On the stability of the quadratic functional equation and its applications.

*(English)*Zbl 1009.39025From the introduction: To quote S. M. Ulam, for very general functional equations, one can ask the following question. When is it true that the solution of an equation differing slightly from a given one, must necessarily be close to the solution of the given equation? Similarly, if we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality lie near the solutions of the equation?

The present paper will provide a solution to Ulam’s problem for the case of the quadratic functional equation.

The quadratic functional equation \[ f(x+y)+f(x-y)-2f(x)-2f(y)=0 \] clearly has \(f(x)=cx^2\) as a solution with \(c\) an arbitrary constant when \(f\) is a real function of a real variable. We are interested in functions \(f: E_1\to E_2\) where both \(E_1\) and \(E_2\) are real vector spaces.

The present paper will provide a solution to Ulam’s problem for the case of the quadratic functional equation.

The quadratic functional equation \[ f(x+y)+f(x-y)-2f(x)-2f(y)=0 \] clearly has \(f(x)=cx^2\) as a solution with \(c\) an arbitrary constant when \(f\) is a real function of a real variable. We are interested in functions \(f: E_1\to E_2\) where both \(E_1\) and \(E_2\) are real vector spaces.