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D’Alembert’s Functional Equation and Superstability Problem in Hypergroups

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Handbook of Functional Equations

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

Abstract

Our main goal is to determine the continuous and bounded complex valued solutions of the functional equation

$$ \langle \delta_{x}\ast \delta_{y},g\rangle +\langle \delta_{x}\ast \delta_{\check{y}},g\rangle =2~g(x)g(y),\;x,y\in X,$$

where X is a hypergroup. The solutions are expressed in terms of 2 -dimensional representations of X. The papers of Davison [10] and Stetkaer [25, 26] are the essential motivation for this first part of the present work and the methods used here are closely related to and inspired by those in [10, 25, 26]. In addition, superstability problem for this functional equation on any hypergroup and without any condition on f is considered.

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Acknowledgement

Our sincere regards and gratitude go to Professor Henrik Stetkær for fruitful discussions and for sending us some of his papers on functional equations.

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Authors and Affiliations

  1. Department of Mathematics, E.N.S.A.M., Moulay Ismail University, B.P.: 15290, Al Mansour, Meknes, Morocco

    D. Zeglami

  2. Department of Mathematics, Ibn Tofail University, 14000, Kenitra, Morocco

    A. Roukbi

  3. Department of Mathematics, National Technical University of Athens, Zofrafou Campus, 15780, Athens, Greece

    Themistocles M. Rassias

Authors
  1. D. Zeglami
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  2. A. Roukbi
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  3. Themistocles M. Rassias
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Corresponding author

Correspondence to D. Zeglami .

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Editors and Affiliations

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

    Themistocles M. Rassias

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Zeglami, D., Roukbi, A., Rassias, T. (2014). D’Alembert’s Functional Equation and Superstability Problem in Hypergroups. 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_17

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