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Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation

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  • El-sayed El-hady

    (Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia
    Basic Science Department, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt)

  • Janusz Brzdęk

    (Faculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, Poland)

Abstract

We prove new results on Ulam stability of the nonhomogeneous Cauchy functional equation f ( x + y ) = f ( x ) + f ( y ) + d ( x , y ) in the class of mappings f from a square symmetric groupoid ( H , + ) into the set of reals R . The mapping d : H 2 → R is assumed to be given and satisfy some weak natural assumption. The equation arises naturally, e.g., in the theory of information in a description of generating functions of branching measures of information. Moreover, we provide a suitable example of application of our results in this area at the very end of this paper. The main tool used in the proofs is the Banach limit.

Suggested Citation

  • El-sayed El-hady & Janusz Brzdęk, 2022. "Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation," Mathematics, MDPI, vol. 10(10), pages 1-15, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:10:p:1695-:d:816215
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    References listed on IDEAS

    as
    1. Soon-Mo Jung, 2011. "Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis," Springer Optimization and Its Applications, Springer, number 978-1-4419-9637-4, December.
    2. Zbigniew Gajda, 1991. "On stability of additive mappings," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 14, pages 1-4, January.
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