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

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 $D^{\ast }$ 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.