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PreprintCC BY 4.0arXiv

Graded-Edge Rakotch Contractions in Generalized Metric Spaces with Weighted Graphs

  • Ismat Beg
  • Muhammad Aslam Javed
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Abstract

We introduce the notion of graded-edge Rakotch contraction mappings on a generalized metric space with a weighted graph. We prove that the successive iterate of every starting point converge uniformly to a common limit, which is the unique fixed point. We derive an explicit iteration-complexity bound and, under a relaxed triangle inequality, a geometric rate whose constant is the graph invariant. Finally, we illustrate the theory on an explicit real graded contraction that realizes the interior optimum, on a quadratic integral equation, and on finite weighted digraphs.

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Cite

BibTeXRIS
BibTeX
@misc{beg2026gradededge,
  title = {{Graded-Edge Rakotch Contractions in Generalized Metric Spaces with Weighted Graphs}},
  author = {Ismat Beg and Muhammad Aslam Javed},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.09996},
  eprint = {2611.09996},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.09996},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Graded-Edge Rakotch Contractions in Generalized Metric Spaces with Weighted Graphs
AU  - Ismat Beg
AU  - Muhammad Aslam Javed
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.09996
UR  - https://arxiv.org/abs/2611.09996
AB  - We introduce the notion of graded-edge Rakotch contraction mappings on a generalized metric space with a weighted graph. We prove that the successive iterate of every starting point converge uniformly to a common limit, which is the unique fixed point. We derive an explicit iteration-complexity bound and, under a relaxed triangle inequality, a geometric rate whose constant is the graph invariant. Finally, we illustrate the theory on an explicit real graded contraction that realizes the interior optimum, on a quadratic integral equation, and on finite weighted digraphs.
ER  -