On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms
This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AGH Univeristy of Science and Technology Press
2020-03-01
|
| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4012.pdf |
| _version_ | 1818613111417995264 |
|---|---|
| author | John R. Graef Said R. Grace Ercan Tunç |
| author_facet | John R. Graef Said R. Grace Ercan Tunç |
| author_sort | John R. Graef |
| collection | DOAJ |
| description | This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained. |
| first_indexed | 2024-12-16T15:56:55Z |
| format | Article |
| id | doaj.art-d778264831474e56b02099a2ef3eab3a |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-16T15:56:55Z |
| publishDate | 2020-03-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-d778264831474e56b02099a2ef3eab3a2022-12-21T22:25:34ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742020-03-01402227239https://doi.org/10.7494/OpMath.2020.40.2.2274012On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative termsJohn R. Graef0https://orcid.org/0000-0002-8149-4633Said R. Grace1https://orcid.org/0000-0001-8783-5227Ercan Tunç2https://orcid.org/0000-0001-8860-608XUniversity of Tennessee at Chattanooga, Department of Mathematics, Chattanooga, TN 37403, USACairo University, Faculty of Engineering, Department of Engineering Mathematics, Orman, Giza 12221, EgyptGaziosmanpasa University, Department of Mathematics, Faculty of Arts and Sciences, 60240, Tokat, TurkeyThis paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4012.pdfintegro-differential equationsfractional differential equationsnonoscillatory solutionsboundednesscaputo derivative |
| spellingShingle | John R. Graef Said R. Grace Ercan Tunç On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms Opuscula Mathematica integro-differential equations fractional differential equations nonoscillatory solutions boundedness caputo derivative |
| title | On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms |
| title_full | On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms |
| title_fullStr | On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms |
| title_full_unstemmed | On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms |
| title_short | On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms |
| title_sort | on the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms |
| topic | integro-differential equations fractional differential equations nonoscillatory solutions boundedness caputo derivative |
| url | https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4012.pdf |
| work_keys_str_mv | AT johnrgraef ontheasymptoticbehaviorofnonoscillatorysolutionsofcertainfractionaldifferentialequationswithpositiveandnegativeterms AT saidrgrace ontheasymptoticbehaviorofnonoscillatorysolutionsofcertainfractionaldifferentialequationswithpositiveandnegativeterms AT ercantunc ontheasymptoticbehaviorofnonoscillatorysolutionsofcertainfractionaldifferentialequationswithpositiveandnegativeterms |