We consider the stability, the superstability and the inverse stability of the functional equations with squares of Cauchy’s, of Jensen’s and of isometry equations and the stability in Ulam-Hyers sense of the alternation of functional equations and of the equation of isometry.

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) $$f(x+y+z)+f(x+y+\sigma (z\left)\right)+f(x+\sigma (y)+z)+f\left(\sigma \right(x)+y+z)=4f\left(x\right)f\left(y\right)f\left(z\right)$$ 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.

In this paper, we study the superstablity problem of the cosine and sine type functional equations: f(xσ(y)a)+f(xya)=2f(x)f(y) $$f\left(x\sigma \right(y\left)a\right)+f\left(xya\right)=2f\left(x\right)f\left(y\right)$$ and f(xσ(y)a)−f(xya)=2f(x)f(y), $$f\left(x\sigma \right(y\left)a\right)-f\left(xya\right)=2f\left(x\right)f\left(y\right),$$ where f : S → ℂ is a complex valued function; S is a semigroup; σ is an involution of S and a is a fixed element in the center of S.