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On new stability results for generalized Cauchy functional equations on groups by using Brzdȩk’s fixed point theorem

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Abstract

In this work we present a new type of stability results for generalized Cauchy functional equation of the form

$$f(ax \ast by) = af(x) \diamond bf(y),$$

where \({a, b \in \mathbb{N}}\) and \({f}\) is a mapping from a commutative semigroup (\({G_1, \ast}\)) to a commutative group (\({G_2, \diamond}\)). Using this form we generalize, extend and complement some earlier classical results concerning the stability of additive Cauchy functional equations. Our results are improvement and generalization of main results of Brzdȩk [Fixed Point Theory Appl. 2013 (2013), doi:10.1186/1687-1812-2013-285:285] and many results in literature. Some of the stability results for many types of functional equations are given here to illustrate the usability of the obtained results.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand

    Laddawan Aiemsomboon & Wutiphol Sintunavarat

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  1. Laddawan Aiemsomboon
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  2. Wutiphol Sintunavarat
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Correspondence to Wutiphol Sintunavarat.

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Aiemsomboon, L., Sintunavarat, W. On new stability results for generalized Cauchy functional equations on groups by using Brzdȩk’s fixed point theorem. J. Fixed Point Theory Appl. 18, 45–59 (2016). https://doi.org/10.1007/s11784-015-0259-7

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  • DOI: https://doi.org/10.1007/s11784-015-0259-7

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