On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems
This article deals with a class of nonlinear fractional differential equations, with initial conditions, involving the Riemann–Liouville fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics>&...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-01-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/2/297 |
| _version_ | 1797439043209789440 |
|---|---|
| author | Luís P. Castro Anabela S. Silva |
| author_facet | Luís P. Castro Anabela S. Silva |
| author_sort | Luís P. Castro |
| collection | DOAJ |
| description | This article deals with a class of nonlinear fractional differential equations, with initial conditions, involving the Riemann–Liouville fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The main objectives are to obtain conditions for the existence and uniqueness of solutions (within appropriate spaces), and to analyze the stabilities of Ulam–Hyers and Ulam–Hyers–Rassias types. In fact, different conditions for the existence and uniqueness of solutions are obtained based on the analysis of an associated class of fractional integral equations and distinct fixed-point arguments. Additionally, using a Bielecki-type metric and some additional contractive arguments, conditions are also obtained to guarantee Ulam–Hyers and Ulam–Hyers–Rassias stabilities for the problems under analysis. Examples are also included to illustrate the theory. |
| first_indexed | 2024-03-09T11:46:04Z |
| format | Article |
| id | doaj.art-0a7bb22b09da4d56b782e886d38dcea8 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:46:04Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-0a7bb22b09da4d56b782e886d38dcea82023-11-30T23:20:14ZengMDPI AGMathematics2227-73902023-01-0111229710.3390/math11020297On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value ProblemsLuís P. Castro0Anabela S. Silva1CIDMA—Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, PortugalCIDMA—Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, PortugalThis article deals with a class of nonlinear fractional differential equations, with initial conditions, involving the Riemann–Liouville fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The main objectives are to obtain conditions for the existence and uniqueness of solutions (within appropriate spaces), and to analyze the stabilities of Ulam–Hyers and Ulam–Hyers–Rassias types. In fact, different conditions for the existence and uniqueness of solutions are obtained based on the analysis of an associated class of fractional integral equations and distinct fixed-point arguments. Additionally, using a Bielecki-type metric and some additional contractive arguments, conditions are also obtained to guarantee Ulam–Hyers and Ulam–Hyers–Rassias stabilities for the problems under analysis. Examples are also included to illustrate the theory.https://www.mdpi.com/2227-7390/11/2/297fractional differential equationsRiemann–Liouville derivativefixed point theoryUlam–Hyers stabilityUlam–Hyers–Rassias stability |
| spellingShingle | Luís P. Castro Anabela S. Silva On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems Mathematics fractional differential equations Riemann–Liouville derivative fixed point theory Ulam–Hyers stability Ulam–Hyers–Rassias stability |
| title | On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems |
| title_full | On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems |
| title_fullStr | On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems |
| title_full_unstemmed | On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems |
| title_short | On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems |
| title_sort | on the existence and stability of solutions for a class of fractional riemann liouville initial value problems |
| topic | fractional differential equations Riemann–Liouville derivative fixed point theory Ulam–Hyers stability Ulam–Hyers–Rassias stability |
| url | https://www.mdpi.com/2227-7390/11/2/297 |
| work_keys_str_mv | AT luispcastro ontheexistenceandstabilityofsolutionsforaclassoffractionalriemannliouvilleinitialvalueproblems AT anabelassilva ontheexistenceandstabilityofsolutionsforaclassoffractionalriemannliouvilleinitialvalueproblems |