Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup>
In a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/1/157 |
| _version_ | 1827597449033678848 |
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| author | Mohamed A. Khamsi Osvaldo D. Méndez |
| author_facet | Mohamed A. Khamsi Osvaldo D. Méndez |
| author_sort | Mohamed A. Khamsi |
| collection | DOAJ |
| description | In a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mrow><mi>p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> obtained under the assumptions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mo>+</mo></msub><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfied by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> will force <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mo>−</mo></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Therefore, the conclusion is well known. In this note, we establish said conclusion without the assumption <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mo>+</mo></msub><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T03:31:32Z |
| format | Article |
| id | doaj.art-74a2c2ed5b5a49a99bb102b994e00059 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T03:31:32Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-74a2c2ed5b5a49a99bb102b994e000592023-12-03T14:54:58ZengMDPI AGMathematics2227-73902022-12-0111115710.3390/math11010157Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup>Mohamed A. Khamsi0Osvaldo D. Méndez1Department of Applied Mathematics and Sciences, Khalifa University, Abu Dhabi 127788, United Arab EmiratesDepartment of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USAIn a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mrow><mi>p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> obtained under the assumptions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mo>+</mo></msub><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfied by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> will force <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mo>−</mo></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Therefore, the conclusion is well known. In this note, we establish said conclusion without the assumption <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mo>+</mo></msub><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/11/1/157electrorheological fluidfixed pointmodular strict convexitymodular vector spacemodular uniform convexityNakano modular |
| spellingShingle | Mohamed A. Khamsi Osvaldo D. Méndez Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup> Mathematics electrorheological fluid fixed point modular strict convexity modular vector space modular uniform convexity Nakano modular |
| title | Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup> |
| title_full | Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup> |
| title_fullStr | Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup> |
| title_full_unstemmed | Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup> |
| title_short | Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i><sup><i>p</i>(·)</sup> |
| title_sort | remark on a fixed point theorem in the lebesgue spaces of variable integrability i l i sup i p i · sup |
| topic | electrorheological fluid fixed point modular strict convexity modular vector space modular uniform convexity Nakano modular |
| url | https://www.mdpi.com/2227-7390/11/1/157 |
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