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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/13/2055 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |