On the superstability of generalized d'Alembert harmonic functions

Iz-iddine EL-Fassi

Abstract


The aim of this paper is to study the superstability problem of the d'Alembert type functional equation

f(x+y+z)+f(x+y+σ(z))+f(x+σ(y)+z)+f(σ(x)+y+z)=4f(x)f(y)f(z)

for all x,y,z ∈ G, where G is an abelian group and σ: G → G is an endomorphism such that σ(σ(x))=x for an unknown function f from G into C or into a commutative semisimple Banach algebra.

Keywords


stability, d’Alembert functional equation

Mathematics Subject Classification (2010)


39B82; 39B52

References


Aoki, Tosio. "On the stability of the linear transformation in Banach spaces." J. Math. Soc. Japan 2 (1950): 64-66.

Badora, Roman. "On the stability of the cosine functional equation." Rocznik Nauk.-Dydakt. Prace Mat. 15 (1998): 5-14.

Badora,Roman, and Roman Ger. "On some trigonometric functional inequalities." In Functional equations—results and advances, 3-15. Vol. 3 of Adv. Math. (Dordr.) Dordrecht: Kluwer Acad. Publ., 2002.

Baker, John A., John W. Lawrence, and Frank A. Zorzitto. "The stability of the equation f(x+y) = f(x)f(y)." Proc. Amer. Math. Soc. 74, no. 2 (1979): 242-246.

Baker, John A. "The stability of the cosine equation." Proc. Amer. Math. Soc. 80, no. 3 (1980): 411-416.

Bouikhalene, Belaid, Elhoucien Elqorachi, and John M. Rassias. "The superstability of d’Alembert’s functional equation on the Heisenberg group." Appl. Math. Lett. 23, no. 1 (2010): 105-109.

Bourgin, D. G. "Approximately isometric and multiplicative transformations on continuous function rings." Duke Math. J. 16 (1949): 385-397.

Brzdęk, Janusz, Włodzimierz Fechner, Mohammad Sal Moslehian, and Justyna Sikorska. "Recent developments of the conditional stability of the homomorphism equation." Banach J. Math. Anal. 9, no. 3, (2015): 278-326.

Gavruta, Pascu. "A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings." J. Math. Anal. Appl. 184, no. 3, (1994): 431-436.

Hyers, Donald H. "On the stability of the linear functional equation." Proc. Nat. Acad. Sci. U. S. A. 27 (1941): 222-224.

Kim, Gwang Hui, and Sang Han Lee. "Stability of the d’Alembert type functional equations." Nonlinear Funct. Anal. Appl. 9, no. 4 (2004): 593-604.

Kim, Gwang Hui. "The stability of d’Alembert and Jensen type functional equations." J. Math. Anal. Appl. 325, no. 1 (2007): 237-248.

Kim, Gwang Hui. "The stability of pexiderized cosine functional equations." Korean J. Math. 16, no. 1 (2008): 103-114.

Kim, Hark-Mahn, Gwang Hui Kim and Mi Hyun Han. "Superstability of approximate d’Alembert harmonic functions." J. Ineq. Appl. 2011, no. 1 (2011): 118.

Rassias, John M. "On approximation of approximately linear mappings by linear mappings." J. Funct. Anal. 46, no. 1 (1982): 126-130.

Rassias, Themistocles M. "On the stability of the linear mapping in Banach spaces." Proc. Amer. Math. Soc. 72, no. 2 (1978): 297-300.

Sinopoulos, Pavlos. "Functional equations on semigroups." Aequationes Math. 59, no. 3 (2000): 255-261.

Székelyhidi, László. "The stability of d’Alembert-type functional equations." Acta Sci. Math. (Szeged) 44, no. 3-4 (1982): 313-320.

Ulam, Stanisław M. "A collection of mathematical problems." Vol 8 of Interscience Tracts in Pure and Applied Mathematics. New York-London: Interscience Publishers, 1960.


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