Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums
The eardrum is one of the most important organs in the body, and disorders such as infection or injury may affect the proper functioning of the eardrum and lead to hearing problems. In this paper, based on a real-world phenomena, we study some mathematical aspects of an abstract fractional $ [\matht...
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AIMS Press
2025-02-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2025108 |
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| author | Reny George Sina Etemad Ivanka Stamova Raaid Alubady |
| author_facet | Reny George Sina Etemad Ivanka Stamova Raaid Alubady |
| author_sort | Reny George |
| collection | DOAJ |
| description | The eardrum is one of the most important organs in the body, and disorders such as infection or injury may affect the proper functioning of the eardrum and lead to hearing problems. In this paper, based on a real-world phenomena, we study some mathematical aspects of an abstract fractional $ [\mathtt{p},\mathtt{q}] $-difference equation with initial conditions. Our initial value problem tries to model a vibrating eardrum by using the newly defined fractional Caputo-type $ [\mathtt{p},\mathtt{q}] $-derivatives in two nonlinear single-valued and set-valued structures. We obtain a general form of the solutions in the framework of a $ [\mathtt{p},\mathtt{q}] $-integral equation, and then we investigate the existence and uniqueness properties with the help of fixed points and the end-points of some special $ \mathtt{β} $-$ \mathtt{α} $-contractions and compact mappings. Finally, we simulate this version of the vibrating eardrum model by giving two numerical examples to validate the established theorems. |
| format | Article |
| id | doaj-art-9d12ad9f11a84baeae2e5343411a1d62 |
| institution | OA Journals |
| issn | 2473-6988 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-9d12ad9f11a84baeae2e5343411a1d622025-08-20T02:08:20ZengAIMS PressAIMS Mathematics2473-69882025-02-011022321234610.3934/math.2025108Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrumsReny George0Sina Etemad1Ivanka Stamova2Raaid Alubady3Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi ArabiaDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, IranDepartment of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USAEngineering Technical College, Al-Ayen Iraqi University, Thi-Qar, IraqThe eardrum is one of the most important organs in the body, and disorders such as infection or injury may affect the proper functioning of the eardrum and lead to hearing problems. In this paper, based on a real-world phenomena, we study some mathematical aspects of an abstract fractional $ [\mathtt{p},\mathtt{q}] $-difference equation with initial conditions. Our initial value problem tries to model a vibrating eardrum by using the newly defined fractional Caputo-type $ [\mathtt{p},\mathtt{q}] $-derivatives in two nonlinear single-valued and set-valued structures. We obtain a general form of the solutions in the framework of a $ [\mathtt{p},\mathtt{q}] $-integral equation, and then we investigate the existence and uniqueness properties with the help of fixed points and the end-points of some special $ \mathtt{β} $-$ \mathtt{α} $-contractions and compact mappings. Finally, we simulate this version of the vibrating eardrum model by giving two numerical examples to validate the established theorems.https://www.aimspress.com/article/doi/10.3934/math.2025108set-valued functionfixed-pointend-point$ [\mathtt{p},\mathtt{q}] $-derivativeinitial problemvibrating eardrum model |
| spellingShingle | Reny George Sina Etemad Ivanka Stamova Raaid Alubady Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums AIMS Mathematics set-valued function fixed-point end-point $ [\mathtt{p},\mathtt{q}] $-derivative initial problem vibrating eardrum model |
| title | Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums |
| title_full | Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums |
| title_fullStr | Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums |
| title_full_unstemmed | Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums |
| title_short | Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums |
| title_sort | existence of solutions for mathtt p mathtt q difference initial value problems application to the mathtt p mathtt q based model of vibrating eardrums |
| topic | set-valued function fixed-point end-point $ [\mathtt{p},\mathtt{q}] $-derivative initial problem vibrating eardrum model |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025108 |
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