Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems

Ahmed Alamer; Faizan Ahmad Khan; Ahmed Alamer; Faizan Ahmad Khan

doi:10.3934/math.2024305

AIMS Mathematics

2024, Volume 9, Issue 3: 6266-6280. doi: 10.3934/math.2024305

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Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems

Ahmed Alamer ^,,
Faizan Ahmad Khan ^,

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Received: 07 December 2023 Revised: 19 January 2024 Accepted: 29 January 2024 Published: 04 February 2024
MSC : 47H10, 54H25, 34B15, 06A75

The area of metric fixed point theory applied to relational metric spaces has received significant attention since the appearance of the relation-theoretic contraction principle. In recent times, a number of fixed point theorems addressing the various contractivity conditions in the relational metric space has been investigated. Such results are extremely advantageous in solving a variety of boundary value problems, matrix equations, and integral equations. This article offerred some fixed point results for a functional contractive mapping depending on a control function due to Boyd and Wong in a metric space endued with a local class of transitive relations. Our findings improved, developed, enhanced, combined and strengthened several fixed point theorems found in the literature. Several illustrative examples were delivered to argue for the reliability of our findings. To verify the relevance of our findings, we conveyed an existence and uniqueness theorem regarding the solution of a first-order boundary value problem.
- fixed points,
- functional contractions,
- control functions,
- binary relations,
- boundary value problems
Citation: Ahmed Alamer, Faizan Ahmad Khan. Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems[J]. AIMS Mathematics, 2024, 9(3): 6266-6280. doi: 10.3934/math.2024305

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Abstract

The area of metric fixed point theory applied to relational metric spaces has received significant attention since the appearance of the relation-theoretic contraction principle. In recent times, a number of fixed point theorems addressing the various contractivity conditions in the relational metric space has been investigated. Such results are extremely advantageous in solving a variety of boundary value problems, matrix equations, and integral equations. This article offerred some fixed point results for a functional contractive mapping depending on a control function due to Boyd and Wong in a metric space endued with a local class of transitive relations. Our findings improved, developed, enhanced, combined and strengthened several fixed point theorems found in the literature. Several illustrative examples were delivered to argue for the reliability of our findings. To verify the relevance of our findings, we conveyed an existence and uniqueness theorem regarding the solution of a first-order boundary value problem.

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