Stability of a mixed type additive and quadratic functional equation in random normed spaces (Q2917786)

The so-called additive-quadratic functional equation: NEWLINE\[NEWLINE f(3x+y)+f(3x-y)=f(x+y)+f(x-y)+2f(3x)-2f(x)\tag{*} NEWLINE\]NEWLINE is considered and its general solution is given for mappings between linear spaces. Namely, \(f\) satisfies (*) if and only if \(f\) is a sum of an additive and quadratic mappings.NEWLINENEWLINEThen the stability of (*) is proved for mappings from a linear space into a complete random normed space. It is proved that an even (resp. odd) approximate solution of (*) can be approximated by a unique quadratic (resp. additive) mapping. Moreover, an arbitrary approximate solution of (*) can be uniquely approximated by a sum of a quadratic and additive mappings. Here, approximation and approximate solutions are defined in terms of random normed spaces.