Representation and Stability of General Nonic Functional Equation

Chang, Ick-Soon; Lee, Yang‐Hi; Roh, Jaiok

doi:10.3390/math11143173

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2024

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“…We refer, for example, to [10][11][12][13][14][15][16]. In addition, there are dozens of published papers that are concerned with the hyperstability of functional equations; see, for instance, [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

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

Asharabi,

Almahalebi

2024

Mathematics

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In this paper, we deduce some hyperstability results for a generalized class of homogeneous Pexiderized functional equations, expressed as ∑ρ∈Γfxρ.y=ℓf(x)+ℓg(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∈Γfxρ.y=ℓf(x)+ℓg(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.

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Section: Introductionmentioning

confidence: 99%

“…Taking g(y) ≡ ∑ ρ∈Γ f (ρ.y) for all y ∈ M in Theorem 2 implies that f is a solution to Equation (20). From the -cancellativity of G, we determine that f is a solution to Equation (19) if and only if f (e) = 0.…”

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confidence: 99%