Measure Zero Stability Problem for Drygas Functional Equation with Complex Involution

Hyperstability for a Generalized Class of Pexiderized Functional Equations on Monoids via Páles’ Approach

“…In 2008, Jung and Lee [29] applied the fixed point method to prove the stability of quadratic functional equations with involutions for a large class of functions. The study of functional equations with involutions has continued to attract the attention of numerous researchers, and the results have been applied in a wide range of mathematical fields, e.g., [30][31][32][33][34][35]. The hyperstability study of this type of functional equations began in 2016 when Almahalebi [25] investigated the hyperstability of σ-Drygas functional equations with an involution.…”

“…Let ϕ : M × M → [0, ∞) be a function and consider the existence of sequence {a n } n∈N in M such that Conditions(34) and(35) hold. Let f : M → G and ψ : M × M → G be functions such thatd f xσ(y) + f xτ(y) , 2 f (x) + f σ(y) + f τ(y) + ψ(x, y) ≤ ϕ(x, y), x, y ∈ M. Suppose that functional equation f xσ(y) + f xτ(y) = 2 f (x) + f σ(y) + f τ(y) + ψ(x, y) admits solution f 0 : M → G. Then, 1.f is a solution of equationf xσ(y) + f xτ(y) + 2 f (e) = 2 f (x) + f σ(y) + f τ(y) + ψ(x, y), x, y ∈ M. 2.f is a solution of equationf xy + f xτ(y) = 2 f (x) + f σ(y) + f τ(y) + ψ(x, y), x, y ∈ M,if and only if f (e) = 0.When Γ = {id M , σ} and 2g(y) ≡ f y + f σ(y) , we notice that Equation (1) becomes the following σ-Drygas functional equation: f xy + f xσ(y) = 2 f (x) + f y + f σ(y) , x, y ∈ M.…”