<p>In this article, we explore a new class of nonlocal boundary value problems defined by coupled multi-term delay Caputo fractional differential equations along with a multipoint-integral boundary problem. For analytical purposes, we reformulate the problem as a fixed-point problem to facilitate the application of fixed-point theory. The existence of solutions is demonstrated using Krasnoselskii’s fixed-point theorem, while the uniqueness of solutions is established through Banach’s fixed-point theorem. We also discuss the stability criteria, including Ulam-Hyers (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13663_2025_789_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="script">UH</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{UH}$</EquationSource> </InlineEquation>), generalized Ulam-Hyers (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13663_2025_789_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="script">GUH</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{GUH}$</EquationSource> </InlineEquation>), Ulam-Hyers-Rassias (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13663_2025_789_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="script">UHR</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{UHR}$</EquationSource> </InlineEquation>), and generalized Ulam-Hyers-Rassias (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13663_2025_789_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="script">GUHR</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{GUHR}$</EquationSource> </InlineEquation>) stability, for solutions of the equation at hand. To illustrate the theoretical results, we present an example.</p>

Existence and stability results for a coupled multi-term Caputo fractional differential equations

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2025年4月14日

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Gunaseelan Mani,
Purushothaman Ganesh,
Pandiarajan Ramasamy,
Sarah Aljohani,
Nabil Mlaiki

摘要

In this article, we explore a new class of nonlocal boundary value problems defined by coupled multi-term delay Caputo fractional differential equations along with a multipoint-integral boundary problem. For analytical purposes, we reformulate the problem as a fixed-point problem to facilitate the application of fixed-point theory. The existence of solutions is demonstrated using Krasnoselskii’s fixed-point theorem, while the uniqueness of solutions is established through Banach’s fixed-point theorem. We also discuss the stability criteria, including Ulam-Hyers ( $UH$ $\mathcal{UH}$ ), generalized Ulam-Hyers ( $GUH$ $\mathcal{GUH}$ ), Ulam-Hyers-Rassias ( $UHR$ $\mathcal{UHR}$ ), and generalized Ulam-Hyers-Rassias ( $GUHR$ $\mathcal{GUHR}$ ) stability, for solutions of the equation at hand. To illustrate the theoretical results, we present an example.