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<p>This manuscript proposes a new structure in fixed point theory by proposing the concept of graphical interpolative contractions. We introduce three types of contractions including graphical Kannan, graphical Reich-Rus-Ćirić and graphical Hardy-Roger interpolative contractions and show suitable fixed point theorems based on graphical structure. A key contribution of this work is the clear demonstration that graphical interpolative contractions do not necessarily satisfy the conditions of classical contractions. Through explicit counterexamples, we prove that graphical metric spaces are not merely extensions of traditional metric spaces but represent a fundamentally distinct setting. To highlight the practical utility of our developed framework, we apply the theoretical framework to real-world problems and successfully solve an integral equation. The results not only exemplify the novelty of graphical contractions but also provide fresh avenues of investigating fixed points in systems that satisfy graphical relations.</p>

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A novel framework for graph-based interpolative contractions with applications to integral equations

  • 期刊论文

  • 发表日期：

    2025年9月3日

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  • Muhammad Umair Saeed,
  • Fahim Ud Din,
  • Ilyas Khan,
  • Haroon Ahmad

摘要

This manuscript proposes a new structure in fixed point theory by proposing the concept of graphical interpolative contractions. We introduce three types of contractions including graphical Kannan, graphical Reich-Rus-Ćirić and graphical Hardy-Roger interpolative contractions and show suitable fixed point theorems based on graphical structure. A key contribution of this work is the clear demonstration that graphical interpolative contractions do not necessarily satisfy the conditions of classical contractions. Through explicit counterexamples, we prove that graphical metric spaces are not merely extensions of traditional metric spaces but represent a fundamentally distinct setting. To highlight the practical utility of our developed framework, we apply the theoretical framework to real-world problems and successfully solve an integral equation. The results not only exemplify the novelty of graphical contractions but also provide fresh avenues of investigating fixed points in systems that satisfy graphical relations.