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<p>We introduce triple-controlled orthogonal <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{S} $</EquationSource> </InlineEquation>-metric type spaces involve triple auxiliary control functions <i>β</i>, <i>μ</i>, &amp; <i>γ</i>, extending the classical controlled <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{S} $</EquationSource> </InlineEquation>-metric type sitting. Furthermore, we formulate <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>,</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\alpha _{s},\bot ) $</EquationSource> </InlineEquation>-admissible function &amp; strengthen Wardowski’s contraction principle, designing <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>−</mo> <mi mathvariant="script">A</mi> <mo>,</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\alpha _{s}-\mathcal{A},\bot ) $</EquationSource> </InlineEquation>-contractive function fitted to the triple-controlled orthogonal structure. On complete spaces of this kind, we establish fixed-point theorems that ensure existence &amp; uniqueness under natural conditions on the control functions and admissibility. As an application, we show that the main result guarantees a unique solution to a class of fractional differential equations.</p>

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Novel results for Wardowski contraction principle in triple-controlled orthogonal \(\mathcal{S} \)-metric space with applications to fractional equations

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  • 发表日期：

    2026年1月13日

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  • Benitha Wises Samuel,
  • Gunaseelan Mani,
  • Santhosh Kumar Gopalakrishnan,
  • Sabri T. M. Thabet,
  • Taoufik Moulahi

摘要

We introduce triple-controlled orthogonal S $\mathcal{S} $ -metric type spaces involve triple auxiliary control functions β, μ, & γ, extending the classical controlled S $\mathcal{S} $ -metric type sitting. Furthermore, we formulate ( α s , ⊥ ) $(\alpha _{s},\bot ) $ -admissible function & strengthen Wardowski’s contraction principle, designing ( α s − A , ⊥ ) $(\alpha _{s}-\mathcal{A},\bot ) $ -contractive function fitted to the triple-controlled orthogonal structure. On complete spaces of this kind, we establish fixed-point theorems that ensure existence & uniqueness under natural conditions on the control functions and admissibility. As an application, we show that the main result guarantees a unique solution to a class of fractional differential equations.