C*-algebra-valued perturbed modular metric spaces and existence results of fourth-order boundary value problems

Mohammed Shehu Shagari; Zulaihatu Tijjani Ahmad; Faryad Ali; Ghada Ali Basendwah; Mohammed A. Al-Kadhi; Akbar Azam; Mohammed Shehu Shagari; Zulaihatu Tijjani Ahmad; Faryad Ali; Ghada Ali Basendwah; Mohammed A. Al-Kadhi; Akbar Azam

doi:10.3934/math.20251088

AIMS Mathematics

2025, Volume 10, Issue 10: 24548-24563. doi: 10.3934/math.20251088

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C*-algebra-valued perturbed modular metric spaces and existence results of fourth-order boundary value problems

1.
Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P. O. BOX 90950, Riyadh, 11623, Saudi Arabia
3.
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4.
Department of Mathematics, Grand Asian University, Sialkot, 7KM, Pasrur Road, Sialkot 51310, Pakistan

Received: 21 August 2025 Revised: 30 September 2025 Accepted: 10 October 2025 Published: 27 October 2025
MSC : 47H09, 47H10, 54E50, 54H25

The primary objective of perturbed metric spaces was to set route for the advancement of problems involving fixed point findings for a distraught structure, where errors inevitably affected the measurement of distance between two points. This objective developed into a broadly applicable context when the range set of the distance function was a $ C^{*} $-algebra. Inspired by this, the notion of $ C^{*} $-algebra-valued perturbed modular metric space was introduced in this paper. Thereafter, some fixed point theorems in the new space were studied, using paired contractive mappings. In order to demonstrate the novelty of the concepts presented herein and to generalize some significant related discoveries in the literature, contrasting examples were given. Under this comparative illustration, numerical and graphical approaches were adopted to study the rate of convergence of Picard-type and paired contractive operators. As an application, new conditions for the existence and uniqueness of a solution to fourth-order boundary value problems were obtained.
- $ C^{*} $-algebra,
- perturbed metric,
- fixed point,
- integral equation
Citation: Mohammed Shehu Shagari, Zulaihatu Tijjani Ahmad, Faryad Ali, Ghada Ali Basendwah, Mohammed A. Al-Kadhi, Akbar Azam. C*-algebra-valued perturbed modular metric spaces and existence results of fourth-order boundary value problems[J]. AIMS Mathematics, 2025, 10(10): 24548-24563. doi: 10.3934/math.20251088

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Abstract

The primary objective of perturbed metric spaces was to set route for the advancement of problems involving fixed point findings for a distraught structure, where errors inevitably affected the measurement of distance between two points. This objective developed into a broadly applicable context when the range set of the distance function was a $ C^{*} $-algebra. Inspired by this, the notion of $ C^{*} $-algebra-valued perturbed modular metric space was introduced in this paper. Thereafter, some fixed point theorems in the new space were studied, using paired contractive mappings. In order to demonstrate the novelty of the concepts presented herein and to generalize some significant related discoveries in the literature, contrasting examples were given. Under this comparative illustration, numerical and graphical approaches were adopted to study the rate of convergence of Picard-type and paired contractive operators. As an application, new conditions for the existence and uniqueness of a solution to fourth-order boundary value problems were obtained.

References

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