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<p>In this work, we aim to explore whether a novel type of fractional space can be defined using Orlicz spaces and fractional calculus. This inquiry is fruitful, as extending classical results to new contexts can lead to a better and deeper understanding of those classical results. Our main objective is to introduce a new fractional Caputo-derivative Orlicz space, denoted by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_692_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {O}^{\alpha ,G}(\Lambda , \mathbb {R}) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">O</mi> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>G</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We are interested in several qualitative properties of this space, such as reflexivity, completeness, and separability. Additionally, we establish a continuous embedding results of this space into a suitable Orlicz space and the space of continuous functions. As an application, we use the mountain-pass theorem (MPT) to ensure the existence of a nontrivial weak solution for a new class of fractional-type problems in Orlicz space.</p>

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On a novel fractional Caputo-derivative Orlicz space

  • 期刊论文

  • 发表日期：

    2025年7月8日

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  • Ayoub Kasmi,
  • El Houssine Azroul,
  • Mohammed Shimi

摘要

In this work, we aim to explore whether a novel type of fractional space can be defined using Orlicz spaces and fractional calculus. This inquiry is fruitful, as extending classical results to new contexts can lead to a better and deeper understanding of those classical results. Our main objective is to introduce a new fractional Caputo-derivative Orlicz space, denoted by \(\mathcal {O}^{\alpha ,G}(\Lambda , \mathbb {R}) \) O α , G ( Λ , R ) . We are interested in several qualitative properties of this space, such as reflexivity, completeness, and separability. Additionally, we establish a continuous embedding results of this space into a suitable Orlicz space and the space of continuous functions. As an application, we use the mountain-pass theorem (MPT) to ensure the existence of a nontrivial weak solution for a new class of fractional-type problems in Orlicz space.