Some convergence results on proximal contractions with application to nonlinear fractional differential equation

Haroon Ahmad; Om Prakash Chauhan; Tania Angelica Lazăr; Vasile Lucian Lazăr; Haroon Ahmad; Om Prakash Chauhan; Tania Angelica Lazăr; Vasile Lucian Lazăr

doi:10.3934/math.2025247

AIMS Mathematics

2025, Volume 10, Issue 3: 5353-5372. doi: 10.3934/math.2025247

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Research article

Some convergence results on proximal contractions with application to nonlinear fractional differential equation

1.
Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan; Email: haroonrao3@gmail.com
2.
Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, India; Email: chauhaan.op@gmail.com
3.
Department of Mathematics, Technical University of Cluj Napoca, 400114 Cluj-Napoca, Romania; Email: tania.lazar@math.utcluj.ro
4.
Department of Economic and Technical Sciences, Vasile Goldiș Western University of Arad, 310025 Arad, Romania; Email: lazar.vasile@uvvg.ro

Received: 08 October 2024 Revised: 27 January 2025 Accepted: 10 February 2025 Published: 10 March 2025
MSC : 34C28, 47H10, 34K37

In this manuscript, we investigated the coincidence points, best proximity points, and fixed-points results endowed with $ F $-contraction within the realm of suprametric spaces. The proximal point results obtained in this work show that our investigation is not purely theoretical; fundamental findings were supplemented with concrete examples that demonstrate their practical ramifications. Furthermore, this paper focuses on boundary value problems (BVPs) related to nonlinear fractional differential equations of order $ 2 < \varpi \leq 3 $. By cleverly translating the BVP into an integral equation, we obtained conditions that confirm the existence and uniqueness of fixed points under $ (\mathscr{F}_{\tau })_{F_{\mathscr{P}}} $-contraction. A relevant part of this work is the approximation of the Green's function, which is critical in proving the existence and uniqueness of solutions. Our work not only adds to the current body of knowledge but also provides strong approaches for dealing with hard mathematical problems in the field of fractional differential equations.
- suprametric space,
- coincidence point,
- best proximity point,
- fixed point,
- Green function,
- fractional boundary value problem
Citation: Haroon Ahmad, Om Prakash Chauhan, Tania Angelica Lazăr, Vasile Lucian Lazăr. Some convergence results on proximal contractions with application to nonlinear fractional differential equation[J]. AIMS Mathematics, 2025, 10(3): 5353-5372. doi: 10.3934/math.2025247

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Abstract

In this manuscript, we investigated the coincidence points, best proximity points, and fixed-points results endowed with $ F $-contraction within the realm of suprametric spaces. The proximal point results obtained in this work show that our investigation is not purely theoretical; fundamental findings were supplemented with concrete examples that demonstrate their practical ramifications. Furthermore, this paper focuses on boundary value problems (BVPs) related to nonlinear fractional differential equations of order $ 2 < \varpi \leq 3 $. By cleverly translating the BVP into an integral equation, we obtained conditions that confirm the existence and uniqueness of fixed points under $ (\mathscr{F}_{\tau })_{F_{\mathscr{P}}} $-contraction. A relevant part of this work is the approximation of the Green's function, which is critical in proving the existence and uniqueness of solutions. Our work not only adds to the current body of knowledge but also provides strong approaches for dealing with hard mathematical problems in the field of fractional differential equations.

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