Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras
Badora (2002) proved the following stability result. Let 𝜀 and 𝛿 be nonnegative real numbers, then for every mapping 𝑓 of a ring ℛ onto a Banach algebra ℬ satisfying ||𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)||≤𝜀 and ||𝑓(𝑥⋅𝑦)−𝑓(𝑥)𝑓(𝑦)||≤𝛿 for all 𝑥,𝑦∈ℛ, there exists a unique ring homomorphism ℎ∶ℛ→ℬ such that ||𝑓(𝑥)−ℎ(𝑥)||≤...
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/653508 |
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| author | Faruk Polat |
| author_facet | Faruk Polat |
| author_sort | Faruk Polat |
| collection | DOAJ |
| description | Badora (2002) proved the following stability result. Let 𝜀 and 𝛿 be nonnegative real numbers, then for every mapping 𝑓 of a ring ℛ onto a Banach algebra ℬ satisfying ||𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)||≤𝜀 and ||𝑓(𝑥⋅𝑦)−𝑓(𝑥)𝑓(𝑦)||≤𝛿 for all 𝑥,𝑦∈ℛ, there exists a unique ring homomorphism ℎ∶ℛ→ℬ such that ||𝑓(𝑥)−ℎ(𝑥)||≤𝜀,𝑥∈ℛ. Moreover, 𝑏⋅(𝑓(𝑥)−ℎ(𝑥))=0,(𝑓(𝑥)−ℎ(𝑥))⋅𝑏=0, for all 𝑥∈ℛ and all 𝑏 from the algebra generated by ℎ(ℛ). In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms. |
| format | Article |
| id | doaj-art-aba610c55646494e8f1702fa8457629e |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-aba610c55646494e8f1702fa8457629e2025-02-03T06:08:26ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/653508653508Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz AlgebrasFaruk Polat0Department of Mathematics, Firat University, 23119 Elazig, TurkeyBadora (2002) proved the following stability result. Let 𝜀 and 𝛿 be nonnegative real numbers, then for every mapping 𝑓 of a ring ℛ onto a Banach algebra ℬ satisfying ||𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)||≤𝜀 and ||𝑓(𝑥⋅𝑦)−𝑓(𝑥)𝑓(𝑦)||≤𝛿 for all 𝑥,𝑦∈ℛ, there exists a unique ring homomorphism ℎ∶ℛ→ℬ such that ||𝑓(𝑥)−ℎ(𝑥)||≤𝜀,𝑥∈ℛ. Moreover, 𝑏⋅(𝑓(𝑥)−ℎ(𝑥))=0,(𝑓(𝑥)−ℎ(𝑥))⋅𝑏=0, for all 𝑥∈ℛ and all 𝑏 from the algebra generated by ℎ(ℛ). In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.http://dx.doi.org/10.1155/2012/653508 |
| spellingShingle | Faruk Polat Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras Abstract and Applied Analysis |
| title | Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras |
| title_full | Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras |
| title_fullStr | Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras |
| title_full_unstemmed | Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras |
| title_short | Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras |
| title_sort | some generalizations of ulam hyers stability functional equations to riesz algebras |
| url | http://dx.doi.org/10.1155/2012/653508 |
| work_keys_str_mv | AT farukpolat somegeneralizationsofulamhyersstabilityfunctionalequationstorieszalgebras |