A Novel Approach to Some Proximal Contractions with Examples of Its Application

A Novel Approach to Some Proximal Contractions with Examples of Its Application

In this article, we will introduce a new generalized proximal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-contraction for multivalued an...

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Bibliographic Details
Main Authors:	Muhammad Zahid, Fahim Ud Din, Luminiţa-Ioana Cotîrlă, Daniel Breaz
Format:	Article
Language:	English
Published:	MDPI AG 2025-05-01
Series:	Axioms
Subjects:	rectangular metric space coincidence best proximity point <i>θ</i>-contraction fixed point equation of motion
Online Access:	https://www.mdpi.com/2075-1680/14/5/382
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Summary:	In this article, we will introduce a new generalized proximal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>-contraction for multivalued and single-valued mappings named <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="script">f</mi><mo>−</mo><msub><mi>θ</mi><mi>κ</mi></msub><mo>)</mo></mrow><msub><mi>C</mi><mi>P</mi></msub></msub></semantics></math></inline-formula>-proximal contraction and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi mathvariant="script">f</mi><mo>−</mo><msub><mi>θ</mi><mi>κ</mi></msub><mo>)</mo></mrow><msub><mi>B</mi><mi>P</mi></msub></msub></semantics></math></inline-formula>-proximal contraction. Using these newly constructed proximal contractions, we will establish new results for the coincidence best proximity point, best proximity point, and fixed point for multivalued mappings in the context of rectangular metric space. Also, we will reduce these contractions for single-valued mappings, named <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>θ</mi><mi>κ</mi></msub><mo>)</mo></mrow><msub><mi>C</mi><mi>P</mi></msub></msub></semantics></math></inline-formula>-proximal contraction and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>θ</mi><mi>κ</mi></msub><mo>)</mo></mrow><msub><mi>B</mi><mi>P</mi></msub></msub></semantics></math></inline-formula>-proximal contraction, to establish results for the coincidence proximity point, best proximity point, and fixed point results. We will give some illustrated examples for our newly generated results with graphical representations. In the last section, we will also find the solution to the equation of motion by using our defined results.
ISSN:	2075-1680