Superstability of the d'Alembert functional equation in L_p^+ spaces



Abstract

Let (X,+,-,0,Σ,μ) be an abelian complete measurable group with μ(X)>0. Let f: X→ℂ be a function. We will show that if A(f)∈Lp+(X×X,ℂ) where
A{f)(x,y) = f{x+y) + f(x-y) - 2f(x)f(y),   x,yX,
then fLp+(X,ℂ) or there exists exactly one function g: X→ℂ with
g{x+y) + g(x-y) - 2g(x)g(y),   x,yX
such that f is equal to g almost everywhere with respect to the measure μ.
Lp+ denotes the space of all functions for which the upper integral of ∥fp is finite.


1. J. Aczél, J. Dhorabres, Functional Equations in Several Variables. Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge 1989.
2. I. Adamaszek, Almost trigonometric functions, Glasnik Mat. 19(39) (1984), 83-104.
3. J.A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), 411-416.
4. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey-London-Singapore-Hong Kong 2002.
5. S. Czerwik, K. Dłutek, Superstability of the equation of quadratic junctionals in L^p spaces, Aequationes Math. 63 (2002), 210-219.
6. S. Czerwik, K. Dłutek, Pexider difference operator in L^p spaces (to appear).
7. P. Gǎvruta, On the stability of some functional equations. In: Th.M. Rassias and J. Tabor (eds), Stability of Mappings of Hyers-Ulam Type, Hadronic Press, Palm Harbor, Florida 1994, 93-98.
8. J. Tabor, Stability of the Cauchy type equation in L_p-norms, Results Math. 32 (1997), 145-158.
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Published : 2004-09-30


PrzybyłaM. J. (2004). Superstability of the d’Alembert functional equation in L_p^+ spaces. Annales Mathematicae Silesianae, 18, 39-47. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14090

Maciej J. Przybyła  maciej_przybyla@bielsko.home.pl
Instytut Matematyki, Politechnika Śląska  Poland



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