Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions
This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><...
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2025-07-01
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| author | Furkan Erkan Nuket Aykut Hamal Sotiris K. Ntouyas Jessada Tariboon |
| author_facet | Furkan Erkan Nuket Aykut Hamal Sotiris K. Ntouyas Jessada Tariboon |
| author_sort | Furkan Erkan |
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| description | This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Hilfer and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’ski<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi mathvariant="normal">i</mi><mo>˘</mo></mover></semantics></math></inline-formula>’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings. |
| format | Article |
| id | doaj-art-e7f4c6657f7849d19a3e0da152b1c1cb |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-e7f4c6657f7849d19a3e0da152b1c1cb2025-08-20T02:45:42ZengMDPI AGFractal and Fractional2504-31102025-07-019743710.3390/fractalfract9070437Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary ConditionsFurkan Erkan0Nuket Aykut Hamal1Sotiris K. Ntouyas2Jessada Tariboon3Department of Mathematics, Ege University, Bornova 35100, Izmir, TürkiyeDepartment of Mathematics, Ege University, Bornova 35100, Izmir, TürkiyeDepartment of Mathematics, University of Ioannina, 451 10 Ioannina, GreeceIntelligent 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 investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Hilfer and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’ski<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi mathvariant="normal">i</mi><mo>˘</mo></mover></semantics></math></inline-formula>’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings.https://www.mdpi.com/2504-3110/9/7/437(<i>k</i>, <i>ψ</i>)-Hilfer fractional derivative(<i>k</i>, <i>ψ</i>)-Caputo fractional derivativefractional differential equationfractional differential inclusionexistenceuniqueness |
| spellingShingle | Furkan Erkan Nuket Aykut Hamal Sotiris K. Ntouyas Jessada Tariboon Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions Fractal and Fractional (<i>k</i>, <i>ψ</i>)-Hilfer fractional derivative (<i>k</i>, <i>ψ</i>)-Caputo fractional derivative fractional differential equation fractional differential inclusion existence uniqueness |
| title | Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions |
| title_full | Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions |
| title_fullStr | Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions |
| title_full_unstemmed | Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions |
| title_short | Existence and Uniqueness Analysis for (<i>k</i>, <i>ψ</i>)-Hilfer and (<i>k</i>, <i>ψ</i>)-Caputo Sequential Fractional Differential Equations and Inclusions with Non-Separated Boundary Conditions |
| title_sort | existence and uniqueness analysis for i k i i ψ i hilfer and i k i i ψ i caputo sequential fractional differential equations and inclusions with non separated boundary conditions |
| topic | (<i>k</i>, <i>ψ</i>)-Hilfer fractional derivative (<i>k</i>, <i>ψ</i>)-Caputo fractional derivative fractional differential equation fractional differential inclusion existence uniqueness |
| url | https://www.mdpi.com/2504-3110/9/7/437 |
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