Journal of Advances in Mathematics and Computer Science

On the Superstability of a Generalization of the Cosine Equation

D. Zeglami *

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco.

S. Kabbaj

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco.

A. Roukbi

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco.

*Author to whom correspondence should be addressed.

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Abstract

The aim of this paper is to investigate the stability problem for the functional equation:
                                      ƒ(xy)+ƒ(xσ(y))=2g(x)ƒ(y),      x,y∈G                      (E_g,ƒ)
and the superstability of the d'Alembert's equation:
                                      ƒ(xy)+ƒ(xσ(y))=2ƒ(x)ƒ(y),      x,y∈G                      (A)
under the conditions from which the differences of each equation are bounded by φ(x), ψ(x) and min(φ(x),ψ(y)) where G is an arbitrary group, not necessarily abelian, ƒ, g are complex valued functions, φ, ψ are real valued functions and σ is an involution of G.

Keywords: Hyers-Ulam stability, Superstability, d'Alembert equation, Wilson's functional equation.

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