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On Ulam Stability of the Inhomogeneous Version of the General Linear Functional Equation

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Abstract

The Ulam stability concerns the following issue: how much an approximate solution to an equation differs from an exact solution to the equation. We prove a general Ulam stability result for the functional equation

$$\begin{aligned} \sum _{i=1}^m A_i f\left( \,\sum _{j=1}^n a_{ij} x_j \right) = D(x_1,\ldots ,x_n), \end{aligned}$$

in the class of functions f mapping a module X, over a commutative ring \({\mathbb {K}}\), into a Banach space Y, where m and n are fixed positive integers, \(a_{ij}\in {\mathbb {K}}\) for every \(i \in \{ 1, \dots , m\}\) and \(j \in \{1, \dots , n\}\), \(A_1,\ldots ,A_m\) are scalars, and the function \(D : X^n \rightarrow Y\) is fixed. In this way we generalize an earlier result of A. Bahyrycz and J. Olko. We also show some interesting consequences of this outcome, including conditions sufficient for the existence of solutions to the equation. Particular cases of the equation that we investigate are for instance the functional equations of Cauchy, Jensen, Jordan–von Neumann, Drygas, Fréchet, Popoviciu, Wright and many others.

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Authors and Affiliations

  1. Faculty of Sciences, Department of Mathematics, Center CeReMAR, Laboratory LMSA, Team GrAAF, Mohammed V University in Rabat, 4 Avenue Ibn Batouta, P.O. Box 1014, , Rabat, Morocco

    Chaimaa Benzarouala

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

    Janusz Brzdęk

  3. Mathematics Department, College of Science, Jouf University, P.O. Box 2014, , Sakaka, Saudi Arabia

    El-sayed El-hady

  4. Basic Science Department, Faculty of Computers and Informatics, Suez Canal University, Ismailia, 41522, Egypt

    El-sayed El-hady

  5. Ecole Normale Supérieure Takaddoum, Department of Mathematics, Center CeReMAR, Laboratory LMSA, Team GrAAF, Mohammed V University in Rabat, P.O. Box 5118, , 10105, Takaddoum, Rabat, Morocco

    Lahbib Oubbi

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  1. Chaimaa Benzarouala
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Benzarouala, C., Brzdęk, J., El-hady, Es. et al. On Ulam Stability of the Inhomogeneous Version of the General Linear Functional Equation. Results Math 78, 76 (2023). https://doi.org/10.1007/s00025-023-01840-7

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