G. S. Saluja, Hemant Kumar Nashine, Reena Jain, Rabha W. Ibrahim, Hossam A. Nabwey
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Nabwey","doi":"10.1155/2024/5108481","DOIUrl":null,"url":null,"abstract":"It has been shown that the findings of <span><svg height=\"9.49473pt\" style=\"vertical-align:-0.2063999pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 7.34169 9.49473\" width=\"7.34169pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>-</span>metric spaces may be deduced from <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 6.25863 8.8423\" width=\"6.25863pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-84\"></use></g></svg>-</span>metric spaces by considering <span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 23.324 11.5564\" width=\"23.324pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-101\"></use></g><g transform=\"matrix(.013,0,0,-0.013,7.215,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,11.713,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,20.36,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"25.4531838 -9.28833 23.25 11.5564\" width=\"23.25pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,25.503,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,32.992,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,41.122,0)\"></path></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"52.3341838 -9.28833 25.02 11.5564\" width=\"25.02pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,52.384,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,61.295,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,65.793,0)\"><use xlink:href=\"#g113-250\"></use></g><g transform=\"matrix(.013,0,0,-0.013,74.44,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"79.5331838 -9.28833 11.61 11.5564\" width=\"11.61pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,79.583,0)\"><use xlink:href=\"#g113-250\"></use></g><g transform=\"matrix(.013,0,0,-0.013,88.229,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"11.5564pt\" style=\"vertical-align:-2.26807pt\" version=\"1.1\" viewbox=\"93.3221838 -9.28833 12.431 11.5564\" width=\"12.431pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,93.372,0)\"><use xlink:href=\"#g113-249\"></use></g><g transform=\"matrix(.013,0,0,-0.013,100.861,0)\"><use xlink:href=\"#g113-42\"></use></g></svg>.</span></span> In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 6.25863 8.8423\" width=\"6.25863pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-84\"></use></g></svg>-</span>metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper’s findings generalize and expand a number of previously published conclusions. In addition, the abstract conclusions are supported by an application of the Riemann-Liouville calculus to a fractional integral problem and a supportive numerical example.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Common Fixed Point Theorems on -Metric Spaces for Integral Type Contractions Involving Rational Terms and Application to Fractional Integral Equation\",\"authors\":\"G. S. Saluja, Hemant Kumar Nashine, Reena Jain, Rabha W. Ibrahim, Hossam A. Nabwey\",\"doi\":\"10.1155/2024/5108481\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It has been shown that the findings of <span><svg height=\\\"9.49473pt\\\" style=\\\"vertical-align:-0.2063999pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 7.34169 9.49473\\\" width=\\\"7.34169pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g></svg>-</span>metric spaces may be deduced from <span><svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 6.25863 8.8423\\\" width=\\\"6.25863pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-84\\\"></use></g></svg>-</span>metric spaces by considering <span><svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 23.324 11.5564\\\" width=\\\"23.324pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-101\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,7.215,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,11.713,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,20.36,0)\\\"></path></g></svg><span></span><svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"25.4531838 -9.28833 23.25 11.5564\\\" width=\\\"23.25pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,25.503,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,32.992,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,41.122,0)\\\"></path></g></svg><span></span><svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"52.3341838 -9.28833 25.02 11.5564\\\" width=\\\"25.02pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,52.384,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,61.295,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,65.793,0)\\\"><use xlink:href=\\\"#g113-250\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,74.44,0)\\\"><use xlink:href=\\\"#g113-45\\\"></use></g></svg><span></span><svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"79.5331838 -9.28833 11.61 11.5564\\\" width=\\\"11.61pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,79.583,0)\\\"><use xlink:href=\\\"#g113-250\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,88.229,0)\\\"><use xlink:href=\\\"#g113-45\\\"></use></g></svg><span></span><span><svg height=\\\"11.5564pt\\\" style=\\\"vertical-align:-2.26807pt\\\" version=\\\"1.1\\\" viewbox=\\\"93.3221838 -9.28833 12.431 11.5564\\\" width=\\\"12.431pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,93.372,0)\\\"><use xlink:href=\\\"#g113-249\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,100.861,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g></svg>.</span></span> In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete <span><svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 6.25863 8.8423\\\" width=\\\"6.25863pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-84\\\"></use></g></svg>-</span>metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper’s findings generalize and expand a number of previously published conclusions. 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Common Fixed Point Theorems on -Metric Spaces for Integral Type Contractions Involving Rational Terms and Application to Fractional Integral Equation
It has been shown that the findings of -metric spaces may be deduced from -metric spaces by considering . In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete -metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper’s findings generalize and expand a number of previously published conclusions. In addition, the abstract conclusions are supported by an application of the Riemann-Liouville calculus to a fractional integral problem and a supportive numerical example.
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