$n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels

Irfan Ullah; Shamoona Jabeen

$n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels

Abstract

This paper establishes novel $n$-tupled fixed point theorems for mappings defined on fuzzy metric spaces. We introduce a generalized rational-type contraction condition to prove the existence and uniqueness of quadruple, quintuple, and, more generally, $n$-tupled fixed points. Our work significantly extends and generalizes several known results in the existing literature. The main theorems are formulated for mappings $F: X^n \to X$ within a complete fuzzy metric space $(X, M, *)$, correcting a common oversight in the framework of $n$-tupled fixed points. To validate the theoretical findings, we provide supportive examples and discuss applications to systems of nonlinear integral equations involving Laplace-type kernels.

Keywords

Fuzzy metric space, $n$-tupled fixed point, Rational-type contractions, Quadruple, Quintuple

Supporting Institution

The authors declare that no external financial support was received for this study.

Project Number

This study was not supported by any funded research project and does not have an assigned project number.

Ethical Statement

The authors declare that this manuscript is an original research work and has been prepared in accordance with accepted academic and ethical standards. The work has not been published previously and is not under consideration for publication elsewhere. No human participants, or personal data were involved in this study. Therefore, ethical approval from an institutional review board or ethics committee was not required. All sources of information used in this manuscript have been properly cited and acknowledged. The authors confirm that there is no plagiarism, data fabrication, data falsification, or inappropriate manipulation of results in this study. The authors declare that there are no conflicts of interest related to this research.

Thanks

Thank you for your support

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Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Irfan Ullah ^*
0009-0003-5191-0872
Pakistan

Shamoona Jabeen
0000-0002-4367-5708
China

Early Pub Date

May 11, 2026

Publication Date

-

Submission Date

December 24, 2025

Acceptance Date

February 27, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

IZ

https://izlik.org/JA96HU74HZ

APA

Ullah, I., & Jabeen, S. (2026). $n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels. Sakarya Journal of Mathematics, Advanced Online Publication, 1-11. https://izlik.org/JA96HU74HZ

AMA

1.Ullah I, Jabeen S. $n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels. Sakarya Journal of Mathematics. 2026;(Advanced Online Publication):1-11. https://izlik.org/JA96HU74HZ

Chicago

Ullah, Irfan, and Shamoona Jabeen. 2026. “$n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces With Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels”. Sakarya Journal of Mathematics, no. Advanced Online Publication: 1-11. https://izlik.org/JA96HU74HZ.

EndNote

Ullah I, Jabeen S (May 1, 2026) $n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels. Sakarya Journal of Mathematics Advanced Online Publication 1–11.

IEEE

[1]I. Ullah and S. Jabeen, “$n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels”, Sakarya Journal of Mathematics, no. Advanced Online Publication, pp. 1–11, May 2026, [Online]. Available: https://izlik.org/JA96HU74HZ

ISNAD

Ullah, Irfan - Jabeen, Shamoona. “$n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces With Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels”. Sakarya Journal of Mathematics. Advanced Online Publication (May 1, 2026): 1-11. https://izlik.org/JA96HU74HZ.

JAMA

1.Ullah I, Jabeen S. $n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels. Sakarya Journal of Mathematics. 2026;:1–11.

MLA

Ullah, Irfan, and Shamoona Jabeen. “$n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces With Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels”. Sakarya Journal of Mathematics, no. Advanced Online Publication, May 2026, pp. 1-11, https://izlik.org/JA96HU74HZ.

Vancouver

1.Irfan Ullah, Shamoona Jabeen. $n$-Tupled Fixed Point Theorems for Rational-Type Contractions in Fuzzy Metric Spaces with Applications to Nonlinear Integral Equations Involving Laplace-Type Kernels. Sakarya Journal of Mathematics [Internet]. 2026 May 1;(Advanced Online Publication):1-11. Available from: https://izlik.org/JA96HU74HZ