The Stability of Functional Equations with a New Direct Method

Dongwen Zhang, Qi Liu, John Michael Rassias and Yongjin Li
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Dongwen Zhang: Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
Qi Liu: Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China
John Michael Rassias: Department of Mathematics and Informatics, National and Kapodistrian University of Athens, 15342 Attikis, Greece
Yongjin Li: Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

Mathematics, 2022, vol. 10, issue 7, 1-14

Abstract: We investigate the Hyers–Ulam stability of an equation involving a single variable of the form ∥ f ( x ) − α f ( k n ( x ) ) − β f ( k n + 1 ( x ) ) ∥ ⩽ u ( x ) where f is an unknown operator from a nonempty set X into a Banach space Y , and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables.

Keywords: Hyers–Ulam stability; functional equations; approximation; the direct method; the convergence series (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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