On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order
In this manuscript we introduce a quadratic integral equation of the Urysohn type of fractional variable order. The existence and uniqueness of solutions of the proposed fractional model are studied by transforming it into an integral equation of fractional constant order. The obtained new results a...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/7/886 |
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| author | Amar Benkerrouche Mohammed Said Souid Gani Stamov Ivanka Stamova |
| author_facet | Amar Benkerrouche Mohammed Said Souid Gani Stamov Ivanka Stamova |
| author_sort | Amar Benkerrouche |
| collection | DOAJ |
| description | In this manuscript we introduce a quadratic integral equation of the Urysohn type of fractional variable order. The existence and uniqueness of solutions of the proposed fractional model are studied by transforming it into an integral equation of fractional constant order. The obtained new results are based on the Schauder’s fixed-point theorem and the Banach contraction principle with the help of piece-wise constant functions. Although the used methods are very powerful, they are not applied to the quadratic integral equation of the Urysohn type of fractional variable order. With this research we extend the applicability of these techniques to the introduced the Urysohn type model of fractional variable order. The applicability of the new results are demonstrated by providing Ulam–Hyers stability criteria and an example. Moreover, the presented results lead to future progress and expansion of the theory of fractional-order models, as well as of the concept of entropy in the framework of fractional calculus. Further, an example is constructed to demonstrate the reasonableness and effectiveness of the observed results. |
| first_indexed | 2024-03-09T03:27:39Z |
| format | Article |
| id | doaj.art-acb135ec38fe426ca77887540e1a2b83 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T03:27:39Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-acb135ec38fe426ca77887540e1a2b832023-12-03T15:00:07ZengMDPI AGEntropy1099-43002022-06-0124788610.3390/e24070886On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable OrderAmar Benkerrouche0Mohammed Said Souid1Gani Stamov2Ivanka Stamova3Department of Mathématiques, Ziane Achour University of Djelfa, Djelfa 17000, AlgeriaDepartment of Economic Sciences, University of Tiaret, Tiaret 14035, AlgeriaDepartment of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USADepartment of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USAIn this manuscript we introduce a quadratic integral equation of the Urysohn type of fractional variable order. The existence and uniqueness of solutions of the proposed fractional model are studied by transforming it into an integral equation of fractional constant order. The obtained new results are based on the Schauder’s fixed-point theorem and the Banach contraction principle with the help of piece-wise constant functions. Although the used methods are very powerful, they are not applied to the quadratic integral equation of the Urysohn type of fractional variable order. With this research we extend the applicability of these techniques to the introduced the Urysohn type model of fractional variable order. The applicability of the new results are demonstrated by providing Ulam–Hyers stability criteria and an example. Moreover, the presented results lead to future progress and expansion of the theory of fractional-order models, as well as of the concept of entropy in the framework of fractional calculus. Further, an example is constructed to demonstrate the reasonableness and effectiveness of the observed results.https://www.mdpi.com/1099-4300/24/7/886fractional derivativevariable-orderfixed-point theoremquadratic integral equationpiece-wise constant functionsuryshon-type integral equations |
| spellingShingle | Amar Benkerrouche Mohammed Said Souid Gani Stamov Ivanka Stamova On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order Entropy fractional derivative variable-order fixed-point theorem quadratic integral equation piece-wise constant functions uryshon-type integral equations |
| title | On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order |
| title_full | On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order |
| title_fullStr | On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order |
| title_full_unstemmed | On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order |
| title_short | On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order |
| title_sort | on the solutions of a quadratic integral equation of the urysohn type of fractional variable order |
| topic | fractional derivative variable-order fixed-point theorem quadratic integral equation piece-wise constant functions uryshon-type integral equations |
| url | https://www.mdpi.com/1099-4300/24/7/886 |
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