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On the stability of the quadratic functional equation and its applications. (English) Zbl 1009.39025
From 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.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B62 Functional inequalities, including subadditivity, convexity, etc.