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Solution of the Ulam stability problem for quartic mappings. (English) Zbl 0951.39008

The author studies the Hyers-Ulam stability of the functional equation

f(x+2y)+f(x-2y)+6f(x)=4f(x+y)+f(x-y)+6f(y)( FE )

using the so called direct method of Hyers. A function F is called a quartic mapping if it satisfies the above functional equation (FE). The author proves the following result: Let X be a normed linear space and Y be a real complete normed linear space. If f:XY satisfies the inequality

f(x+2y)+f(x-2y)+6f(x)-4f(x+y)+f(x-y)+6f(y)ε( FI )

for all x,yX with a constant ε0 (independent of x and y), then there exists a unique quartic function F:XY such that F(x)-f(x)17 180ε. This result is obtained through six lemmas.


MSC:
39B82Stability, separation, extension, and related topics
39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)
39B52Functional equations for functions with more general domains and/or ranges