On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential Equation
In this work, we address a nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential e...
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2025-06-01
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| author | Cemil Tunç Fehaid Salem Alshammari Fahir Talay Akyıldız |
| author_facet | Cemil Tunç Fehaid Salem Alshammari Fahir Talay Akyıldız |
| author_sort | Cemil Tunç |
| collection | DOAJ |
| description | In this work, we address a nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equation that incorporates <i>n</i>-multiple-variable time delays. Employing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability, and Ulam–Hyers stability of the proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equation through the fixed-point approach. In this study, we enhance and generalize existing results in the literature on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equations involving <i>n</i>-multiple-variable time delays. This study provides novel theoretical insights that deepen the qualitative understanding of fractional calculus. |
| format | Article |
| id | doaj-art-aeeb2d11703145159d414cdc7b229b08 |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-aeeb2d11703145159d414cdc7b229b082025-08-20T02:45:42ZengMDPI AGFractal and Fractional2504-31102025-06-019740910.3390/fractalfract9070409On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential EquationCemil Tunç0Fehaid Salem Alshammari1Fahir Talay Akyıldız2Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yıl University, 65080 Van, TurkeyDepartment of Mathematics and Statistics, Faculty of Sciences, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi ArabiaDepartment of Mathematics and Statistics, Faculty of Sciences, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi ArabiaIn this work, we address a nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equation that incorporates <i>n</i>-multiple-variable time delays. Employing the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability, and Ulam–Hyers stability of the proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equation through the fixed-point approach. In this study, we enhance and generalize existing results in the literature on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Hilfer fractional-order Volterra integro-differential equations involving <i>n</i>-multiple-variable time delays. This study provides novel theoretical insights that deepen the qualitative understanding of fractional calculus.https://www.mdpi.com/2504-3110/9/7/409ψ-Hilfer FrOVI-DEunique solutionUHR stabilitysemi-UHR stabilityUH stabilityfixed-point approach |
| spellingShingle | Cemil Tunç Fehaid Salem Alshammari Fahir Talay Akyıldız On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential Equation Fractal and Fractional ψ-Hilfer FrOVI-DE unique solution UHR stability semi-UHR stability UH stability fixed-point approach |
| title | On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential Equation |
| title_full | On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential Equation |
| title_fullStr | On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential Equation |
| title_full_unstemmed | On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential Equation |
| title_short | On the Existence of Solutions and Ulam-Type Stability for a Nonlinear <i>ψ</i>-Hilfer Fractional-Order Delay Integro-Differential Equation |
| title_sort | on the existence of solutions and ulam type stability for a nonlinear i ψ i hilfer fractional order delay integro differential equation |
| topic | ψ-Hilfer FrOVI-DE unique solution UHR stability semi-UHR stability UH stability fixed-point approach |
| url | https://www.mdpi.com/2504-3110/9/7/409 |
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