Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications
The <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-Hausdorff–Pompeiu b-metric for <inline-formula><math display="inline">&l...
| Asıl Yazarlar: | , |
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| Materyal Türü: | Makale |
| Dil: | English |
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MDPI AG
2020-12-01
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| Seri Bilgileri: | Mathematics |
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| Online Erişim: | https://www.mdpi.com/2227-7390/9/1/12 |
| _version_ | 1827699489788395520 |
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| author | Reny George Hemanth Kumar Pathak |
| author_facet | Reny George Hemanth Kumar Pathak |
| author_sort | Reny George |
| collection | DOAJ |
| description | The <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-Hausdorff–Pompeiu b-metric for <inline-formula><math display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> is introduced as a new variant of the Hausdorff–Pompeiu b-metric <i>H</i>. Various types of multi-valued <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-contractions are introduced and fixed point theorems are proved for such contractions in a b-metric space. The multi-valued Nadler contraction, Czervik contraction, q-quasi contraction, Hardy Rogers contraction, weak quasi contraction and Ciric contraction existing in literature are all one or the other type of multi-valued <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-contraction but the converse is not necessarily true. Proper examples are given in support of our claim. As applications of our results, we have proved the existence of a unique multi-valued fractal of an iterated multifunction system defined on a b-metric space and an existence theorem of Filippov type for an integral inclusion problem by introducing a generalized norm on the space of selections of the multifunction. |
| first_indexed | 2024-03-10T13:50:31Z |
| format | Article |
| id | doaj.art-932e7f1f94e94c549938e1a0a0a8737c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T13:50:31Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-932e7f1f94e94c549938e1a0a0a8737c2023-11-21T02:10:35ZengMDPI AGMathematics2227-73902020-12-01911210.3390/math9010012Some New Extensions of Multivalued Contractions in a b-metric Space and Its ApplicationsReny George0Hemanth Kumar Pathak1Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaSOS in Mathematics, Pt. Ravishankar Shukla University, Raipur 492010, IndiaThe <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-Hausdorff–Pompeiu b-metric for <inline-formula><math display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> is introduced as a new variant of the Hausdorff–Pompeiu b-metric <i>H</i>. Various types of multi-valued <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-contractions are introduced and fixed point theorems are proved for such contractions in a b-metric space. The multi-valued Nadler contraction, Czervik contraction, q-quasi contraction, Hardy Rogers contraction, weak quasi contraction and Ciric contraction existing in literature are all one or the other type of multi-valued <inline-formula><math display="inline"><semantics><msup><mi>H</mi><mi>β</mi></msup></semantics></math></inline-formula>-contraction but the converse is not necessarily true. Proper examples are given in support of our claim. As applications of our results, we have proved the existence of a unique multi-valued fractal of an iterated multifunction system defined on a b-metric space and an existence theorem of Filippov type for an integral inclusion problem by introducing a generalized norm on the space of selections of the multifunction.https://www.mdpi.com/2227-7390/9/1/12b-metric space<i>H<sup>β</sup></i>-Hausdorff–Pompeiu b-metricmulti-valued fractaliterated multifunction systemintegral inclusion |
| spellingShingle | Reny George Hemanth Kumar Pathak Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications Mathematics b-metric space <i>H<sup>β</sup></i>-Hausdorff–Pompeiu b-metric multi-valued fractal iterated multifunction system integral inclusion |
| title | Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications |
| title_full | Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications |
| title_fullStr | Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications |
| title_full_unstemmed | Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications |
| title_short | Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications |
| title_sort | some new extensions of multivalued contractions in a b metric space and its applications |
| topic | b-metric space <i>H<sup>β</sup></i>-Hausdorff–Pompeiu b-metric multi-valued fractal iterated multifunction system integral inclusion |
| url | https://www.mdpi.com/2227-7390/9/1/12 |
| work_keys_str_mv | AT renygeorge somenewextensionsofmultivaluedcontractionsinabmetricspaceanditsapplications AT hemanthkumarpathak somenewextensionsofmultivaluedcontractionsinabmetricspaceanditsapplications |