to Y\), a quadratic function \(Q: X\to Y\), and an element \(B\in Y\). (\(Q\) quadratic means that \(Q(x+ y)+ Q(x- y)= 2Q(x)+ 2Q(y)\) for all \(x,y\in X\).) Hyers-Ulam-Rassias stability of this so-called quadratic Jensen type functional equation is also established.
quadratic Jensen type functional equation
scientific article; zbMATH DE number 1797527