Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation

El-sayed El-hady and Janusz Brzdęk
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El-sayed El-hady: Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia
Janusz Brzdęk: Faculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, Poland

Mathematics, 2022, vol. 10, issue 10, 1-15

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.

Keywords: Banach limit; Ulam stability; nonhomogeneous Cauchy functional equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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