On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings
The purpose of the paper is to establish a sufficient condition for the existence of a solution to the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">T<...
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MDPI AG
2023-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/8/1852 |
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| author | Huaping Huang Subhadip Pal Ashis Bera Lakshmi Kanta Dey |
| author_facet | Huaping Huang Subhadip Pal Ashis Bera Lakshmi Kanta Dey |
| author_sort | Huaping Huang |
| collection | DOAJ |
| description | The purpose of the paper is to establish a sufficient condition for the existence of a solution to the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">T</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi mathvariant="script">C</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>u</mi></mrow></semantics></math></inline-formula> using Kannan-type equicontractive mappings, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">T</mi><mo>:</mo><mi mathvariant="script">A</mi><mo>×</mo><mover><mrow><mi mathvariant="script">C</mi><mo>(</mo><mi mathvariant="script">A</mi><mo>)</mo></mrow><mo>¯</mo></mover><mo>→</mo><mi mathvariant="double-struck">Y</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">C</mi></semantics></math></inline-formula> is a compact mapping, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is a bounded, closed and convex subset of a Banach space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Y</mi></semantics></math></inline-formula>. To achieve this objective, the authors have presented Sadovskii’s theorem, which utilizes the measure of noncompactness. The relevance of the obtained results has been illustrated through the consideration of various initial value problems. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:46:51Z |
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| spelling | doaj.art-8770e74630a0474fa50e194437c1c0102023-11-17T20:17:28ZengMDPI AGMathematics2227-73902023-04-01118185210.3390/math11081852On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive MappingsHuaping Huang0Subhadip Pal1Ashis Bera2Lakshmi Kanta Dey3School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404020, ChinaDepartment of Mathematics, National Institute of Technology Durgapur, Durgapur 713209, IndiaDepartment of Mathematics, School of Advanced Sciences, VIT Chennai, Chennai 600127, IndiaDepartment of Mathematics, National Institute of Technology Durgapur, Durgapur 713209, IndiaThe purpose of the paper is to establish a sufficient condition for the existence of a solution to the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">T</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi mathvariant="script">C</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>u</mi></mrow></semantics></math></inline-formula> using Kannan-type equicontractive mappings, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">T</mi><mo>:</mo><mi mathvariant="script">A</mi><mo>×</mo><mover><mrow><mi mathvariant="script">C</mi><mo>(</mo><mi mathvariant="script">A</mi><mo>)</mo></mrow><mo>¯</mo></mover><mo>→</mo><mi mathvariant="double-struck">Y</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">C</mi></semantics></math></inline-formula> is a compact mapping, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is a bounded, closed and convex subset of a Banach space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Y</mi></semantics></math></inline-formula>. To achieve this objective, the authors have presented Sadovskii’s theorem, which utilizes the measure of noncompactness. The relevance of the obtained results has been illustrated through the consideration of various initial value problems.https://www.mdpi.com/2227-7390/11/8/1852Hausdorff measure of noncompactnesscompact mappingKannan-type equicontractioninitial value problem |
| spellingShingle | Huaping Huang Subhadip Pal Ashis Bera Lakshmi Kanta Dey On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings Mathematics Hausdorff measure of noncompactness compact mapping Kannan-type equicontraction initial value problem |
| title | On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings |
| title_full | On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings |
| title_fullStr | On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings |
| title_full_unstemmed | On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings |
| title_short | On the Extended Version of Krasnoselśkiĭ’s Theorem for Kannan-Type Equicontractive Mappings |
| title_sort | on the extended version of krasnoselskii s theorem for kannan type equicontractive mappings |
| topic | Hausdorff measure of noncompactness compact mapping Kannan-type equicontraction initial value problem |
| url | https://www.mdpi.com/2227-7390/11/8/1852 |
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