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

  • Chaimaa Benzarouala
  • Lahbib OubbiEmail author
Chapter

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:
$$\displaystyle \sum _{i=1}^m A_i f(\sum _{j=1}^n a_{ij} x_j) + A = 0,\qquad (x_1, x_2, \dots , x_n) \in X^n, $$
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}, Ai and aij are scalars.

Keywords

Hyers-Ulam stability Hyperstability Functional equation Fixed point theorem 

Mathematics Subject Classification (2010)

Primary 39B82 47H14 47J20; Secondary 39B62 47H10 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Team GrAAF, Laboratory LMSA, Center CeReMar, Faculty of SciencesMohammed V University in RabatRabatMorocco
  2. 2.Department of Mathematics, Team GrAAF, Laboratory LMSA, Center CeReMarEcole Normale Supérieure, Mohammed V University in RabatTakaddoum, RabatMorocco

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