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<p>The notions of graphical and pointwise limits of a sequence of set-valued functions defined from one metric space into another were studied by Aubin and Frankowska. In this study, by using the concept of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1812_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \beta\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>-density of subsets of natural numbers, we introduce the notions of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1812_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(St^{\alpha \beta }_{\gamma }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msubsup> <mi>t</mi> <mi>γ</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>-graphical and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1812_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(St^{\alpha \beta }_{\gamma }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msubsup> <mi>t</mi> <mi>γ</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>-pointwise limits of a sequence of set-valued functions defined from one probabilistic normed space into another. Subsequently, the article introduces the notions of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1812_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(St^{\alpha \beta }_{\gamma }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msubsup> <mi>t</mi> <mi>γ</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>-graph and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1812_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(St^{\alpha \beta }_{\gamma }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msubsup> <mi>t</mi> <mi>γ</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>-pointwise convergence of these sequences. Moreover, we look at the correspondence between these convergences and establish some associated theorems.</p>

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On \(\varvec{St^{\alpha \beta }_{\gamma }}\)-Graph and \(\varvec{St^{\alpha \beta }_{\gamma }}\)-Pointwise Convergence of Sequences of Set-Valued Functions Defined on Probabilistic Normed Spaces

  • 期刊论文

  • 发表日期：

    2025年4月10日

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  • SK Ashadul Rahaman,
  • Mohammad Mursaleen

摘要

The notions of graphical and pointwise limits of a sequence of set-valued functions defined from one metric space into another were studied by Aubin and Frankowska. In this study, by using the concept of \(\alpha \beta\) α β -density of subsets of natural numbers, we introduce the notions of \(St^{\alpha \beta }_{\gamma }\) S t γ α β -graphical and \(St^{\alpha \beta }_{\gamma }\) S t γ α β -pointwise limits of a sequence of set-valued functions defined from one probabilistic normed space into another. Subsequently, the article introduces the notions of \(St^{\alpha \beta }_{\gamma }\) S t γ α β -graph and \(St^{\alpha \beta }_{\gamma }\) S t γ α β -pointwise convergence of these sequences. Moreover, we look at the correspondence between these convergences and establish some associated theorems.