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The generalized Hyers--Ulam stability of a class of functional equations

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Summary: We generalize some results concerning Hyers-Ulam stability of functional equations. Our intention is to include a possibly most general class of functional equations, whose proof of stability runs by classical Hyers' method. The idea of this generalization comes from G. L. Forti and Z. Kominek.
Artur Grabiec
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Generalized Dichotomies and Hyers–Ulam Stability

Results in Mathematics, 2023
Consider \[ x^\prime=A(t)x+f(t,x),\,t\geq 0,\tag{1} \] where \(A:\mathbb{R}_+\to \mathbb{R}^{n\times n}\) and \(f:\mathbb{R}_+\times \mathbb{R}^n\to \mathbb{R}^n\) are continuous mappings. It is known that if the corresponding linear differential equation \(x^\prime=A(t)x\) has a uniform exponential dichotomy and \(f(t,x)\) is Lipschitz in \(x ...
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Hyers–Ulam stability of generalized Wilson’s and d’Alembert’s functional equations

Afrika Matematika, 2013
The authors investigate the Hyers-Ulam stability for the following functional equation \[ \sum_{\varphi \in \Phi} \int_K f(xk\varphi(y)k^{-1})\;dw_K(k)=|\Phi|f(x)g(y), \quad x,y \in G, \leqno (E) \] where \(G\) is a locally compact group, \(K\) is a compact subgroup of \(G\), \(w_K\) is the normalized Haar measure of \(K\), \(\Phi\) is a finite group ...
Zeglami, D., Roukbi, A., Kabbaj, S.
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On Hyers–Ulam stability of generalized linear functional equation and its induced Hyers–Ulam programming problem

Aequationes mathematicae, 2015
The Hyers-Ulam stability of a generalized linear functional equation is investigated. The connection between the existence of a solution of a certain inequality and the stability problem is also examined.
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Hyers–Ulam Stability of General Jensen-Type Mappings in Banach Algebras

Results in Mathematics, 2014
Let \(S\) be a connected subset of the circle \(\{v \in \mathbb{C}: |v| = 1 \}\) such that \(1 \in S \neq \{1 \}\). Suppose that \(X\) is a complex algebra, \(Y\) is a complex Banach algebra and \(\alpha, \beta \in \mathbb{R} \setminus \{0 \}\). Assume that \(f: X \to Y\) is a mapping for which there exist a constant \(L < 1\) and a function \(\varphi:
Lu, Gang, Park, Choonkil
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On Hyers--Ulam stability of the generalized Cauchy and Wilson equations

Publicationes Mathematicae Debrecen, 2005
Let \(G\) be a topological group and \(\mu\) a compactly supported measure on \(G\). Moreover, let \(\sigma\) denote a continuous involution on \(G\). The authors consider the following functional equations: \[ \begin{aligned} &\int_G f(xty) d\mu(t)=g(x)f(y),\\ &\int_G f(xty) d\mu(t)+\int_G f(xt\sigma(y)) d\mu(t)=2f(x)g(y).\\ \end{aligned} \] The first
Elqorachi, Elhoucien, Akkouchi, Mohamed
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