Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation
We prove new results on Ulam stability of the nonhomogeneous Cauchy functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><...
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MDPI AG
2022-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/10/1695 |
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| author | El-sayed El-hady Janusz Brzdęk |
| author_facet | El-sayed El-hady Janusz Brzdęk |
| author_sort | El-sayed El-hady |
| collection | DOAJ |
| description | We prove new results on Ulam stability of the nonhomogeneous Cauchy functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the class of mappings <i>f</i> from a square symmetric groupoid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</mo><mo>+</mo><mo>)</mo></mrow></semantics></math></inline-formula> into the set of reals <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>. The mapping <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>:</mo><msup><mi>H</mi><mn>2</mn></msup><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> is assumed to be given and satisfy some weak natural assumption. The equation arises naturally, e.g., in the theory of information in a description of generating functions of branching measures of information. Moreover, we provide a suitable example of application of our results in this area at the very end of this paper. The main tool used in the proofs is the Banach limit. |
| first_indexed | 2024-03-10T03:28:57Z |
| format | Article |
| id | doaj.art-709774f129434c2a9ca55bc405da7e69 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:28:57Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-709774f129434c2a9ca55bc405da7e692023-11-23T12:01:04ZengMDPI AGMathematics2227-73902022-05-011010169510.3390/math10101695Banach Limit and Ulam Stability of Nonhomogeneous Cauchy EquationEl-sayed El-hady0Janusz Brzdęk1Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi ArabiaFaculty of Applied Mathematics, AGH University of Science and Technology, Mickiewicza 30, 30-059 Kraków, PolandWe prove new results on Ulam stability of the nonhomogeneous Cauchy functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the class of mappings <i>f</i> from a square symmetric groupoid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</mo><mo>+</mo><mo>)</mo></mrow></semantics></math></inline-formula> into the set of reals <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">R</mi></semantics></math></inline-formula>. The mapping <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>:</mo><msup><mi>H</mi><mn>2</mn></msup><mo>→</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> is assumed to be given and satisfy some weak natural assumption. The equation arises naturally, e.g., in the theory of information in a description of generating functions of branching measures of information. Moreover, we provide a suitable example of application of our results in this area at the very end of this paper. The main tool used in the proofs is the Banach limit.https://www.mdpi.com/2227-7390/10/10/1695Banach limitUlam stabilitynonhomogeneous Cauchy functional equation |
| spellingShingle | El-sayed El-hady Janusz Brzdęk Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation Mathematics Banach limit Ulam stability nonhomogeneous Cauchy functional equation |
| title | Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation |
| title_full | Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation |
| title_fullStr | Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation |
| title_full_unstemmed | Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation |
| title_short | Banach Limit and Ulam Stability of Nonhomogeneous Cauchy Equation |
| title_sort | banach limit and ulam stability of nonhomogeneous cauchy equation |
| topic | Banach limit Ulam stability nonhomogeneous Cauchy functional equation |
| url | https://www.mdpi.com/2227-7390/10/10/1695 |
| work_keys_str_mv | AT elsayedelhady banachlimitandulamstabilityofnonhomogeneouscauchyequation AT januszbrzdek banachlimitandulamstabilityofnonhomogeneouscauchyequation |