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

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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Bibliographic Details
Main Authors: Furkan Erkan, Nuket Aykut Hamal, Sotiris K. Ntouyas, Jessada Tariboon
Format: Article
Language:English
Published: MDPI AG 2025-07-01
Series:Fractal and Fractional
Subjects:
(<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
Online Access:https://www.mdpi.com/2504-3110/9/7/437
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Summary: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.
ISSN:2504-3110

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