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On the stability of the quadratic mapping in Banach modules. (English) Zbl 1017.39010

Let B be a unital Banach algebra with norm |·|, B 1 ={aB:|a|=1} and let B M 1 , B M 2 be left Banach B-modules. A quadratic mapping Q: B M 1 B M 2 is called B-quadratic if Q(ax)=a 2 Q(x) for all aB, x B M 1 . Let φ: B M 1 × B M 2 [0,) be a function such that one of the series n=1 2 -2n φ(2 n-1 x,2 n-1 x) and n=1 2 2n-2 φ(2 -n x,2 -n x) converges for every x B M 1 . Denote by φ ˜(x) the sum of the convergent series. The following theorem is proved:

Theorem. Let f: B M 1 B M 2 be a mapping such that f(0)=0 and

f (ax+ay) + f (ax-ay) - 2 a 2 f (x) - 2 a 2 f (y)φ(x,y)

for all aB 1 and all x,y B M 1 . If f(tx) is continuous in t for each fixed x B M 1 , then there exists a unique B-quadratic mapping Q: B M 1 B M 2 such that f ( x ) - Q ( x )φ ˜(x) for all x B M 1 .

The similar results are obtained for the other functional equations:

f(ax+y)+f(ax-y)=2a 2 f(x)+2f(y),a 2 f(x+y)+a 2 f(x-y)=2f(ax)+2f(ay),f(ax+ay)+f(ax-ay)=2a 2 g(x)+2a 2 g(y),a 2 f(x+y)+a 2 f(x-y)=2g(ax)+2g(ay)

and for the classical quadratic functional equation.


MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
47B48Operators on Banach algebras
47H99Nonlinear operators
46H25Normed modules and Banach modules, topological modules