Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces
The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display=&qu...
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MDPI AG
2023-06-01
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| Online Access: | https://www.mdpi.com/2075-1680/12/6/549 |
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| author | Safeer Hussain Khan Abdullah Eqal Al-Mazrooei Abdul Latif |
| author_facet | Safeer Hussain Khan Abdullah Eqal Al-Mazrooei Abdul Latif |
| author_sort | Safeer Hussain Khan |
| collection | DOAJ |
| description | The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-contractions and enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-contractions. We make a first ever attempt to prove existence results for enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings and deduce the result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings. Note that even <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples. |
| first_indexed | 2024-03-11T02:46:44Z |
| format | Article |
| id | doaj.art-5f29d6c490004b159923d9770d39b3ba |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T02:46:44Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj.art-5f29d6c490004b159923d9770d39b3ba2023-11-18T09:16:37ZengMDPI AGAxioms2075-16802023-06-0112654910.3390/axioms12060549Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function SpacesSafeer Hussain Khan0Abdullah Eqal Al-Mazrooei1Abdul Latif2Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54000, PakistanDepartment of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaThe idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-contractions and enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-contractions. We make a first ever attempt to prove existence results for enriched <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings and deduce the result for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings. Note that even <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples.https://www.mdpi.com/2075-1680/12/6/549fixed pointenriched ρ-contractionenriched ρ-Kannan mappingiterative processmodular function space |
| spellingShingle | Safeer Hussain Khan Abdullah Eqal Al-Mazrooei Abdul Latif Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces Axioms fixed point enriched ρ-contraction enriched ρ-Kannan mapping iterative process modular function space |
| title | Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces |
| title_full | Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces |
| title_fullStr | Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces |
| title_full_unstemmed | Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces |
| title_short | Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces |
| title_sort | banach contraction principle type results for some enriched mappings in modular function spaces |
| topic | fixed point enriched ρ-contraction enriched ρ-Kannan mapping iterative process modular function space |
| url | https://www.mdpi.com/2075-1680/12/6/549 |
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