Application of fixed point result to the boundary value problem using the $ M $-type generalized contraction condition for best proximity point considerations

Rajagopalan Ramaswamy; Penumarthy Parvateesam Murthy; Pushplata Sahu; Rayan Abdulrahman Alkhowaiter; Ola Ashour Abdelnaby; Gunaseelan Mani; Rajagopalan Ramaswamy; Penumarthy Parvateesam Murthy; Pushplata Sahu; Rayan Abdulrahman Alkhowaiter; Ola Ashour Abdelnaby; Gunaseelan Mani

doi:10.3934/math.2025613

AIMS Mathematics

2025, Volume 10, Issue 6: 13622-13639. doi: 10.3934/math.2025613

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Application of fixed point result to the boundary value problem using the $ M $-type generalized contraction condition for best proximity point considerations

1.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
2.
Department of Mathematics, Guru Ghasidas Vishwavidyalay, Bilaspur, Chhattisgarh, 495009, India
3.
Department of Mathematics, Pt Sundarlal Sharma (Open) University, Bilaspur 495009, (C.G), India
4.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, India

Received: 21 February 2025 Revised: 06 May 2025 Accepted: 27 May 2025 Published: 13 June 2025
MSC : : 47H10, 54H25

This paper serves a dual purpose: to introduce a contraction condition and to demonstrate its application. We established a new framework for obtaining best proximity points in complete metric spaces, extending and generalizing several existing fixed point theorems. From this foundational result, multiple corollaries were derived, providing broader applicability in various mathematical settings. To validate the theoretical development, we applied our results to boundary value problems and dynamic market equilibrium models, illustrating both the mathematical robustness and real-world relevance of the proposed method.
- complete metric space,
- $ M $-type generalized contraction,
- $ P $-property,
- best proximity point
Citation: Rajagopalan Ramaswamy, Penumarthy Parvateesam Murthy, Pushplata Sahu, Rayan Abdulrahman Alkhowaiter, Ola Ashour Abdelnaby, Gunaseelan Mani. Application of fixed point result to the boundary value problem using the $ M $-type generalized contraction condition for best proximity point considerations[J]. AIMS Mathematics, 2025, 10(6): 13622-13639. doi: 10.3934/math.2025613

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Abstract

This paper serves a dual purpose: to introduce a contraction condition and to demonstrate its application. We established a new framework for obtaining best proximity points in complete metric spaces, extending and generalizing several existing fixed point theorems. From this foundational result, multiple corollaries were derived, providing broader applicability in various mathematical settings. To validate the theoretical development, we applied our results to boundary value problems and dynamic market equilibrium models, illustrating both the mathematical robustness and real-world relevance of the proposed method.

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