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<p>The Hardy–Rogers contraction generalizes several classical extensions of the Banach fixed point theorem, including those due to Kannan, Chatterjea, and Reich, by incorporating finite linear combinations of six basic metric distances. All these classical contraction conditions share the inherent restriction of involving only finitely many fixed metric distances between the points <i>e</i>, <i>h</i>, and their first iterates Φ<i>e</i>, Φ<i>h</i>. We introduce a novel framework called permutation contraction that removes this finite-set restriction by admitting infinite series of weighted orbital distances <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi mathvariant="script">D</mi><mo stretchy="false">(</mo><msup><mi mathvariant="normal">Φ</mi><mi>p</mi></msup><mi>e</mi><mo>,</mo><msup><mi mathvariant="normal">Φ</mi><mi>q</mi></msup><mi>h</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$\mathcal{D}(\Phi ^{p} e, \Phi ^{q} h)$</EquationSource></InlineEquation> with arbitrary permutations of iteration indices. Our main results (Theorems 3.1 and 3.6) establish existence and uniqueness of fixed points under this generalized setting, with classical contraction conditions recovered as finite truncations. An application to nonlinear integral equations demonstrates the practical utility of this extended theory.</p>

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A generalized fixed point theorem via permutation contractions with an application to integral equations

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  • 发表日期：

    2026年6月3日

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  • Irshad Ayoob,
  • Nabil Mlaiki

摘要

The Hardy–Rogers contraction generalizes several classical extensions of the Banach fixed point theorem, including those due to Kannan, Chatterjea, and Reich, by incorporating finite linear combinations of six basic metric distances. All these classical contraction conditions share the inherent restriction of involving only finitely many fixed metric distances between the points e, h, and their first iterates Φe, Φh. We introduce a novel framework called permutation contraction that removes this finite-set restriction by admitting infinite series of weighted orbital distances D(Φpe,Φqh)$\mathcal{D}(\Phi ^{p} e, \Phi ^{q} h)$ with arbitrary permutations of iteration indices. Our main results (Theorems 3.1 and 3.6) establish existence and uniqueness of fixed points under this generalized setting, with classical contraction conditions recovered as finite truncations. An application to nonlinear integral equations demonstrates the practical utility of this extended theory.