Proximal Contractions for Multivalued Mappings with an Application to 2D Volterra Integral Equations

Haroon Ahmad, Mudasir Younis (), Hami Gündoǧdu, Nisha Barley and Vijay Kumar Patel
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Haroon Ahmad: Abdus Salam School of Mathematical Science, Government College University, Lahore 54600, Pakistan
Mudasir Younis: Department of Mathematics, Sakarya University, 54050 Sakarya, Turkey
Hami Gündoǧdu: Department of Mathematics, Sakarya University, 54050 Sakarya, Turkey
Nisha Barley: Dr. Radhabai Government Naveen Kanya College, Raipur 492001, India
Vijay Kumar Patel: School of Advanced Sciences and Languages, VIT Bhopal University, Bhopal 466114, India

Mathematics, 2024, vol. 12, issue 23, 1-21

Abstract: In this paper, we delve into the ideas of Geraghty-type proximal contractions and their relation to multivalued, single-valued, and self mappings. We begin by introducing the notions of ( ψ ω ) M C P -proximal Geraghty contraction and rational ( ψ ω ) R M C P -proximal Geraghty contraction for multivalued mappings, aimed at establishing coincidence point results. To enhance our understanding and illustrate the concepts, practical examples are provided with each definition. This study extends these contractions to single-valued mappings with the introduction of ( ψ ω ) S C P -proximal Geraghty contraction and rational ( ψ ω ) R S C P -proximal Geraghty contraction, supported by relevant examples to reinforce the main results. Then, we explore ( ψ ω ) S F P Geraghty contraction and rational ( ψ ω ) R S F P contraction for self-mappings, obtaining fixed point theorems and clearly illustrating them through examples. Finally, we apply the theoretical framework developed to investigate the existence and uniqueness of solutions to certain two-dimensional Volterra integral equations. Specifically, we consider the transformation of first-kind Volterra integral equations, which play crucial roles in modeling memory in diverse scientific fields like biology, physics, and engineering. This approach provides a powerful tool for solving difficult integral equations and furthering applied mathematics research.

Keywords: suprametric space; coincidence point ( C P ); best proximity point ( BP P ); multivalued Geraghty proximal contractions; fixed point; 2D Volterra integral equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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