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We prove some stability and hyperstability results for the well-known Fréchet equation stemming from one of the characterizations of the inner product spaces. As the main tool, we use a fixed point theorem for the function spaces. We finish the paper with some new inequalities characterizing the inner product spaces.

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Abstract. Let A be a subgroup of a commutative group (G, +) and P be a commutative ring. We give the full description of functions g : G → P satisfyingThus we obtain a family of functions depicting evolutions of quite arbitrary functions g 0 : G → P into cosine functions g : G → P , i.e., solutions of the d'Alembert (cosine) functional equationWe also show that every function g : G → P , fulfilling (A), is a solution of (B) if and only if A = G. Mathematics Subject Classification (2010). 39B52, 39B55.

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