A novel modified Picard-Krasnoselskii iterative algorithm with optimized convergence
*Ritu SaharanCorresponding authorritusaharan.rs@gmail.comDepartment of MathematicsUniversity Institute of Sciences (UIS)Chandigarh UniversityMohali, Punjab, 140413, IndiaView full profile → , Naveen Kumarimnaveenphd@gmail.comDepartment of MathematicsUniversity Institute of Sciences (UIS)Chandigarh UniversityMohali, Punjab, 140413, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Feb 2026
- Published Online:
- 25 Aug 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2634
- Pages:
- 2487–2493
Abstract
We present a novel fixed-point iterative algorithm, termed as Modified Picard-Krasnoselskii Hybrid Algorithm, abbreviated as PKm algorithm (m denotes the modification level). This new process is developed by modifying the classical Picard-Krasnoselskii hybrid iteration, aiming to achieve improved convergence characteristics. The motivation for this development stems from the need to optimize the convergence of well-known iterative processes such as Picard, Mann, Noor, and Ishikawa iterations, as well as their hybrid variants including Picard-Noor, Picard-Ishikawa, and Picard-Mann processes. The proposed algorithm is rigorously analyzed within the framework of fixed-point theory, and its superiority is established in the sense of Berinde’s [1] definition of faster convergence. Theoretical results are examined numerically, which clearly demonstrates that the proposed algorithm achieves faster convergence than its predecessors when applied to a suitable test function. The comparison is made based on the iterations count required and the accuracy of the approximate fixed point. Our findings confirm that the Modified Picard-Krasnoselskii Hybrid iterative algorithm not only retains the stability and applicability of traditional methods but also exhibits significantly improved convergence behavior. This contribution opens new avenues for the development of even more efficient iterative processes and their applications in nonlinear functional analysis and related computational problems.
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References
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