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

In this paper, we deduce some hyperstability results for a generalized class of homogeneous Pexiderized functional equations, expressed as ∑ ρ ∈ Γ f x ρ . y = ℓ f ( x ) + ℓ g ( y ) , x , y ∈ M , which is inspired by the concept of Ulam stability. Indeed, we prove that function f that approximately satisfies an equation can, under certain conditions, be considered an exact solution. Domain M is a monoid (semigroup with a neutral element), Γ is a finite subgroup of the automorphisms group of M , ℓ is the cardinality of Γ , and f , g : M → G such that ( G , + ) denotes an ℓ -cancellative commutative group. We also examine the hyperstability of the given equation in its inhomogeneous version ∑ ρ ∈ Γ f x ρ . y = ℓ f ( x ) + ℓ g ( y ) + ψ ( x , y ) , x , y ∈ M , where ψ : M × M → G . Additionally, we apply the main results to elucidate the hyperstability of various functional equations with involutions.