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Remarks on Stability of the Equation of Homomorphism for Square Symmetric Groupoids

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Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 96))

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Abstract

Let \((G,\star)\) and \((H,\circ)\) be square symmetric groupoids and \(S\subset G\) be nonempty. We present some remarks on stability of the following conditional equation of homomorphism

$$f(x\star y)=f(x)\circ f(y) \qquad x,y\in S, x\star y\in S\;,$$

in the class of functions mapping S into H. In particular, we consider the situation where \(H=\mathbb{R}\) and

$$-\nu(x,y)\le h(x\star y)-h(x)\circ h(y) \le \mu(x,y) \qquad x,y\in S, x\star y\in S\;,$$

with some functions \(\mu,\nu:S^2\to [0,\infty)\).

Mathematics Subject Classification (2010) 39B22, 39B52, 39B82.

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Authors and Affiliations

  1. Department of Mathematics, Pedagogical University, Podchora¸żych 2, 30-084, Kraków, Poland

    Anna Bahyrycz & Janusz Brzdȩk

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  1. Anna Bahyrycz
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  2. Janusz Brzdȩk
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Correspondence to Anna Bahyrycz .

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Bahyrycz, A., Brzdȩk, J. (2014). Remarks on Stability of the Equation of Homomorphism for Square Symmetric Groupoids. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 96. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1286-5_2

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