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A Functional Equation for the Cosine on Semigroups with Endomorphisms

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Abstract

Let S be a semigroup, and let \(\mathbb {F}\) be a quadratically closed field of characteristic ≠ 2 with identity element 1. We describe, in terms of multiplicative functions of S, the solutions \(f:S\rightarrow \mathbb {F}\) of the new functional equation

$$ f(x\phi (y))+\mu (y)f(\varphi (y)x)=2f(x)f(y),\quad x,y\in S, $$

where \(\phi ,\varphi :S\rightarrow S\) are two endomorphisms that need not be involutive and \(\mu :S\rightarrow \mathbb {F}\) is a multiplicative map such that μ(xφ(x)) = 1 for all xS. Significant consequences of this result are presented.

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References

  1. Aczél, J.: Lectures on Functional Equations and Their Applications. Mathematics in Science and Engineering. Academic Press, New York (1966)

    Google Scholar 

  2. D’Alembert, J.: Addition au mémoire sur la courbe que forme une corde tendue mise en vibration. Hist. Acad. Berlin 6, 355–360 (1750)

    Google Scholar 

  3. Akkaoui, A., El Fatini, M., Fadli, B.: A variant of d’Alembert’s functional equation on semigroups with endomorphisms. Ann. Math. Sil. 36, 1–14 (2022)

    MathSciNet  Google Scholar 

  4. Ayoubi, M., Zeglami, D.: D’alembert’s functional equations on monoids with an anti-endomorphism. Results Math. 75, 74 (2020)

    Article  MathSciNet  Google Scholar 

  5. Ayoubi, M., Zeglami, D., Mouzoun, A.: D’alembert’s functional equation on monoids with both an endomorphism and an anti-endomorphism. Publ. Math. Debr. 99, 413–429 (2021)

    Article  MathSciNet  Google Scholar 

  6. Ayoubi, M., Zeglami, D.: The algebraic small dimension lemma with an anti-homomorphism on semigroups. Results Math. 76, 66 (2021)

    Article  MathSciNet  Google Scholar 

  7. Ayoubi, M., Zeglami, D.: A variant of d’Alembert’s functional equation on semigroups with an anti-endomorphism. Aequat. Math. 96, 549–565 (2022)

    Article  MathSciNet  Google Scholar 

  8. Ayoubi, M., Zeglami, D.: D’alembert μ-functions on semigroups. Arch. Math. 118, 239–245 (2022)

    Article  MathSciNet  Google Scholar 

  9. Davison, T.M.K.: D’alembert’s functional equation on topological monoids. Publ. Math. Debr. 75, 41–66 (2009)

    Article  MathSciNet  Google Scholar 

  10. Ebanks, B.: A Fully pexiderized variant of d’Alembert’s functional equations on monoids. Results Math. 76, 17 (2021)

    Article  MathSciNet  Google Scholar 

  11. Fadli, B., Kabbaj, S., Sabour, Kh., Zeglami, D.: Functional equations on semigroups with an endomorphism. Acta. Math. Hungar. 150, 363–371 (2016)

    Article  MathSciNet  Google Scholar 

  12. Fadli, B., Zeglami, D., Kabbaj, S.: An integral functional equation on groups under two measures. Proyecc. (Antofagasta) 37, 565–581 (2018)

    Article  MathSciNet  Google Scholar 

  13. Kannappan, Pl.: The functional equation f(xy) + f(xy− 1) = 2f(x)f(y) for groups. Proc. Amer. Math. Soc. 19, 69–74 (1968)

    MathSciNet  Google Scholar 

  14. Kannappan, Pl.: A functional equation for the cosine. Can. Math. Bull. 11, 495–498 (1968)

    Article  Google Scholar 

  15. Sabour, Kh., Fadli, B., Kabbaj, S.: Wilson’s functional equation on monoids with involutive automorphisms. Aequat. Math. 90, 189–196 (2016)

    Article  MathSciNet  Google Scholar 

  16. Sabour, Kh., Charifi, A., Kabbaj, S.: On a variant of μ-Wilson’s functional equation with an endomorphism. In: Anastassiou, G. A., Rassias, J. M. (eds.) Frontiers in Functional Equations and Analytic Inequalities, pp 93–111. Springer, Cham (2019)

  17. Stetkær, H.: D’alembert’s functional equation on groups. Banach Cent. Publ. 99, 173–191 (2013)

    Article  MathSciNet  Google Scholar 

  18. Stetkær, H.: Functional Equations on Groups. World Scientific Publishing Company, Singapore (2013)

    Book  Google Scholar 

  19. Stetkær, H.: A variant of d’Alembert’s functional equation. Aequat. Math. 89, 657–662 (2015)

    Article  MathSciNet  Google Scholar 

  20. Stetkær, H.: The small dimension lemma and d’Alembert’s equation on semigroups. Aequat. Math. 95, 281–299 (2021)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank the referee for his helpful comments which improved the presentation of these results.

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Correspondence to Mohamed Ayoubi.

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Ayoubi, M., Zeglami, D. A Functional Equation for the Cosine on Semigroups with Endomorphisms. Vietnam J. Math. 52, 149–157 (2024). https://doi.org/10.1007/s10013-022-00587-y

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