Fuzziness and randomness in fixed point theory: Measurable selections and applications

Akbar Azam; Faryad Ali; Sehar Afsheen; Mohammed Shehu Shagari; Akbar Azam; Faryad Ali; Sehar Afsheen; Mohammed Shehu Shagari

doi:10.3934/math.2025864

AIMS Mathematics

2025, Volume 10, Issue 8: 19335-19356. doi: 10.3934/math.2025864

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Research article

Fuzziness and randomness in fixed point theory: Measurable selections and applications

1.
Department of Mathematics, Grand Asian University, Sialkot 7KM, Pasrur Road, 51310, Sialkot, Pakistan; akbarazam@gaus.edu.pk or akbarazam@yahoo.com
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P. O. BOX 90950, Riyadh, 11623, Saudi Arabia; faali@imamu.edu.sa
3.
Department of Mathematics and Statistics, International Islamic University, H-10, 44000, Islamabad, Pakistan; sehar.phdma144@iiu.edu.pk
4.
Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria; shagaris@ymail.com

Received: 22 May 2025 Revised: 28 July 2025 Accepted: 12 August 2025 Published: 25 August 2025
MSC : 46S40, 47H10, 54H25

This paper presents a novel fixed point theorem for measurable selections arising from pairs of fuzzy set-valued operators defined on Polish spaces. By establishing conditions under which such selections exist, we provided a rigorous framework for analyzing the solvability of random multivalued operator equations in fuzzy environments. Our approach seamlessly integrated fuzziness and randomness, extending classical fixed point theory into a more realistic setting where uncertainty is both probabilistic and vague. To demonstrate the utility and applicability of our results, we constructed well-structured and insightful examples rooted in engineering-inspired scenarios. These examples not only validate the theoretical framework but also highlight its effectiveness in modeling complex systems affected by dual sources of uncertainty.
- fixed point,
- coincidence point,
- random operators,
- contractive-type mapping,
- fuzzy mapping
Citation: Akbar Azam, Faryad Ali, Sehar Afsheen, Mohammed Shehu Shagari. Fuzziness and randomness in fixed point theory: Measurable selections and applications[J]. AIMS Mathematics, 2025, 10(8): 19335-19356. doi: 10.3934/math.2025864

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Abstract

This paper presents a novel fixed point theorem for measurable selections arising from pairs of fuzzy set-valued operators defined on Polish spaces. By establishing conditions under which such selections exist, we provided a rigorous framework for analyzing the solvability of random multivalued operator equations in fuzzy environments. Our approach seamlessly integrated fuzziness and randomness, extending classical fixed point theory into a more realistic setting where uncertainty is both probabilistic and vague. To demonstrate the utility and applicability of our results, we constructed well-structured and insightful examples rooted in engineering-inspired scenarios. These examples not only validate the theoretical framework but also highlight its effectiveness in modeling complex systems affected by dual sources of uncertainty.

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