Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition
This paper deals with the following boundary value problem Dαut=ft,ut,t∈0,1,u0=u1=Dα−3u0=u′1=0, where 3<α≤4,Dα is the Riemann-Liouville fractional derivative, and the nonlinearity f, which could be singular at both t=0 and t=1, is required to be continuous on 0,1×ℝ satisfying a mild Lipschitz ass...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2021-01-01
|
| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/6666015 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832547020615712768 |
|---|---|
| author | Imed Bachar Habib Mâagli Hassan Eltayeb |
| author_facet | Imed Bachar Habib Mâagli Hassan Eltayeb |
| author_sort | Imed Bachar |
| collection | DOAJ |
| description | This paper deals with the following boundary value problem Dαut=ft,ut,t∈0,1,u0=u1=Dα−3u0=u′1=0, where 3<α≤4,Dα is the Riemann-Liouville fractional derivative, and the nonlinearity f, which could be singular at both t=0 and t=1, is required to be continuous on 0,1×ℝ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying ut≤cωt,for t∈0,1 and c>0, where ωt≔tα−21−t2. |
| format | Article |
| id | doaj-art-85b67eb7802c44138e44ffad87189550 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-85b67eb7802c44138e44ffad871895502025-02-03T06:46:15ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/66660156666015Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz ConditionImed Bachar0Habib Mâagli1Hassan Eltayeb2College of Science Mathematics Department, King Saud University, P.O.Box 2455, Riyadh 11451, Saudi ArabiaCollege of Sciences and Arts, Rabigh Campus, Department of Mathematics, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi ArabiaCollege of Science, Mathematics Department, King Saud University, P.O.Box 2455, Riyadh 11451, Saudi ArabiaThis paper deals with the following boundary value problem Dαut=ft,ut,t∈0,1,u0=u1=Dα−3u0=u′1=0, where 3<α≤4,Dα is the Riemann-Liouville fractional derivative, and the nonlinearity f, which could be singular at both t=0 and t=1, is required to be continuous on 0,1×ℝ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying ut≤cωt,for t∈0,1 and c>0, where ωt≔tα−21−t2.http://dx.doi.org/10.1155/2021/6666015 |
| spellingShingle | Imed Bachar Habib Mâagli Hassan Eltayeb Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition Journal of Function Spaces |
| title | Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition |
| title_full | Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition |
| title_fullStr | Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition |
| title_full_unstemmed | Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition |
| title_short | Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition |
| title_sort | existence and uniqueness of solutions for fractional boundary value problems under mild lipschitz condition |
| url | http://dx.doi.org/10.1155/2021/6666015 |
| work_keys_str_mv | AT imedbachar existenceanduniquenessofsolutionsforfractionalboundaryvalueproblemsundermildlipschitzcondition AT habibmaagli existenceanduniquenessofsolutionsforfractionalboundaryvalueproblemsundermildlipschitzcondition AT hassaneltayeb existenceanduniquenessofsolutionsforfractionalboundaryvalueproblemsundermildlipschitzcondition |