Remarks on the stability of some quadratic functional equations

Abstract

Stability problems concerning the functional equations of the form f(2x + y) = 4f(x) + f(y) + f(x + y) − f(x − y), and f(2x + y) + f(2x − y) = 8f(x) + 2f(y) are investigated. We prove that if the norm of the difference between the LHS and the RHS of one of equations (1) or (2), calculated for a function g is say, dominated by a function ' in two variables having some standard properties then there exists a unique solution f of this equation and the norm of the difference between g and f is controlled by a function depending on '.