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<p>We elicit an extension of the most celebrated and elegant contraction principle, due to Banach, to generalized complete spaces. In fact, in light of the fact that an almost acyclic (noncyclic) contraction is a natural generalization of the contraction self-mapping, the proposed generalization is presented as a fixed point theorem for an almost acyclic contraction in the setting of a generalized complete space. As a consequence, we obtain a best proximity point theorem for almost cyclic contraction mappings to render optimal approximate solutions, known as best proximity points, to the fixed point equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_200_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(Tx=x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> wherein <i>T</i> is an almost cyclic contraction.</p>

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An extension of the contraction principle to render optimal approximate solutions

  • 期刊论文

  • 发表日期：

    2025年9月12日

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  • S. Sadiq Basha

摘要

We elicit an extension of the most celebrated and elegant contraction principle, due to Banach, to generalized complete spaces. In fact, in light of the fact that an almost acyclic (noncyclic) contraction is a natural generalization of the contraction self-mapping, the proposed generalization is presented as a fixed point theorem for an almost acyclic contraction in the setting of a generalized complete space. As a consequence, we obtain a best proximity point theorem for almost cyclic contraction mappings to render optimal approximate solutions, known as best proximity points, to the fixed point equation \(Tx=x\) T x = x wherein T is an almost cyclic contraction.