A Purely Fixed Point Approach to the Ulam-Hyers Stability and Hyperstability of a General Functional Equation

Chaimaa Benzarouala and Lahbib Oubbi ()
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Chaimaa Benzarouala: Mohammed V University in Rabat, Department of Mathematics, Team GrAAF, Laboratory LMSA, Center CeReMar, Faculty of Sciences
Lahbib Oubbi: Ecole Normale Supérieure, Mohammed V University in Rabat, Department of Mathematics, Team GrAAF, Laboratory LMSA, Center CeReMar

Chapter Chapter 2 in Ulam Type Stability, 2019, pp 47-56 from Springer

Abstract: Abstract In this paper, using a purely fixed point approach, we produce a new proof of the Ulam-Hyers stability and hyperstability of the general functional equation: ∑ i = 1 m A i f ( ∑ j = 1 n a i j x j ) + A = 0 , ( x 1 , x 2 , … , x n ) ∈ X n , $$\displaystyle \sum _{i=1}^m A_i f(\sum _{j=1}^n a_{ij} x_j) + A = 0,\qquad (x_1, x_2, \dots , x_n) \in X^n, $$ considered in Bahyrycz and Olko (Aequationes Math 89:1461, 2015. https://doi.org/10.1007/s00010-014-0317-z ), and in Bahyrycz and Olko (Aequationes Math 90:527, 2016. https://doi.org/10.1007/s00010-016-0418-y ). Here m and n are positive integers, f is a mapping from a vector space X into a Banach space (Y, ∥ ∥), A ∈ Y and, for every i ∈{1, 2, …, m} and j ∈{1, …, n}, A i and a ij are scalars.

Keywords: Hyers-Ulam stability; Hyperstability; Functional equation; Fixed point theorem; Primary 39B82, 47H14, 47J20; Secondary 39B62, 47H10 (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-28972-0_2

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DOI: 10.1007/978-3-030-28972-0_2

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