On assessing convergence and stability of a novel iterative method for fixed-point problems

Aftab Hussain; Danish Ali; Amer Hassan Albargi; Aftab Hussain; Danish Ali; Amer Hassan Albargi

doi:10.3934/math.2025687

AIMS Mathematics

2025, Volume 10, Issue 7: 15333-15357. doi: 10.3934/math.2025687

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On assessing convergence and stability of a novel iterative method for fixed-point problems

1.
Faculty of Science, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
2.
Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced 37008 Salamanca, Spain

Received: 06 November 2024 Revised: 07 May 2025 Accepted: 21 May 2025 Published: 03 July 2025
MSC : 47H09, 47H10

Fixed-point theory, a major field of mathematics, analyzes outcomes that remain unchanged under particular operators, featuring multiple applications in mathematics, physics, engineering, computer science, and economics. This study presents the $ D^{*} $ iteration technique, a robust and effective iterative scheme for approximating fixed points in Suzuki generalized nonexpansive mappings. Within the context of uniformly convex Banach spaces, the novel scheme's weak and strong convergence properties were carefully addressed. The efficiency of this approach was demonstrated through detailed theoretical, numerical, and graphical assessments. Additionally, the stability of the iterative process was established. The method is used to generalize and enhance previous findings by approximating solutions for a fractional differential problem.
- differential equations,
- weak convergence,
- strong convergence,
- iteration process
Citation: Aftab Hussain, Danish Ali, Amer Hassan Albargi. On assessing convergence and stability of a novel iterative method for fixed-point problems[J]. AIMS Mathematics, 2025, 10(7): 15333-15357. doi: 10.3934/math.2025687

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Abstract

Fixed-point theory, a major field of mathematics, analyzes outcomes that remain unchanged under particular operators, featuring multiple applications in mathematics, physics, engineering, computer science, and economics. This study presents the $ D^{*} $ iteration technique, a robust and effective iterative scheme for approximating fixed points in Suzuki generalized nonexpansive mappings. Within the context of uniformly convex Banach spaces, the novel scheme's weak and strong convergence properties were carefully addressed. The efficiency of this approach was demonstrated through detailed theoretical, numerical, and graphical assessments. Additionally, the stability of the iterative process was established. The method is used to generalize and enhance previous findings by approximating solutions for a fractional differential problem.

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