Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems
This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow>&...
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2025-06-01
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| author | Sotiris K. Ntouyas Bashir Ahmad Jessada Tariboon |
| author_facet | Sotiris K. Ntouyas Bashir Ahmad Jessada Tariboon |
| author_sort | Sotiris K. Ntouyas |
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| description | This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Hilfer and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Caputo fractional derivative operators, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mn>1</mn></msub></semantics></math></inline-formula> and the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. Also the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mn>2</mn></msub></semantics></math></inline-formula>-Riemann–Liouville, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>k</mi><mn>2</mn></msub></semantics></math></inline-formula>-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. |
| format | Article |
| id | doaj-art-5c7e7790d9fa4ca2b88552ed4b0c842e |
| institution | Kabale University |
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| language | English |
| publishDate | 2025-06-01 |
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| spelling | doaj-art-5c7e7790d9fa4ca2b88552ed4b0c842e2025-08-20T03:28:32ZengMDPI AGMathematics2227-73902025-06-011313205510.3390/math13132055Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value ProblemsSotiris K. Ntouyas0Bashir Ahmad1Jessada Tariboon2Department of Mathematics, University of Ioannina, 451 10 Ioannina, GreeceNonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaIntelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, ThailandThis paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Hilfer and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Caputo fractional derivative operators, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>1</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mn>1</mn></msub></semantics></math></inline-formula> and the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. Also the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>k</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ψ</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula>-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mn>2</mn></msub></semantics></math></inline-formula>-Riemann–Liouville, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>k</mi><mn>2</mn></msub></semantics></math></inline-formula>-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study.https://www.mdpi.com/2227-7390/13/13/2055Hilfer and Caputo fractional derivative operatorsinclusionsintegral boundary conditionsexistencefixed-point theorems for multivalued maps |
| spellingShingle | Sotiris K. Ntouyas Bashir Ahmad Jessada Tariboon Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems Mathematics Hilfer and Caputo fractional derivative operators inclusions integral boundary conditions existence fixed-point theorems for multivalued maps |
| title | Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems |
| title_full | Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems |
| title_fullStr | Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems |
| title_full_unstemmed | Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems |
| title_short | Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems |
| title_sort | nonlocal nonlinear fractional order sequential hilfer caputo multivalued boundary value problems |
| topic | Hilfer and Caputo fractional derivative operators inclusions integral boundary conditions existence fixed-point theorems for multivalued maps |
| url | https://www.mdpi.com/2227-7390/13/13/2055 |
| work_keys_str_mv | AT sotiriskntouyas nonlocalnonlinearfractionalordersequentialhilfercaputomultivaluedboundaryvalueproblems AT bashirahmad nonlocalnonlinearfractionalordersequentialhilfercaputomultivaluedboundaryvalueproblems AT jessadatariboon nonlocalnonlinearfractionalordersequentialhilfercaputomultivaluedboundaryvalueproblems |