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Baire category theorem and approximation of the Pexider quadratic functional equation on a set of Lebesgue measure zero

Baire category theorem and approximation of the Pexider quadratic functional equation on a set of... Let ℝ be the set of real numbers and Y be a Banach space. We investigate the Hyers- Ulam stability theorem when f , g ∶ ℝ → Y satisfy the following Pexider quadratic inequality ‖f (x + y)+ f (x − y)− 2g(x)− 2f (y)‖ ≤ , in a set Ω ⊂ ℝ of Lebesgue measure m(Ω) = 0. Keywords Pexider quadratic functional equation · Stability · First category Lebesgue measure · Baire category theorem · Banach space. Mathematics Subject Classification Primary 39B82 · 39B52 · 46Bxx 1 Introduction In 1940, Ulam [30] gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems and among those the following question concerning the stability of homomorphisms: Let G be a group and let (G , d) be a metric group. Given 𝜖> 0, does there exist 1 2 𝛿> 0 such that if a mapping h ∶ G → G satisfies the inequality 1 2 d(h(xy), h(x)h(y)) ≤ * Iz-iddine EL-Fassi Izidd-math@hotmail.fr Abdellatif Chahbi ab_1980@live.fr Samir Kabbaj samkabbaj@yahoo.fr Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 133, Kenitra, Morocco Vol.:(0123456789) 1 3 Iz. EL-Fassi et al. for all x, y ∈ G http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

Baire category theorem and approximation of the Pexider quadratic functional equation on a set of Lebesgue measure zero

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Forum D'Analystes, Chennai
Subject
Mathematics; Analysis; Functional Analysis; Abstract Harmonic Analysis; Special Functions; Fourier Analysis; Measure and Integration
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-018-0112-7
Publisher site
See Article on Publisher Site

Abstract

Let ℝ be the set of real numbers and Y be a Banach space. We investigate the Hyers- Ulam stability theorem when f , g ∶ ℝ → Y satisfy the following Pexider quadratic inequality ‖f (x + y)+ f (x − y)− 2g(x)− 2f (y)‖ ≤ , in a set Ω ⊂ ℝ of Lebesgue measure m(Ω) = 0. Keywords Pexider quadratic functional equation · Stability · First category Lebesgue measure · Baire category theorem · Banach space. Mathematics Subject Classification Primary 39B82 · 39B52 · 46Bxx 1 Introduction In 1940, Ulam [30] gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems and among those the following question concerning the stability of homomorphisms: Let G be a group and let (G , d) be a metric group. Given 𝜖> 0, does there exist 1 2 𝛿> 0 such that if a mapping h ∶ G → G satisfies the inequality 1 2 d(h(xy), h(x)h(y)) ≤ * Iz-iddine EL-Fassi Izidd-math@hotmail.fr Abdellatif Chahbi ab_1980@live.fr Samir Kabbaj samkabbaj@yahoo.fr Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 133, Kenitra, Morocco Vol.:(0123456789) 1 3 Iz. EL-Fassi et al. for all x, y ∈ G

Journal

The Journal of AnalysisSpringer Journals

Published: Jun 27, 2018

References