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On stability of additive mappings. (English) Zbl 0739.39013

Let E 1 , E 2 be real normed spaces with E 2 complete, and let p, ε be real numbers with ε0. When f:E 1 E 2 satisfies the inequality f(x+y)-f(x)-f(y)ε(x p +y p ) for all x,yE, it was shown by T. M. Rassias [Proc. Amer. Math. Soc. 72, 299-300 (1978; Zbl 0398.47040)] that there exists a unique additive mapping T:E 1 E 2 such that f(x)-T(x)δx p for all xE 1 , providing that p<1, where δ=2ε/(2-2 p ).

The relationship between f and T was given by the formula T(x)=lim n 2 -n f(2 n x). Rassias also proved that if the mapping from to E 2 given by tf(tx) is continuous for each fixed xE, then T is linear.

In the present paper the author extends these results to the case p>1, but now the additive mapping T is given by T(x)=lim n 2 n f(2 -n x), and the corresponding value of δ is δ=2ε/(2 p -2). The author also gives a counterexample to show that the theorem is false for the case p=1, and any choice of δ>0 when ε>0.

Reviewer: Prof.D.H.Hyers

MSC:
39B72Systems of functional equations and inequalities
39B52Functional equations for functions with more general domains and/or ranges