On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point Theorems
Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as <inline-formula>&l...
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MDPI AG
2023-09-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/10/1826 |
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| author | Victory Asem Yumnam Mahendra Singh Mohammad Saeed Khan Salvatore Sessa |
| author_facet | Victory Asem Yumnam Mahendra Singh Mohammad Saeed Khan Salvatore Sessa |
| author_sort | Victory Asem |
| collection | DOAJ |
| description | Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Liptschitz mappings and studied not only the modified form of the Lipschitz condition, but also the behavior of a finite number of their iterates. The purpose of this paper is to discuss the various types of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-contractions with cyclic representation that extend the results due to Banach, Kannan, and Chatterjea. Moreover, based on such types of contractions and the property of symmetry, we obtain some related fixed-point results in the setting of metric spaces. Some examples are studied to illustrate the validity of our obtained results. As an application of our results, we establish the existence of the solution to a class of Fredholm integral equations. |
| first_indexed | 2024-03-10T20:51:46Z |
| format | Article |
| id | doaj.art-5e6fec77ff0848768943edd0a56e90a4 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T20:51:46Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-5e6fec77ff0848768943edd0a56e90a42023-11-19T18:17:19ZengMDPI AGSymmetry2073-89942023-09-011510182610.3390/sym15101826On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point TheoremsVictory Asem0Yumnam Mahendra Singh1Mohammad Saeed Khan2Salvatore Sessa3Department of Mathematics, Manipur University, Canchipur 795003, Manipur, IndiaDepartment of Basic Sciences and Humanities, Manipur Institute of Technology, A Constituent College of Manipur University, Takyelpat 795004, Manipur, IndiaDepartment of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa 0208, South AfricaDepartment of Architecture, Federico II Naples University, Via Toledo 402, 80134 Naples, ItalyLipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Liptschitz mappings and studied not only the modified form of the Lipschitz condition, but also the behavior of a finite number of their iterates. The purpose of this paper is to discuss the various types of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-contractions with cyclic representation that extend the results due to Banach, Kannan, and Chatterjea. Moreover, based on such types of contractions and the property of symmetry, we obtain some related fixed-point results in the setting of metric spaces. Some examples are studied to illustrate the validity of our obtained results. As an application of our results, we establish the existence of the solution to a class of Fredholm integral equations.https://www.mdpi.com/2073-8994/15/10/1826fixed point(<i>α</i>,<i>p</i>)-cyclic contraction(<i>α</i>,<i>p</i>)-Kannan-type contraction(<i>α</i>,<i>p</i>)-Chatterjea-type contraction |
| spellingShingle | Victory Asem Yumnam Mahendra Singh Mohammad Saeed Khan Salvatore Sessa On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point Theorems Symmetry fixed point (<i>α</i>,<i>p</i>)-cyclic contraction (<i>α</i>,<i>p</i>)-Kannan-type contraction (<i>α</i>,<i>p</i>)-Chatterjea-type contraction |
| title | On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point Theorems |
| title_full | On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point Theorems |
| title_fullStr | On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point Theorems |
| title_full_unstemmed | On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point Theorems |
| title_short | On (<i>α</i>,<i>p</i>)-Cyclic Contractions and Related Fixed Point Theorems |
| title_sort | on i α i i p i cyclic contractions and related fixed point theorems |
| topic | fixed point (<i>α</i>,<i>p</i>)-cyclic contraction (<i>α</i>,<i>p</i>)-Kannan-type contraction (<i>α</i>,<i>p</i>)-Chatterjea-type contraction |
| url | https://www.mdpi.com/2073-8994/15/10/1826 |
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