Modified $ \Re $-rational contractions and fixed point results with applications to boundary value problems

Hasanen A. Hammad; Manal Elzain Mohamed Abdalla; Hasanen A. Hammad; Manal Elzain Mohamed Abdalla

doi:10.3934/math.2025911

AIMS Mathematics

2025, Volume 10, Issue 9: 20385-20411. doi: 10.3934/math.2025911

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Research article

Modified $ \Re $-rational contractions and fixed point results with applications to boundary value problems

Hasanen A. Hammad ^{1
,
,},
Manal Elzain Mohamed Abdalla ²

1.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
2.
Department of Mathematics, Applied College, King Khalid University, Saudi Arabia

Received: 19 June 2025 Revised: 13 August 2025 Accepted: 22 August 2025 Published: 05 September 2025
MSC : 47H09, 47H10

This manuscript introduces modified $ \Re $-rational contractions for single self-maps, leveraging $ \omega $-distance in a relational-theoretic metric space. This novel approach establishes the existence and uniqueness of a fixed point for self-maps, specifically by applying the locally $ \Theta $-transitivity property. We support our theoretical advancements with compelling examples and demonstrate their practical significance by solving a fourth-order boundary value problem related to transverse oscillation in a homogeneous bar and a first-order periodic boundary value problem.
- fixed point technique,
- $ \Re $-rational contraction mapping,
- transverse oscillation bar,
- boundary value problem,
- evaluation metrics
Citation: Hasanen A. Hammad, Manal Elzain Mohamed Abdalla. Modified $ \Re $-rational contractions and fixed point results with applications to boundary value problems[J]. AIMS Mathematics, 2025, 10(9): 20385-20411. doi: 10.3934/math.2025911

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Abstract

This manuscript introduces modified $ \Re $-rational contractions for single self-maps, leveraging $ \omega $-distance in a relational-theoretic metric space. This novel approach establishes the existence and uniqueness of a fixed point for self-maps, specifically by applying the locally $ \Theta $-transitivity property. We support our theoretical advancements with compelling examples and demonstrate their practical significance by solving a fourth-order boundary value problem related to transverse oscillation in a homogeneous bar and a first-order periodic boundary value problem.

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